Prove two triangles congruent using SSS, SAS, ASA, and RHS criteria, and apply congruency to solve geometric problems.
Two triangles are congruent if they satisfy SSS (all sides equal), SAS (two sides and included angle equal), ASA (two angles and included side equal), or RHS (right angle, hypotenuse, side, for right triangles).
Triangle ABC has sides 5,6,7 cm and triangle DEF has sides 5,6,7 cm. Are they congruent?
Yes, by SSS congruency
Two right triangles have equal hypotenuses (10 cm) and one equal leg (6 cm). Are they congruent?
Yes, by RHS congruency
Once two triangles are proven congruent, their Corresponding Parts are Congruent (CPCT) — this lets us conclude equal angles or sides that weren't given directly.
Triangles ABC and PQR are congruent by SAS, with AB=PQ, angle A=angle P given. What can we conclude about angle B and angle Q using CPCT?
Angle B = Angle Q, since corresponding parts of congruent triangles are equal
Why is CPCT useful in geometric proofs?
It lets us establish equal sides or angles that weren't directly given, once congruency is proven, which helps build further proofs
1. Two triangles have sides 4,5,6 cm and 4,5,6 cm. Are they congruent? By which criterion?
2. Two triangles share two equal sides and the included angle. Which criterion proves congruency?
3. Two right triangles have equal hypotenuse and equal one side. Which criterion applies?
4. Two triangles have two equal angles and the included side equal. Which criterion applies?
5. If triangle ABC≅triangle XYZ by SSS, and AB=XY, BC=YZ, CA=ZX, what can we say about angle A and angle X by CPCT?