Congruent Triangles Worksheet with Answers – Geometry (Free PDF)

Problem 1 Corresponding parts ★★★

△PQR ≅ △XYZ, with PQ = 9.5, QR = 12, m∠P = 51° and m∠Q = 77°. Find XY, YZ, m∠X, m∠Y and m∠Z.

Problem 2 Name the criterion ★★★

Name the congruence criterion (SSS, SAS, ASA, AAS, HL or none) that each description supports.

1. Three pairs of sides are congruent.
2. Two pairs of sides and the angles between them are congruent.
3. Two pairs of angles and the sides between them are congruent.
4. Two pairs of angles and a pair of sides that is not between them are congruent.
5. Two right triangles have congruent hypotenuses and one pair of congruent legs.

Problem 3 Why SSA is not enough ★★★

Two triangles each have a side of 6, a side of 9 and a 40° angle that is not between those sides. A student says they must be congruent. Is the student right?

Problem 4 Solve for x ★★★

△ABC ≅ △DEF. AB = 4x + 1 and DE = 7x − 11. Find x and AB.

Problem 5 Shared side ★★★

In the figure, AB ≅ CB and AD ≅ CD. Name a third pair of congruent parts and the criterion that proves △ABD ≅ △CBD.

Problem 6 Read the marks ★★★

Use the marks to decide which criterion proves the triangles congruent.

Problem 7 Angle sum in congruent triangles ★★★

△ABC ≅ △DEF, with m∠A = (4x + 3)°, m∠D = (6x − 17)° and m∠B = 62°. Find x and m∠F.

Problem 8 Perimeter of congruent triangles ★★★

△JKL ≅ △MNP. JK = 2x + 1, KL = 3x − 2, JL = 4x − 5, MN = 13 and NP = 16. Find x, JL and the perimeter of △JKL.

Problem 9 ASA or AAS ★★★

In △ABC and △DEF, decide whether each set of givens proves congruence, and name the criterion.

1. ∠A ≅ ∠D, AB ≅ DE, ∠B ≅ ∠E
2. ∠A ≅ ∠D, ∠B ≅ ∠E, BC ≅ EF
3. ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F

Problem 10 Hypotenuse and leg ★★★

△ABC and △DEF are right triangles with right angles at B and E. AC = DF = 13 and AB = DE = 5. Prove the triangles are congruent, then find BC and EF.

Problem 11 Congruence from coordinates ★★★

△ABC has A(1, 1), B(5, 1) and C(5, 4). △DEF has D(−3, −2), E(−3, −6) and F(−6, −6). Show that the triangles are congruent.

Problem 12 The ambiguous SSA case ★★★

In a triangle, ∠A = 40°, AB = 10 and BC = 7. The height from B to the line through A and C is 10 sin 40° ≈ 6.4. Explain why two non-congruent triangles fit, and estimate both lengths of AC.

Problem 13 Prove angles congruent with SSS ★★★

Given: AB ≅ CB and AD ≅ CD. Prove: ∠BAD ≅ ∠BCD. Write a two-column proof.

Problem 14 Crossing midpoints ★★★

Segments AC and BD intersect at O, and O is the midpoint of both. Prove: △AOB ≅ △COD.

Problem 15 Roof truss braces ★★★

A roof truss has two triangular braces, △PQR and △PSR, sharing the beam PR. PQ = PS = 18 feet and QR = SR = 14.5 feet. Show that the braces are congruent, and find m∠PQR when m∠PSR = 112°.

Problem 16 Matching ramps ★★★

A skate park builds two right-triangle ramps. Each ramp has a hypotenuse of 10 feet and a base leg of 6 feet. Show the ramps are congruent, find the height of each, and find the area of one triangular side.

Problem 17 Bisector and perpendicular ★★★

Given: BD bisects ∠ABC and BD ⊥ AC at D. Prove: AD ≅ CD. Write a two-column proof.

Problem 18 Prove sides congruent with AAS ★★★

Given: ∠B ≅ ∠E, ∠C ≅ ∠F and AB ≅ DE. Prove: AC ≅ DF. Write a two-column proof.

Problem 19 Congruent triangles with algebra ★★★

△ABC ≅ △XYZ. The sides of △ABC are AB = 3x − 1, BC = 2x + 5 and CA = x + 12. In △XYZ, XY = 17, YZ = 17 and XZ = 18. Find x, the perimeter, and say what type of triangle this is by its sides.

Problem 20 Overlapping triangles ★★★

Points D and E lie on AB and AC. Given: AB ≅ AC and AD ≅ AE. Prove: BE ≅ CD. Write a two-column proof.