Geometry Congruent Triangles and Rigid Motions Worksheet Solutions (PDF)

Geometry math worksheet solutions: Congruent Triangles and Rigid Motions – Maths PDF

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This congruence practice solutions for Geometry covers congruent triangles and rigid motions (Common Core G-CO.B.7, G-CO.B.8), with a free PDF to print.

Use this answer key to check the 20 problems on congruent triangles. Every solution shows the formula or theorem used, each step of the work, and the final answer in bold. If an answer differs from yours, compare the steps line by line to find where the work changed.

Geometry · Common Core State Standards: G-CO.B.7, G-CO.B.8

Congruence practice solutions: exercises 1 to 10

Exercise 1 – Corresponding parts

A matches D, so \(m\angle D = 48^{\circ}\). B matches E and C matches F, so \(EF = BC = 9\).

m∠D = 48°, EF = 9

Exercise 2 – Name the test: SSS

SSS: three pairs of congruent sides.

Each pair of corresponding sides is marked congruent (three pairs).

SSS

Exercise 3 – Name the test: SAS

Two pairs of sides are marked and the 50° angle is between them.

SAS

Exercise 4 – Name the test: ASA

Two pairs of angles are marked and the side between them is marked.

ASA

Exercise 5 – Name the test: AAS

QR is opposite P and YZ is opposite X, so the side is not between the two angles.

Two angles and a non-included side give AAS.

Yes, by AAS

Exercise 6 – Name the test: HL

Right triangles with a congruent hypotenuse and a congruent leg: HL.

With leg 5 and hypotenuse 13, the other leg is \(\sqrt{13^{2} – 5^{2}} = \sqrt{144} = 12\).

HL; 12

Exercise 7 – Why SSA fails

The side of length 6 can swing to meet the base in two different places, giving two different triangles.

SSA is not a valid test.

Not enough information: SSA does not prove congruence

Exercise 8 – Angles with x

Corresponding angles are equal.

\(3x + 5 = 5x – 21\)

\(26 = 2x\)

\(x = 13\)

\(m\angle A = 3(13) + 5 = 44^{\circ}\)

x = 13, m∠A = 44°

Exercise 9 – Sides with x

\(2x + 3 = 4x – 9\)

\(12 = 2x\)

\(x = 6\)

\(AB = 2(6) + 3 = 15\)

x = 6, AB = 15

Exercise 10 – Missing side

The side opposite the same vertex must match: EF corresponds to BC, so \(EF = 12\).

\(9 + 12 + 15 = 36\)

EF = 12, perimeter 36

Exercises 11 and up

Exercise 11 – Translation shows congruence

A translation preserves side lengths.

ABC: \(AB = 4\), \(AC = 3\), \(BC = \sqrt{16 + 9} = 5\).

DEF: D(5, 1), E(9, 1), F(5, 4), so \(DE = 4\), \(DF = 3\), \(EF = \sqrt{16 + 9} = 5\).

Both have sides 3, 4, 5, so the triangles are congruent by SSS

Exercise 12 – Reflection and distance

Reflection across the y-axis.

A′(-1, 1), B′(-5, 1), C′(-2, 4).

\(AC = \sqrt{1^{2} + 3^{2}} = \sqrt{10}\) and \(A^{\prime}C^{\prime} = \sqrt{1^{2} + 3^{2}} = \sqrt{10}\).

\(BC = \sqrt{3^{2} + 3^{2}} = 3\sqrt{2}\) and \(B^{\prime}C^{\prime} = \sqrt{3^{2} + 3^{2}} = 3\sqrt{2}\).

The lengths match, so the triangles are congruent

Exercise 13 – Which are congruent?

Triangles 1 and 2 have the same three side lengths, so they are congruent by SSS.

Triangle 3 has a side of 6, so it is not congruent to the others.

Triangles 1 and 2, by SSS

Exercise 14 – Included angle?

In triangle A the angle is included; in triangle B it is not.

That is the SSA pattern, which does not prove congruence.

No, not necessarily

Exercise 15 – Choose the missing fact

ASA needs the other angle at the ends of AB: \(m\angle A = m\angle D\).

SAS needs the other side that makes the included angle B: \(BC = EF\).

ASA: m∠A = m∠D; SAS: BC = EF

Exercise 16 – Bridge truss

SSS applies: AB = AD, CB = CD, AC = AC.

AB = AD (given), CB = CD (given), AC = AC (shared). So the triangles are congruent by SSS.

By CPCTC, \(\angle BAC\) matches \(\angle DAC\), so it is also \(38^{\circ}\).

SSS; m∠BAC = 38°

Exercise 17 – River width

Triangles TAM and PBM each have a right angle (at A and B).

AM = BM = 30 yards (she walked equal distances).

The vertical angles at M are congruent.

So the triangles are congruent by ASA, and by CPCTC \(AT = BP = 42\).

AT = 42 yards

Exercise 18 – Ladders

Each ladder forms a right triangle with hypotenuse 13 and a leg of 5, so the triangles are congruent by HL.

The height is the other leg: \(\sqrt{13^{2} – 5^{2}} = 12\).

Both reach 12 feet

Exercise 19 – Triangular flags

Each flag is a 18-24-30 right triangle.

Each flag has two sides and the included angle equal, so any two flags are congruent by SAS.

Longest side: \(\sqrt{18^{2} + 24^{2}} = \sqrt{900} = 30\).

Congruent by SAS; 30 inches

Exercise 20 – Crossing braces

AE = DE (given), BE = CE (given), \(\angle AEB\) and \(\angle DEC\) are vertical angles, so they are congruent.

The triangles are congruent by SAS, and by CPCTC \(DC = AB = 9\).

SAS; DC = 9

Key facts behind this congruence practice solutions

Key facts

Congruent triangles have three pairs of congruent sides and three pairs of congruent angles.
Valid tests: SSS, SAS, ASA, AAS and, for right triangles, HL.
SSA and AAA do not prove congruence.
CPCTC: after proving triangles congruent, corresponding parts are congruent.
A rigid motion maps a figure onto a congruent figure.

How to use this congruence practice solutions

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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