Question

Given:$\angle \text{l}\cong \angle 2;\angle 3\cong \angle 4$

Prove:$\Delta ZXY$is isosceles.

Short answer

By proving the congruency of$\Delta WZX\cong \Delta WZY$, it is proven that$\Delta ZXY$is an Isosceles Triangle.

Step-by-step solution

- Define Congruent Triangles 

The two triangles are said to be congruent if they are copies of each other and if their vertices are superposed, then say that corresponding angles and the sides of the triangles are congruent.

Congruency of triangles can be proved by SSS, SAS and ASA, AAS and HL postulates.

- State the theorem used 

The Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

Converse of Isosceles Theorem: If the two angles of a triangle are congruent, then the sides opposite to those angles are congruent.

- State the proof 

To prove that two segments or two angles are congruent, firstly identity the two triangles in which that two segments or angles are corresponding parts. After that, prove that the identified triangles are congruent and then, state that the two parts are congruent using the reason “Corresponding parts of$\cong$triangles are$\cong$”.

Given:$\angle 1\cong \angle 2;\angle 3\cong \angle 4$

Prove:$\Delta ZXY$is isosceles

Proof:

1. $\angle 1\cong \angle 2$

Reason: Given

1. $\overline{WX}\cong \overline{WZ}$

Reason: Converse of isosceles theorem

1. $\angle 3\cong \angle 4$

Reason: Given

1. $\overline{WX}\cong \overline{WZ}$

Reason: Reflexive property

1. $\Delta WZX\cong \Delta WZY$

Reason: SAS postulate

1. $\Delta ZXY$is isosceles

Reason: Definition of isosceles triangle.

Therefore, it is proven that $\Delta ZXY$is an Isosceles Triangle.