49
Constructing an angle equal to a given angle
Textbook: pp. 128–129
GoalKnow how to construct an angle equal to a given angle.
New words
angle construction · burchak yasasharc · yoyintersection of circles · aylana kesishuvijustification by SSS · TTT bilan asoslash
Explanation
To construct on ray O an angle equal to a given angle A: 1) draw a circle of any radius centred at A meeting the sides at B and C; 2) draw a circle of the same radius centred at O meeting the ray at C₁; 3) draw a circle centred at C₁ with radius BC meeting the previous circle at B₁; 4) draw ray OB₁. Justification: in △ABC and △OB₁C₁, AB = OB₁, AC = OC₁ and BC = B₁C₁ by construction, so they are equal by SSS and ∠B₁OC₁ = ∠A. The problem has two solutions, one in each half-plane of the ray. An angle equal to the sum or difference of two angles is constructed the same way.
Worked examples
To construct 60°: take C₁ on the ray, draw circles of radius OC₁ centred at O and C₁, and call an intersection point B₁; then △OB₁C₁ is equilateral.
To construct the sum of 30° and 50°, lay them side by side on a ray; the result is 80°.
Class activity
“Copy the angle”: copy an angle onto another ray with a compass (with an adult) and check with a protractor.
Practice
1
Two angles are 25° and 40°. The angle equal to their sum is how many degrees?
65
2
Two angles are 70° and 20°. The angle equal to their difference is how many degrees?
50
3
How many solutions are there when copying an angle onto a ray?
2
4
Why is the constructed angle equal to the given one?
The two triangles have three equal sides (SSS), so the corresponding angles are equal.