Question

$ \text{AB}=\text{AC} $ and $ \angle \text{B}=\angle \text{C} $ . Prove that $ \Delta \text{ABE}\cong \Delta \text{ACD} $ .

Answer 1

Hint: We have an isosceles triangle $ \Delta \text{ABC} $ . We can use the given information to see which angles and sides in the figure are congruent to each other. In triangles $ \Delta \text{ACD} $ and $ \Delta \text{ABE} $ , we will check if there are any congruent angles and sides. We will use one of the tests for congruency of triangles to prove that $ \Delta \text{ABE}\cong \Delta \text{ACD} $ .

Complete step by step answer:
We have an isosceles triangle $ \Delta \text{ABC} $ . We are given that $ \text{AB}=\text{AC} $ and $ \angle \text{B}=\angle \text{C} $ . Segment CD is perpendicular to side AB. Similarly, segment BE is perpendicular to side AC. Therefore, we have $ \angle \text{ADC}=\angle \text{AEB}=90{}^\circ $ . Next, we see that segment AB is a side of $ \Delta \text{ABE} $ . Similarly, segment AC is a side of $ \Delta \text{ACD} $ . We know that $ \text{AB}=\text{AC} $ . Therefore, side AB and side AC are congruent sides in triangles $ \Delta \text{ABE} $ and $ \Delta \text{ACD} $ . From the figure, we can see that $ \angle \text{A} $ is a common angle in triangles $ \Delta \text{ABE} $ and $ \Delta \text{ACD} $ . Hence, we have the angles $ \angle \text{BAE}=\angle \text{CAD} $ .
This implies that we have two angles and a side that are equal in the two triangles $ \Delta \text{ACD} $ and $ \Delta \text{ABE} $ . Therefore, by the A-S-A test, we can conclude that triangles $ \Delta \text{ACD} $ and $ \Delta \text{ABE} $ are congruent. Hence, proved.

Note:
There are different tests to prove the congruence of two triangles. These tests involve showing the equality of angles and sides in the pair of triangles. Some of the other tests are the A-A-A test, S-S-S test, S-A-S test, S-S-A test. It is important to understand the geometry of triangles for such type of questions. Using the properties of the triangles, we can prove their congruence.