Chapter 5: Q8 (page 169)

In Exercises 5-10 quad. PQRS is a parallelogram. Find the values of a, b, x and y.

Short Answer

Expert verified

The values of a, b, x and y are $9, 11, 33 °$ and $27 °$ respectively.

Step by step solution

Step 1. Consider a following figure.

Step 2. Apply property of parallelogram.

Diagonals of a parallelogram bisect each other.

Step 3. Description of step.

In the given figure, $\bar{P R}$ and role="math" localid="1637641507892" $\bar{Q S}$ bisect each other and O is the midpoint of $\bar{P R}$ and $\bar{Q S}$ implies that

$\begin{matrix} P O = O R \\ b = 11 \end{matrix}$

And

$\begin{matrix} Q O = O S \\ a = 9 \end{matrix}$

Step 4. Description of step.

Both pairs of opposite sides of a parallelogram are parallel, that is, $\bar{P Q} | | \bar{S R}$ and $\bar{S Q}$ is a transversal then angle made by a transversal with respect to two lines are equal.

$∠ P Q S ≅ ∠ Q S R$

Therefore

$\begin{matrix} m ∠ P Q S = m ∠ Q S R \\ x = 33 ° \end{matrix}$

Step 5. Description of step.

Both pairs of opposite sides of a parallelogram are parallel, that is, $\bar{P Q} | | \bar{S R}$ and $\bar{P S}$ is a transversal then same side interior angles are supplementary.

$\begin{matrix} m ∠ Q P S + m ∠ P S R = 180 \\ 50 + y + 70 + 33 = 180 \\ y = 180 - 153 \\ = 27 ° \end{matrix}$