Geometry Congruent Triangles and Rigid Motions Worksheet (PDF)

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

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

These 20 problems practice congruent triangles, corresponding parts and congruence tests (Common Core G-CO.B.7, G-CO.B.8). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are proofs and applied problems with trusses, ladders and surveying.

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

Congruence practice worksheet: exercises 1 to 10

Exercise 1 – Corresponding parts

Triangle ABC is congruent to triangle DEF with vertices matching in order. If \(m\angle A = 48^{\circ}\) and \(BC = 9\), find \(m\angle D\) and \(EF\).

Exercise 2 – Name the test: SSS

SSS: three pairs of congruent sides.

Name the test that proves that triangle ABC is congruent to triangle DEF.

Exercise 3 – Name the test: SAS

SAS: two sides and the angle between them.

Name the test that proves the triangles congruent.

Exercise 4 – Name the test: ASA

ASA: two angles and the side between them.

Name the test that proves the triangles congruent.

Exercise 5 – Name the test: AAS

Triangle PQR has \(m\angle P = 40^{\circ}\), \(m\angle Q = 75^{\circ}\) and \(QR = 8\). Triangle XYZ has \(m\angle X = 40^{\circ}\), \(m\angle Y = 75^{\circ}\) and \(YZ = 8\). Is there enough information to prove them congruent?

Exercise 6 – Name the test: HL

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

Both triangles are right triangles. Which test proves them congruent, and what is the length of the missing leg AC if the hypotenuse BC is 13?

Exercise 7 – Why SSA fails

Two triangles each have sides of length 6 and 8, and a 30° angle that is not between them. Explain why this is not enough to prove congruence.

Exercise 8 – Angles with x

Triangle ABC is congruent to triangle PQR. \(m\angle A = (3x + 5)^{\circ}\) and \(m\angle P = (5x – 21)^{\circ}\). Find \(x\) and \(m\angle A\).

Exercise 9 – Sides with x

Triangle ABC is congruent to triangle DEF. \(AB = 2x + 3\) and \(DE = 4x – 9\). Find \(x\) and \(AB\).

Exercise 10 – Missing side

Triangle ABC has sides AB = 9, BC = 12 and CA = 15. Triangle DEF is congruent to it with DE = 9 and DF = 15. Find EF and the perimeter of DEF.

Exercises 11 and up

Exercise 11 – Translation shows congruence

Triangle ABC and its translation.

Triangle ABC has A(0, 0), B(4, 0) and C(0, 3). It is translated by (x, y) → (x + 5, y + 1) to give DEF. Find the side lengths of both triangles and name the congruence test that applies.

Exercise 12 – Reflection and distance

Triangle ABC has A(1, 1), B(5, 1) and C(2, 4). Reflect it across the y-axis to get A′B′C′. Show that AC = A′C′ and BC = B′C′.

Exercise 13 – Which are congruent?

Triangle 1 has sides 3, 4, 5. Triangle 2 has sides 5, 3, 4. Triangle 3 has sides 3, 4, 6. Which pair is congruent, and by which test?

Exercise 14 – Included angle?

Triangle A has sides 6 and 8 with a 50° angle between them. Triangle B has sides 6 and 8 with a 50° angle that is opposite the side of length 8. Are the triangles necessarily congruent?

Exercise 15 – Choose the missing fact

Triangles ABC and DEF have \(AB = DE\) and \(m\angle B = m\angle E\). Name one more fact that proves them congruent by ASA, and one that proves them congruent by SAS.

Exercise 16 – Bridge truss

Truss with shared side AC.

A truss has AB = AD and CB = CD, and they share side AC. Prove that triangle ABC is congruent to triangle ADC, and find \(m\angle BAC\) if \(m\angle DAC = 38^{\circ}\).

Exercise 17 – River width

Stakes A, M and B with congruent triangles across the river.

Sofia wants the width of a river. She stands at A directly across from a tree T, walks 30 yards along the bank to M, then another 30 yards to B. She turns 90° and walks away from the river until the tree, M and P line up. She measures BP = 42 yards. Explain why the river width AT is 42 yards.

Exercise 18 – Ladders

Two ladders each 13 feet long lean against a vertical wall. Each one has its base 5 feet from the wall. Show the ladders reach the same height, and find that height.

Exercise 19 – Triangular flags

A school makes triangular flags. Each flag has sides of 18 inches and 24 inches with a 90° angle between them. Show that any two flags are congruent and find the longest side.

Exercise 20 – Crossing braces

Segments AD and BC cross at E, which is the midpoint of both. AE = ED = 7, BE = EC = 5 and AB = 9. Prove that triangle AEB is congruent to triangle DEC and find DC.

Key facts behind this congruence practice worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the congruence practice worksheet as a PDF with the button below and keep it with your notes.

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