Lý thuyết Góc (Cánh diều 2024) Toán 6

Lý thuyết Toán lớp 6 Bài 5: Góc

A. Lý thuyết Góc

1. Khái niệm góc

Góc là hình gồm hai tia chung gốc.

Chú ý:

- Góc xOy (hoặc góc yOx) được kí hiệu là $\widehat{xOy}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$ (hoặc $\widehat{yOx}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$).

- Hai tia Ox và Oy được gọi là hai cạnh của góc. Gốc chung O của hai tia được gọi là đỉnh của góc.

Ví dụ 1. Điền vào chỗ trống trong các phát biểu sau:

a) Góc tạo bởi hai tia Om và …… gọi là góc mOn, kí hiệu ……

b) Góc MNP có đỉnh là …. và cạnh là ……………. Kí hiệu là……..

c) Hai đường thẳng AB và CD cắt nhau tai điểm O. Các góc khác góc bẹt là: ……………

Hướng dẫn giải

a) Góc tạo bởi hai tia Om và On gọi là góc mOn, kí hiệu $\widehat{mOn}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$.

b) Góc MNP có đỉnh là N và cạnh là NM và NP. Kí hiệu là $\widehat{MNP}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$.

c) Hai đường thẳng AB và CD cắt nhau tai điểm O. Các góc khác góc bẹt là: $\widehat{AOB}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$ và  $\widehat{COD}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$

2. Điểm nằm trong góc

Điểm M như trong hình trên (không thuộc tia Ox, Oy) được gọi là điểm nằm trong góc xOy hay điểm trong của góc xOy.

Ví dụ 2. Cho hình vẽ:

Trong hình vẽ trên, điểm B nằm trong góc aMb.

3. Số đo của góc

a) Đo góc

Thước đo góc có dạng nửa hình tròn và được chia đều thành 180 phần bằng nhau, mỗi phần ứng với 1°.

Cách đo góc:

Đo góc xOy:

- Bước 1: Đặt thước đo góc sao cho tâm của thước trùng với đỉnh O của góc. Vạch 0 của thước nằm trên cạnh Ox.

- Bước 2: Xác định xem cạnh Oy đi qua vạch chỉ số nào trên thước đo góc, ta sẽ được số đo của góc đó.

Chú ý:

- Mỗi góc có một số đo.

- Nếu số đo của góc xOy là n° thì ta kí hiệu $\widehat{xOy}=n°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$ hoặc $\widehat{yOx}=n°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$.

- Chúng ta chỉ xét các góc có số đo không vượt quá 180°.

Ví dụ 3.

Đo góc xOy trong hình vẽ trên:

Bước 1: Đặt thước đo góc sao cho tâm của thước trùng với đỉnh của góc. Vạch 0 của thước nằm trên cạnh Ox.

Bước 2: Xác định xem cạnh Oy đi qua vạch chia độ nào để biết số đo của góc.

Ta thấy tia Oy đi qua vạch chỉ số 35 nên ta được $\widehat{xOy}=35°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">35</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$.

Chú ý: Trong một hình có nhiều góc, người ta thường vẽ thêm một hay nhiều vòng cung nhỏ nối hai cạnh của góc đó để dễ thấy góc mà ta đang xét tới. Khi cần phân biệt các góc có chung một đỉnh, chẳng hạn chung đỉnh O như hình vẽ dưới đây, ta dùng kí hiệu $\widehat{O_1}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}_{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">1</span></span></span>}}}$,  $\widehat{O_2}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}_{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">2</span></span></span>}}}$.

b) So sánh hai góc

Ta có thể so sánh góc dựa vào số đo của chúng.

- Nếu hai góc A và B có số đo bằng nhau thì ta nói hai góc đó bằng nhau và kí hiệu $\hat{A}=\hat{B}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$.

- Nếu góc A có số đo nhỏ hơn số đo của góc B thì ta nói góc A nhỏ hơn góc B và kí hiệu $\hat{A}<\hat{B}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;"><</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$.

- Nếu góc A có số đo lớn hơn số đo của góc B thì ta nói góc A lớn hơn góc B và kí hiệu $\hat{A}>\hat{B}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">></span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$.

Ví dụ 4. Cho $\widehat{mOn}=30°,\widehat{xAy}=80°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">30</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span><span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">,</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">80</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$.

Vì 30° < 80° nên $\widehat{mOn}<\widehat{xAy}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;"><</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$

4. Góc vuông, góc nhọn, góc tù, góc bẹt

- Góc nhọn là góc có số đo lớn hơn 0° và nhỏ hơn 90°.

- Góc vuông là góc có số đo bằng 90°.

- Góc tù là góc có số đo lớn hơn 90° và nhỏ hơn 180°.

- Góc bẹt là góc có số đo bằng 180°.

Ví dụ 5. Hãy cho biết trong các góc sau, góc nào là góc nhọn, góc vuông, góc tù hay góc bẹt?

$\widehat{xOy}=37°,\widehat{BCD}=97°,\widehat{mAn}=90°,\widehat{IHK}=180°,\widehat{MPQ}=173°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">37</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span><span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">,</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">97</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span><span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">,</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">90</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span><span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">,</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">180</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span><span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">,</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">173</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$

Hướng dẫn giải

Ta có:

+) 0° < 37° < 90° nên $\widehat{xOy}=37°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">37</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$ là góc nhọn;

+) 90° < 97° < 180° nên $\widehat{BCD}=97°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">97</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$ là góc tù;

+) $\widehat{mAn}=90°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">90</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$ nên $\widehat{mAn}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$ là góc vuông;

+) $\widehat{IHK}=180°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">180</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$ nên $\widehat{IHK}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$ là góc bẹt;

+) 90° < 173° < 180° nên $\widehat{MPQ}=173°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">173</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$ là góc tù.

B. Bài tập tự luyện

Bài 1. Tính số đo góc tạo bởi kim giờ và kim phút của đồng hồ lúc: 2 giờ, 5 giờ, 18 giờ, 12 giờ, 9 giờ, 2 giờ 30 phút, 5 giờ 30 phút.

Hướng dẫn giải

Lúc 1 giờ thì góc tạo bởi kim giờ và kim phút là 30°, nên khi kim giờ và kim phút cách nhau 1 số thì số đo góc giữa kim giờ va kim phút là 30°, do đó:

+ Lúc 2 giờ góc tạo bởi kim giờ và kim phút là 60°;

+ Lúc 5 giờ góc tạo bởi kim giờ và kim phút là 150°;

+ Lúc 18 giờ cũng chính là 6 giờ, góc tạo bởi kim giờ và kim phút là 180°;

+ Lúc 12 giờ góc tạo bởi kim giờ và kim phút là 0°;

+ Lúc 9 giờ góc tạo bởi kim giờ và kim phút là 270°;

+ Lúc 2 giờ 30 phút góc tạo bởi kim giờ và kim phút là 165°;

+ Lúc 5 giờ 30 phút góc tạo bởi kim giờ và kim phút là 15°.

Bài 2. Quan sát hình vẽ rồi điền vào bảng sau các góc có trong hình vẽ.

Hướng dẫn giải

Bài 3. Cho hình vẽ:

Đọc tên các điểm nằm trong góc OAE trên hình vẽ.

Hướng dẫn giải

Trong hình vẽ trên, các điểm nằm trong góc OAE là: điểm B, điểm C.

Bài 4. Trong hình vẽ sau, cho $\Delta ABC$$\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}$ đều và $\widehat{DBC}=20°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">20</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$, $\widehat{ACB}=60°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">60</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$.

a) Kể tên các góc có trong hình vẽ trên, những góc nào có số đo bằng 60°?

b) Điểm D có nằm trong góc ABC không? Điểm C có nằm trong góc ACB không ?

c) Em hãy dự đoán số đo góc ABD và sử dụng thước đo góc để kiểm tra lại dự đoán của mình?

Hướng dẫn giải

a) Trong hình vẽ trên có các góc là: $\widehat{ABC},\widehat{ABD};\widehat{CBD};\widehat{ACB};\widehat{CAB};\widehat{CAD};\widehat{BAD}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">,</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">;</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">;</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">;</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">;</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">;</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$.

Vì tam giác ABC là tam giác đều nên $\widehat{ABC}=\widehat{BCA}=\widehat{CAB}$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}$ 

Mà $\widehat{ACB}=60°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">60</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$ nên $\widehat{ABC}=\widehat{BCA}=\widehat{CAB}=60°a$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">60</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}$.

b) Điểm D có nằm trong góc ABC, điểm C không nằm trong góc ACB.

c) Dự đoán số đo góc $\widehat{ABD}=40°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">40</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$

Kiểm tra bằng thước đo góc

Bước 1: Đặt thước đo góc sao cho tâm của thước trùng với đỉnh B của góc . Vạch 0 của thước nằm trên cạnh AB.

Bước 2: Xác định cạnh AD đi qua vạch chia độ 40° nên $\widehat{ABD}=40°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">40</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$.

Bài 5. Cho tia Ax. Vẽ tia Ay sao cho $\widehat{xAy}=125°.$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">125</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span><span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">.</span></span></span>$ 

Hướng dẫn giải

Để vẽ góc xAy có số đo bằng 125° ta làm như sau:

Bước 1: Vẽ tia Ax

Bước 2: Đặt thước đo góc sao cho tâm của thước trùng với A, vạch 0 của thước nằm trên tia Ax.

Bước 3: Đánh dấu một điểm trên vạch chia độ của thước tương ứng với số chỉ 125 độ, kẻ tia Ay đi qua điểm đã đánh dấu.

Ta có $\widehat{xAy}=125°$$\stackrel{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">^</span></span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">�</span></span></span>}}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">=</span></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">125</span></span></span>}<span\; style="font-size:18px;"><span\; style="font-size:11px;"><span\; style="line-height:2;">°</span></span></span>$ đã được vẽ.