Identify whether the triangles below are congruent. What conditions did you use to establish their congruence? Express the congruence.

Question

Homi · Accepted Answer

Understand the Question

Two triangles are congruent by the Side-Angle-Side (SAS) criterion if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle.
To express congruence correctly, the vertices of both triangles must be written in corresponding order.

Step 1 · Compare Corresponding Parts

From $\triangle ABC$ and $\triangle XZY$ : Diagram 1

Comparing the given sides and angles:

\begin{aligned} AB &= XZ = 7\text{ cm} \\[0.4em] \angle ABC &= \angle XZY = 47^\circ \\[0.4em] BC &= ZY = 5\text{ cm} \end{aligned}

Here, two sides and the included angle between them are equal in both triangles.

Step 2 · Establish Congruence and Vertex Correspondence

Since two sides and the included angle of $\triangle ABC$ are equal to the corresponding two sides and included angle of $\triangle XZY$ , the triangles are congruent by the SAS (Side-Angle-Side) criterion.

The correspondence between vertices is:

$A \leftrightarrow X$
$B \leftrightarrow Z$
$C \leftrightarrow Y$

Therefore, the congruence relation is: $\triangle ABC \cong \triangle XZY$

Answer

Yes, the triangles are congruent by the SAS criterion, expressed as $\triangle ABC \cong \triangle XZY$ .

Common Mistakes

Incorrect Vertex Order: Writing $\triangle ABC \cong \triangle XYZ$ instead of matching the equal angles at vertices $B$ and $Z$ , which gives $\triangle ABC \cong \triangle XZY$ .
Non-Included Angle: Assuming SAS congruence when the given angle is not between the two known sides.

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