Triangles are said to be congruent if they measure same in size. There are various ways to determine the congruency of two triangles.
Practice Questions on Congruence of Triangles - Solved
1. 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?
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.
2. 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?
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.
3: 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?
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.
4: In △ABC and △DEF, ∠A=30°, ∠D=30°, ∠B=45°, ∠E=45°, and BC=10 cm, EF=10 cm. Are the triangles are congruent?
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.
5: 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?
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.