Congruent figures drills

Identify congruent figures and corresponding parts.

Free, timed SHSAT practice sets on congruent figures— every question includes a full worked explanation, and there's no account required.

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How to solve congruent figures problems

Congruent figures are exact copies: every matching side and every matching angle is equal. The perimeter and area are equal too.

The key is the order of the letters. In \(\triangle ABC \cong \triangle DEF\), the first letters match, then the second, then the third:

If \(\triangle PQR \cong \triangle XYZ\) and \(QR = 9\), then \(YZ = 9\), because Q and Y are both second and R and Z are both third.

Common traps: matching sides by where they appear in a picture or by size, instead of by letter order, and choosing "cannot be determined". Congruent means same size, so every part of the copy is known. For a perimeter, add the three sides of one triangle only.

Worked examples

Example 1.
Triangle ABC is congruent to triangle DEF (\(\triangle ABC \cong \triangle DEF\)). If \(AB = 15\) in, \(BC = 19\) in, and \(CA = 20\) in, what is the length of \(DE\)?
Show answer and explanation
Answer:
C

The congruence statement matches vertices in order (\(A \leftrightarrow D\), \(B \leftrightarrow E\), \(C \leftrightarrow F\)), so \(DE\) matches \(AB = 15\) in. (Matching \(DE\) with \(BC\) gives 19 in. Matching \(DE\) with \(CA\) gives 20 in. Thinking the length of a side of DEF must be given forgets that matching sides of congruent triangles are equal.)

Example 2.
Triangle ABC is congruent to triangle DEF (\(\triangle ABC \cong \triangle DEF\)). If \(m\angle A = 75^\circ\), \(m\angle B = 47^\circ\), and \(m\angle C = 58^\circ\), what is \(m\angle E\)?
Show answer and explanation
Answer:
D

The congruence statement matches \(B \leftrightarrow E\), so \(m\angle E = m\angle B = 47^\circ\). (Matching \(\angle E\) with \(\angle A\) gives \(75^\circ\). Matching \(\angle E\) with \(\angle C\) gives \(58^\circ\). Thinking the angle of DEF must be given forgets that matching angles of congruent triangles are equal.)

Example 3.
Triangle PQR is congruent to triangle XYZ. Triangle PQR has sides 12 yd, 16 yd, and 12 yd. What is the perimeter of triangle XYZ?
Show answer and explanation
Answer:
A

Congruent figures have equal matching sides, so triangle XYZ has the same perimeter: \(12 + 16 + 12 = 40\) yd. (Adding the perimeters of both triangles gives 80. Leaving out the third side, \(12 + 16\), gives 28. Thinking the sides of XYZ must be given forgets that they equal the sides of PQR.)