314. Matching Sides & Angles · How similarity pairs up triangle parts
Matching colors always pair corresponding sides and angles, no matter how you reshape the triangle.
What this lesson covers
Try to break it
Drag A, B, or C. The colour-coded sides stay paired up: AB ↔ PQ (blue), BC ↔ QR (amber), AC ↔ PR (cyan). Angles pair as A ↔ P, B ↔ Q, C ↔ R. The three ratios AB/PQ, BC/QR, AC/PR always read the same number — the scale factor. Try to break one ratio while keeping the others; impossible.
How you build it
Build a triangle similar to ABC, matching corresponding parts.
- A is at (1, 1). P corresponds to A and is twice as far from O along ray OA, so double each coordinate: P = (2, 2).
- B is at (4, 1). Q corresponds to B — double each coordinate along ray OB: Q = (8, 2).
- C is at (2, 3). R corresponds to C — double each coordinate along ray OC: R = (4, 6).
- Join P to Q. PQ corresponds to AB (the blue side), and PQ = 2·AB.
- Join Q to R. QR corresponds to BC (the orange side).
- Join R to P. RP corresponds to CA (the green side). Matching A→P, B→Q, C→R pairs every side and angle, and every side of PQR is 2× its match in ABC.
The proof, step by step
Prove that similar triangles have equal corresponding angles and proportional corresponding sides.
- By definition, similar triangles have equal corresponding angles.
- The ratio of their corresponding sides is constant (proportional).
- Sides opposite to equal angles are corresponding sides.
- Angles opposite to proportional sides are corresponding angles.
Worked example
In ΔABC ~ ΔPQR, if ∠A = 50° and ∠B = 70°, what is the measure of the angle corresponding to ∠C in ΔPQR?
∠C = 180° - (50° + 70°) = 60°. In similar triangles, corresponding angles are equal, so the angle corresponding to ∠C is also 60°.
- 50°
- 60° — correct
- 70°
- 80°