After proving that triangles are congruent, use the fact that “corresponding sides and angles of congruent triangles are equal” to derive another equation.
Example ①
In quadrilateral ABCD, AB=CB and AD=CD. Prove that ∠ABD=∠CBD.
Answer: In △ABD and △CBD / AB=CB (Given) ① / AD=CD (Given) ② / BD=BD (Common side) ③ / From ①②③, by SSS, △ABD≅△CBD. Since corresponding angles of congruent triangles are equal, ∠ABD=∠CBD.
In quadrilateral ABCD, AB=CB and AD=CD. Prove that ∠ABD=∠CBD.
Answer: In △ABD and △CBD / AB=CB (Given) ① / AD=CD (Given) ② / BD=BD (Common side) ③ / From ①②③, by SSS, △ABD≅△CBD. Since corresponding angles of congruent triangles are equal, ∠ABD=∠CBD.
Key Points
- The phrase “from △○≅△□ (from corresponding sides/angles)” is necessary
- Write the sides and angles in the order of their corresponding vertices
Practice Problem
- In △ABC, AB=AC. If M is the midpoint of BC, prove that AM⊥BC.
Answer and Explanation
- Answer: In △ABM and △ACM / AB=AC (Given) ① / BM=CM (M is the midpoint) ② / AM=AM (Common side) ③ / From ①②③, by SSS, △ABM≅△ACM. ∠AMB=∠AMC. Since ∠AMB+∠AMC=180° (a straight line), ∠AMB=90°. Therefore AM⊥BC.