Practice Questions on Congruence of Triangles

Triangles are said to be congruent if they measure same in size. There are various ways to determine the congruency of two triangles. This article provides practice questions on the congruency of triangles. It also has practice questions to learn how to solve the questions.

There are several criteria used to determine if two triangles are congruent:

SSS (Side-Side-Side) Congruence: If sides AB, BC, and AC, of one triangle, are just like sides AB, BC, and AC, of another triangle, then the triangles are congruent.

SAS (Side-Angle-Side) Congruence: If a given side and the included angle from the first triangle were equal to the other side and the included angle of another triangle, these triangles will be considered congruent.

ASA (Angle-Side-Angle) Congruence: When the special property of two angles and the side that is between them with one triangle is the same as that of the two angles and the side that is between them with the second triangle, then the triangles are congruent.

AAS (Angle-Angle-Side) Congruence: Suppose two pairs of angles and a non-included side of the first triangle are equal to two pairs of angles and a non-included side of the other triangle. Consequently, these triangles are congruent.

HL (Hypotenuse-Leg) Congruence: Since the hypotenuse and one leg of a triangle are equal to the other triangle’s hypotenuse and one leg, then the two triangles are congruent. This is also called RHS congruence rule.

Q1. Given triangles are △ABC and △DEF with AB=5 cm, DE=5 cm, BC=7 cm, EF=7 cm, and CA=8 cm, FD=8 cm. Are these triangles congruent or not?

Solution:

Given: AB = DE = 5 cm, BC = EF = 7 cm, CA = FD = 8 cm.

To Prove: △ABC ≅ △DEF.

Proof:

In triangles △ABC and △DEF,

AB = DE (Given)

BC = EF (Given)

CA = FD (Given)

Since all three sides of △ABC are equal to the corresponding sides of △DEF, by the SSS (Side-Side-Side) congruence criterion, △ABC ≅ △DEF.

Conclusion: Yes, the triangles are congruent by SSS.

Q2. In △ABC and △DEF, AB=6 cm, DE=6 cm, ∠B=45°, ∠E=45°, and BC=8 cm, EF=8 cm. Are the triangles are congruent?

Solution:

Given: AB = DE = 6 cm, ∠B = ∠E = 45°, BC = EF = 8 cm.

To Prove: △ABC ≅ △DEF.

Proof:

In triangles △ABC and △DEF,

AB = DE (Given)

∠B = ∠E (Given)

BC = EF (Given)

Since two sides and the included angle of △ABC are equal to the corresponding sides and included angle of △DEF, by the SAS (Side-Angle-Side) congruence criterion, △ABC ≅ △DEF.

Conclusion: Yes, the triangles are congruent by SAS.

Q3: Given △ABC and △DEF with ∠A=60°, ∠D=60°, ∠B=50°, ∠E=50°, and AB=4 cm, DE=4 cm. Are the triangles are congruent?

Solution:

Given: ∠A = ∠D = 60°, ∠B = ∠E = 50°, AB = DE = 4 cm.

To Prove: △ABC ≅ △DEF.

Proof:

In triangles △ABC and △DEF,

∠A = ∠D (Given)

∠B = ∠E (Given)

AB = DE (Given)

Since two angles and the included side of △ABC is equal to the corresponding angles and included side of △DEF, by the ASA (Angle-Side-Angle) congruence criterion, △ABC ≅ △DEF.

Conclusion: Yes, the triangles are congruent by ASA.

Q4: In △ABC and △DEF, ∠A=30°, ∠D=30°, ∠B=45°, ∠E=45°, and BC=10 cm, EF=10 cm. Are the triangles are congruent?

Solution:

Given: ∠A = ∠D = 30°, ∠B = ∠E = 45°, BC = EF = 10 cm.

To Prove: △ABC ≅ △DEF.

Proof:

In triangles △ABC and △DEF,

∠A = ∠D (Given)

∠B = ∠E (Given)

BC = EF (Given)

Since two angles and a non-included side of △ABC are equal to the corresponding angles and side of △DEF, by the AAS (Angle-Angle-Side) congruence criterion, △ABC ≅ △DEF.

Conclusion: Yes, the triangles are congruent by AAS.

Q5: Given right triangles △ABC and △DEF with hypotenuse AC=13 cm, DF=13 cm, and leg BC=12 cm, EF=12 cm. Are the triangles are congruent?

Solution:

Given: AC = DF = 13 cm, BC = EF = 12 cm.

To Prove: △ABC ≅ △DEF.

Proof:

In right triangles △ABC and △DEF,

AC = DF (Hypotenuse given)

BC = EF (Leg given)

Since the hypotenuse and one leg of △ABC are equal to the hypotenuse and one leg of △DEF, by the HL (Hypotenuse-Leg) congruence criterion, △ABC ≅ △DEF.

Conclusion: Yes, the triangles are congruent by HL.

Q1: Given triangles △PQR and △STU with PQ=8 cm, ST=8 cm, QR=10 cm, TU=10 cm, and PR=12 cm, SU=12 cm. Are the triangles congruent? Prove your answer.

Q2: In △MNO and △XYZ, if MN=5 cm, XY=5 cm, ∠N=∠Y=40°, and NO=7 cm, XZ=7 cm, are the triangles congruent? Provide the proof.

Q3: Given △ABC and △DEF with ∠A=45°, ∠D=45°, ∠C=90°, ∠F=90°, and AB=9 cm, DE=9 cm. Are the triangles congruent? Prove your answer.

Q4: In △JKL and △MNO, if ∠J=∠M=35°, ∠K=∠N=55°, and KL=11 cm, NO=11 cm, are the triangles congruent? Provide the proof.

Q5: Given right triangles △GHI and △JKL with hypotenuse GH=17 cm, JK=17 cm, and leg HI=15 cm, KL=15 cm. Are the triangles congruent? Prove your answer.

Learn More,

• SAS Congruence Rule
• ASA Congruence Rule
• Side Side Side
• RHS Congruence Rule

What is the meaning of congruency of two triangles?

If two triangles are congruent it means if they placed on one another they will completely coincide each other

Who are the ways to determine congruence of triangles?

The main criteria for triangle congruence are:

• SSS (Side-Side-Side) Congruence: The three corresponding sides are all of equal lengths.
• SAS (Side-Angle-Side) Congruence: There are 2 sides and 1 interior angle which are same.
• ASA (Angle-Side-Angle) Congruence: Two angles are equated and the including side is equal.
• AAS (Angle-Angle-Side) Congruence: Two angles and a side in common are the equal.
• HL (Hypotenuse-Leg) Congruence: The side that is opposite to an acute angle and one leg of a triangle are equal in the right triangles.

What is SSS Congruency?

SSS Congruency states that if three sides of a triangle are equal to corresponding three sides of other triangle then the two triangles are said to be congruent

What are ASA and AAS congruency?

ASA (Angle-Side-Angle) congruency says that two triangles are congruent if two angles and included sides are equal while AAS (Angle-Angle-Side) states that if two angles and any one side is equal the two triangles are congruent

Can two pairs of triangles be congruent will only two parts being equal?

No, it will be impossible to say that two particular triangles are congruent merely because two corresponding sides are equal.

What is RHS congruence?

If you have two right-angled triangles and know that one of their sides and hypotenuses are equal to the corresponding sides and hypotenuses of the other triangle, as well as one of their acute angles, the triangles are RHS congruent.

Is it possible to draw the same triangle with an equal area?

Not necessarily. In triangles, the same area is found for triangles with different shapes and sizes. Correspondence implies that the matched sides and angles need to be equal to those of the other figure.