Fundamental Concepts
20. Congruent Angles · Same measure, same shape
The angles are congruent when their measures are equal.
Two angles are congruent when they have the same measure. Drag Z to change ∠YXZ — when its degree value matches ∠AOB, the two angles are congruent regardless of where they sit on the page.
What this lesson covers
Try to break it
When the measures are different, the angles are not congruent. Drag Z further to see how the measure changes!
How you build it
Make two congruent angles.
- Place vertex O for the first angle.
- Place point A to aim the base arm.
- Draw ray OA (Ray tool: O then A).
- With centre O, draw an arc across ray OA. This locks the compass opening for every arc that follows.
- With centre P and the same radius, draw an arc crossing the first arc at Q. OP = PQ = OQ, so triangle OPQ is equilateral and angle POQ = 60 degrees.
- Draw ray OQ. The first angle ∠AOQ is 60 degrees.
- Place vertex R for the second angle, away from the first.
- Place point S to aim the second base arm.
- Draw ray RS (Ray tool: R then S).
- With centre R and the same compass opening, draw an arc across ray RS at T.
- With centre T and the same radius, draw an arc crossing the previous arc at U. Triangle RTU is equilateral, so angle TRU = 60 degrees.
- Draw ray RU. ∠SRU is also 60 degrees — equal measure means the two angles are congruent.
The proof, step by step
Prove that ∠AOB and ∠YXZ are congruent.
- ∠AOB measures 60°.
- ∠YXZ also measures 60° — the same measure as ∠AOB.
- Two angles are congruent when they have the same measure.
- Therefore, ∠AOB and ∠YXZ are congruent.
Worked example
In the given figure, ∠AOB and ∠YXZ are congruent angles. If the measure of ∠AOB is 60°, what is the measure of ∠YXZ?
Since ∠AOB and ∠YXZ are congruent angles, they have the same measure. Therefore, the measure of ∠YXZ is also 60°.
- 30°
- 60° — correct
- 90°
- 120°