Question

Given:$\overline{\mathrm{OP}}\cong \overline{\mathrm{OQ}}\mathbf{;}\angle \mathbf{3}\cong \angle \mathbf{4}$

Prove:$\angle \mathbf{5}\cong \angle \mathbf{6}$

Short answer

It is proved that $\angle 5\cong \angle 6$.

Step-by-step solution

Step 1. Vertically opposite angles.

From the figure it can be observed that,$\angle POS$ and$\angle QOR$ are vertically opposite angles. Therefore, $\angle POS\cong \angle QOR$.

Step 2. ASA postulate.

If two angles and included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Here, $\angle 3\cong \angle 4$,$\overline{OP}\cong \overline{OQ}$ and $\angle POS\cong \angle QOR$, then by ASA postulate width="112" height="20" role="math">ΔPOS≅ΔQOR.

Step 3. Description of step.

As$\Delta POS\cong \Delta QOR$, implies that sides of congruent angles are congruent, that is, $\overline{OS}\cong \overline{OR}$.

Step 4. Isosceles triangle theorem.

If two sides of triangle are congruent then the angles opposite to those sides are congruent.

Consider$\mathrm{\Delta }OSR,$as $\overline{OS}\cong \overline{OR}$, therefore, by isosceles triangle theorem, $\angle 5\cong \angle 6$.

Hence it is proved that, $\angle 5\cong \angle 6$.