Question

Given: triangle ABC is isosceles with vertex B, angle ABD is congruent to angle CBEProve: triangle BED is isosceles(ignore the congruent markings)

Answer 1

Here, we want to prove that triangle BED is an isosceles triangle

From the question, we are given two things to work with

That triangle ABC is isosceles and angle ABD is equal to CBE

From the vertex B, we can see that we have 3 angles

That is angle B is divided into three angles 1, 2 and 3

From the congruency statement given;

`\begin{gathered} 1\text{ + 2 = 2 + 3} \\ \text{Thus;} \\ 1\text{ = 3} \end{gathered}`

We can see from above that angle 1 and angle 3 are equal

Since 1 and 3 are equal, and also, angle marked A and C are equal;

Then, we have two completely congruent triangles

Triangle ABE and CDB

Since these two triangles are completely congrunet, due to the fact that all angles are equal, then the sides facig the angles are equal

Thus;

`BE\text{ = BD}`

Since the two sides are equal, we can conclude that the base angles are equal, and so , the triangle BED is isosceles