Triangle Congruence Proofs Using SAS and CPCT

Published on October 4, 2026

Geometry

Learn how to prove triangle congruence using the SAS criterion and apply CPCT properties with clear, step-by-step geometric solutions.

Question

Solve with explanation

Answer

Here are the step-by-step solutions for the problems provided in the image.

Solution for Question 9

Given:

(indicated by the square symbols in the figure)

To prove:

Proof:
In

and

(Given)
(Given)
and . Since (Given), then , which means .

By the SAS (Side-Angle-Side) congruence criterion, since two sides and the included angle of

are equal to the corresponding sides and included angle of

Solution for Question 10 (Case Based Question)

Given:

(AD is the angle bisector of )

(a) Name the two triangles which are congruent in the given figure.
The two congruent triangles are

and

(b) Which congruence rule is used to prove that the two triangles are congruent?
We have:

(Given)
(Given)
(Common side)
Therefore, the SAS (Side-Angle-Side) congruence rule is used.

, by CPCT (Corresponding Parts of Congruent Triangles):

© Is there any other congruence criteria that can be used to prove that the triangles are congruent? Justify your answer.
Yes, the SSS (Side-Side-Side) or ASA (Angle-Side-Angle) criteria can also be used:

SSS: Since and is common, and we can prove (because in an isosceles triangle, the angle bisector of the vertex angle is also the median), all three sides are equal.
ASA: Since , then (angles opposite to equal sides). Given and is common, we can use ASA (or AAS).

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