A congruence proof is written in a set format. A proof is a logical explanation written from the “given conditions” to the “conclusion.”
How to Write a Proof (Template)
In △○○○ and △□□□ ① [Write the basis for the equal sides or angles] … (Basis) ② [Write the basis for the equal sides or angles] … (Basis) ③ [Write the basis for the equal sides or angles] … (Basis) From ①②③, by [the congruence condition], △○○○ ≅ △□□□
Example ①
In quadrilateral ABCD, AB=CD and AD=CB. Prove that △ABD ≅ △CDB.
Answer:
In △ABD and △CDB
AB = CD … (Given) ①
AD = CB … (Given) ②
BD = DB … (Common side) ③
From ①②③, since three pairs of corresponding sides are equal, △ABD ≅ △CDB
In quadrilateral ABCD, AB=CD and AD=CB. Prove that △ABD ≅ △CDB.
Answer:
In △ABD and △CDB
AB = CD … (Given) ①
AD = CB … (Given) ②
BD = DB … (Common side) ③
From ①②③, since three pairs of corresponding sides are equal, △ABD ≅ △CDB
Key Points
- Possible bases include “given,” “common side,” “vertical angles,” and “alternate/corresponding angles of parallel lines”
- When using congruence, clearly write “because △○○≅△□□”
- The conclusion may be written as “therefore ∠○=∠□”
Practice Problem
- Let M be the midpoint of side BC of △ABC, and take point D on the extension of AM such that AM=MD. Prove that △ABM ≅ △DCM.
Answer and Explanation
- Answer: In △ABM and △DCM / BM=CM (M is the midpoint) ① / AM=DM (Given) ② / ∠AMB=∠DMC (Vertical angles) ③ / From ①②③, by SAS, △ABM≅△DCM.