Lý thuyết Hình thang – Hình thang cân (Chân trời sáng tạo 2024) Toán 8

Lý thuyết Toán lớp 8 Bài 3: Hình thang – Hình thang cân

A. Lý thuyết Hình thang – Hình thang cân

1. Khái niệm

Hình thang là tứ giác có hai cạnh đối song song.

Hình thang cân là hình thang có hai góc kề một đáy bằng nhau.

 2. Tính chất của hình thang cân

+ Hai cạnh bên bằng nhau.

+ Hai đường chéo bằng nhau.

Ví dụ:

 Hình thang cân EFGH có hai cạnh bên EH = FG, hai đường chéo EG = FH.

3. Dấu hiệu nhận biết hình thang cân

- Hình thang có hai góc kề một đáy bằng nhau là hình thang cân.

- Hình thang có hai đường chéo bằng nhau là hình thang cân.

B. Bài tập Hình thang – Hình thang cân

Bài 1. Cho hình thang cân ABCD (AB // CD). Hai tia phân giác của A và B cắt nhau tại điểm K nằm trên cạnh đáy DC. Từ K kẻ đoạn thẳng KM vuông góc với AB tại M.

a) Chứng minh ∆ABK là tam giác cân.

b) Chứng minh AM = BM.

Hướng dẫn giải

a) Do AK là tia phân giác của $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}}$ nên $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">D</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">K</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">K</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\frac{\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">D</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">2</span></span></span>}}$.

BK là tia phân giác của $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}}$ nên $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">C</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">K</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">K</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\frac{\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">C</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">2</span></span></span>}}$.

Mà ABCD là hình thang cân nên $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">D</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">C</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}}$ (tính chất hình thang cân).

Do đó $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">K</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">K</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}}$, suy ra ∆ABK là tam giác cân tại K.

b) Vì ∆ABK là tam giác cân nên KM là đường cao và cũng là đường trung tuyến ứng với cạnh đáy AB.

Do vậy M là trung điểm của AB nên AM = BM.

Bài 2. Tính số đo các góc chưa biết của hình thang IJGH (IJ // GH) trong các trường hợp sau:

a) $\hat{H}=50°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">50</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ , $\hat{J}=110°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">110</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ .

b) IJGH là hình thang cân và $\hat{H}=30°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">30</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ .

c) $\hat{H}=90°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">90</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ , $\hat{G}=50°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">50</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ .

Hướng dẫn giải

a)

Do IJ // GH nên ta có:

$\hat{I}+\hat{H}=180°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">+</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">180</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$, suy ra $\hat{I}=180°-\hat{H}=180°-50°=130°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">180</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">-</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">180</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">-</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">50</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">130</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$

$\hat{J}+\hat{G}=180°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">+</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">180</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$, suy ra $\hat{G}=180°-\hat{J}=180°-110°=70°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">180</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">-</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">180</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">-</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">110</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">70</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$

Vậy hình thang IJGH có: $\hat{I}=130°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">130</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$; $\hat{G}=70°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">70</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$.

b)

Hình thang cân IJGH (IJ // GH) có $\hat{I}=\hat{J}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}$ và $\hat{G}=\hat{H}=30°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">30</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ (tính chất hình thang cân).

Áp dụng định lí tổng các góc của một tứ giác, ta có:

$\hat{I}+\hat{J}+\hat{G}+\hat{H}=360°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">+</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">+</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">+</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">360</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$

Suy ra $2\hat{I}=360°-2\hat{H}=360°-2.30°=300°$$\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">2</span></span></span></span>}\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">360</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">-</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">2</span></span></span></span>}\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">360</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">-</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">2.30</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">300</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$

Do đó $\hat{I}=\hat{J}=150°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">150</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ .

Vậy hình thang cân IJGH có: $\hat{G}=30°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">30</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$; $\hat{I}=\hat{J}=150°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">150</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$.

c)

$\hat{H}=90°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">90</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ nên hình thang IJGH là hình thang vuông, suy ra $\hat{H}=\hat{I}=90°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">90</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$

Áp dụng định lí tổng các góc của một tứ giác, ta có:

$\hat{J}=360°-\left(\hat{I}+\hat{G}+\hat{H}\right)=360°-\left(90°+50°+90°\right)=360°-230°=130°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">360</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">-</span></span></span></span>\left(\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">+</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">+</span></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}\right)<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">360</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">-</span></span></span></span>\left(\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">90</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">+</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">50</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">+</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">90</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>\right)<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">360</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">-</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">230</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">130</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$

Vậy hình thang vuông IJGH có: $\hat{I}=90°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">90</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ , $\hat{J}=130°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">^</span></span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">�</span></span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">=</span></span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">130</span></span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;"><span\; style="line-height:2;">°</span></span></span></span>$ .

Bài 3. Cho tam giác cân EFG có EF = EG. Trên các cạnh EF và EG, lần lượt lấy các điểm H và I sao cho EH = EI.

a) Chứng minh HIGF là hình thang.

b) Chứng minh HIGF là hình thang cân.

Hướng dẫn giải

Ta có EH = EI nên ∆EHI là tam giác cân tại E.

Suy ra $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">H</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">I</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">I</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">H</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">180</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">°</span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">-</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">2</span></span></span>}}$ (1)

Lại có EF = EG nên ∆EFG là tam giác cân tại E.

Suy ra $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">F</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">G</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">G</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">F</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">180</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">°</span></span></span><span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">-</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">2</span></span></span>}}$ (2)

Từ (1) và (2) suy ra $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">H</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">I</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">F</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">G</span></span></span>}}$.

Mà $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">H</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">I</span></span></span>}}$ và $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">E</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">F</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">G</span></span></span>}}$ ở vị trí đồng vị nên HI // FG (dấu hiệu nhận biết).

Suy ra tứ giác HIGF là hình thang.

b) Vì ∆EFG là tam giác cân nên $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">F</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">G</span></span></span>}}$.

Suy ra hình thang HIGF là hình thang cân.

Bài 4. Cho hình thang ABCD (AD // BC) có AC = BD.

a) Hình thang ABCD là hình thang gì? Vì sao?

b) Chứng minh $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">D</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">D</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">C</span></span></span>}}$ .

Hướng dẫn giải

a) Vì hai đường chéo của hình thang ABCD là AC và BD bằng nhau nên hình thang ABCD là hình thang cân.

b) Theo tính chất hình thang cân, ta có: AB = DC

Xét ∆ABD và ∆DCA có:

AD là cạnh chung;

AB = DC;

BD = AC.

Suy ra ∆ABD = ∆DCA (c.c.c)

Do đó $\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">D</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">B</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">D</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">A</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:10px;"><span\; style="line-height:2;">C</span></span></span>}}$ (hai cạnh tương ứng).

Video bài giảng Toán 8 Bài 3: Hình thang – Hình thang cân - Chân trời sáng tạo