Question

For the following figure, given that$\overline{PR}\left|\right|\overline{VO};\overline{RO}\left|\right|\overline{PV};\overline{PR}\cong \overline{RO}$ . Prove that$\angle 1$and$\angle 2$are complementary.

Short answer

Therefore$\angle 1$and$\angle 2$ are complementary.

Step-by-step solution

Step 1- Check the figure.

Consider the figure.

Step 2- Apply the following concepts.

• Alternate angles are equal to each other.

• The addition of all angles of a triangle is 180.

• Rhombus’s diagonals are perpendicular.

• Two angles of the rhombus bisected by each diagonal of a rhombus.

• the parallelogram is a rhombus if two consecutive sides of a parallelogram are equal.

Step 3- Step description.

Consider the quadrilateral$PROV$.

Since the opposite sides of a quadrilateral are parallel then it is a parallelogram.

Here consider that$\overline{PR}\left|\right|\overline{VO};\overline{RO}\left|\right|\overline{PV}$ thus$PROV$ is a parallelogram.

Also$\overline{PR}\cong \overline{RO}$ thus the parallelogram$PROV$ is a rhombus.

Step 4- Step description

From the figure$\overline{RV}\perp \overline{PO}$, so that $\angle REO={90}^{\circ }$.

Consider the triangle$\angle REO$.

Since the sum of all the angles in a triangle is 180 degrees thus

$\begin{array}{l}m\angle REO+m\angle EOR+m\angle ORE={180}^{\circ }\\ 90+m\angle EOR+m\angle 2=180\\ m\angle EOR+m\angle 2=180-90\\ m\angle EOR+m\angle 2=90\text{(1)}\end{array}$

Also, from the figure$\overline{PR}\left|\right|\overline{VO};\overline{RO}\left|\right|\overline{PV}$and $\overline{RV}$is a transversal therefore

$\begin{array}{l}m\angle PRV\cong m\angle RVO\\ m\angle PRV=m\angle 1\text{(2)}\end{array}$

The diagonal$\overline{RV}$bisects the angle$\angle PRO$.

That is$m\angle PRV=m\angle VRO\text{(3)}$

Use (1), (2) and (3) as follows:

$m\angle 1+m\angle 2=90$

Therefore, the angles are complementary.