# Congruent triangle problems – Set 03

In this chapter we will solve questions related to congruent triangles.

All the questions are to the standard of grade 09.

Question 01
In the below figure AD = BC. Prove that line CD bisects AB.

Given:
∠OAD = ∠OBC = 90 degree

To prove:
OA = OB

Solution

∠BOC = ∠DOA ( vertically opposite angle)
∠OBC = ∠OAD = 90 degree
BC = DA

By AAS congruency, both the triangles are congruent.
So OB = OA

Hence Proved

Question 02
In the below figure parallel lines l & m are intersected by parallel transversal p and q. Prove that triangle ABC and CDA are congruent.

Solution
Consider triangle ABC and CDA;

∠BAC = ∠ACD ( alternate interior angle )
AC = CA ( common side )
∠ACB = ∠DAC ( alternate interior angle )

By ASA congruency, both triangles are congruent.
Hence Proved.

Question 03
ABC is an isosceles triangle in which AB = AC. AD is the altitude. Prove that AD bisect BC.

Given;
AB = AC
∠B = ∠C (angle opposite to equal sides are equal)

To prove;
BD = DC

Proof:
Consider triangle ABD and ACD

AB = AC

By RHS congruency, both triangles are congruent.
So, BD = DC

Hence Proved

Question 04
ABC & DBC are isosceles triangle.
Prove that;
(i) triangle ABD & ACD are congruent
(ii) triangle ABE & ACE are congruent

Given:
AB = AC
DB = DC

(i) Prove triangle ABD & ACD are congruent

Consider triangle ABD & ACD;
AB = AC ( given )
DB = DC ( given )
AD = DA ( common side )

By SSS congruency condition, both the triangles are congruent.

(ii) Prove that triangle ABE & ACE are congruent

Consider triangle ABE & ACE;
AB = AC
∠ABE = ∠ACE ( angle opposite to equal sides are equal)
AE = EA (common side)

By SAS congruency, both the triangles are congruent.

Question 05
In the below figure AB = CB and ∠x = ∠y. Prove that AE = CD

Solution

AB is a straight line and we know that sum of adjacent angle in straight line measure 180 degree.

∠BDC + ∠x = 180

∠BDC = 180 – ∠x

Similarly we can write the following;
∠ AEB + ∠y = 180

∠AEB = 180 – ∠y

Since ∠x =y, we can say that ∠BDC = ∠ AEB

Now consider triangle ABE and CBD;

AB = CB ( given )
∠B = ∠B (common angle)
∠AEB = ∠BDC (proved above)

By AAS congruency, both triangles are congruent.
So, AE = CD

Hence Proved.

Question 06
In the below figure, line l bisect ∠A into two equal parts. Prove that triangle AQB & APB are congruent.

Solution:
Consider triangle AQB and APB;

∠QAB = ∠PAB ( line l is angle bisector)
∠ AQB = ∠ APB = 90 degree
AB = BA (common side)

By AAS congruency, both triangles are congruent.

Hence Proved.