Question

$ \Delta ABC $ and $ \Delta DBC $ are two isosceles triangles on the same $ BC $ vertex and $ A $ and $ D $ are on the same side of $ BC $ . If $ AD $ is extended to intersect $ BC $ at $ P $ , show that $ \Delta ABC\cong \Delta DBC $ .

Answer 1

Hint: For answering this question we will compare the sides of the given two triangles $ \Delta ABC $ and $ \Delta DBC $ using the given information in the question and prove that they are similarly based on the side-side-side congruence one among the four congruence tests which are Side-side-side, Side-angle-side, Angle-side-angle, and Angle-angle-side.

Complete step by step answer:
Now considering the question we have two isosceles triangles $ \Delta ABC $ and $ \Delta DBC $ so we have to prove that they are similar that is $ \Delta ABC\cong \Delta DBC $ .
We know that $ \Delta ABC $ and $ \Delta DBC $ are two isosceles triangles on the same $ BC $ vertex, $ A $ and $ D $ are on the same side of $ BC $ . If $ AD $ is extended it intersects $ BC $ at $ P $ .
In $ \Delta ABC $ , $ AB=AC $ since it is an isosceles triangle.
In $ \Delta DBC $ , $ DB=DC $ since it is also an isosceles triangle.
And $ BC $ is the common side in both the triangles $ \Delta ABC $ and $ \Delta DBC $ .
From the basic concept, we know that we have four congruence tests which are side-side-side, side-angle-side, angle-side-angle, and angle-angle-side.
From side-side-side congruence the two triangles are similar, that is $ \Delta ABC\cong \Delta DBC $ .
Hence, proved that when $ \Delta ABC $ and $ \Delta DBC $ are two isosceles triangles on the same $ BC $ vertex, $ A $ and $ D $ are on the same side of $ BC $ . If $ AD $ is extended to intersect $ BC $ at $ P $ , those two are similar, that is $ \Delta ABC\cong \Delta DBC $ .

Note:
 While answering questions of this type we should be sure of the corresponding sides of the triangles. If we had mistaken one side with another we cannot prove that they are similar like if we had mistaken the $ BC $ with $ DB $ we will not be able to answer the question.