Chapter 10.6, Problem 42P

A parallelogram has a height of 5 cm and side measures of 8 cm and 12 cm. What are the measures of the angles?

To determine

To calculate: The measures of the angle of a parallelogram

Answer to Problem 42P

The degrees of all angles of parallelogram are $\left(\angle A\text{ and }\angle C={141.3}^{\circ }\right)$ and $\left(\angle B\text{ and }\angle D={38.7}^{\circ }\right)$

Explanation of Solution

Given information:

The height of the parallelogram is 5 cm and side measures of 8 cm and 12 cm.

Consider the diagram given in the question

Formula used:

  $\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}}$

Calculation:

Assume the angles of a parallelogram as $\angle A,\angle B,\angle C\text{ and}\angle D$ and height is $AP$

For a parallelogram, assume hypotenuse as $AD$ and perpendicular as $AP$

Now,

  $\begin{array}{l}\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}}\\ \sin \theta =\frac{AP}{AD}\\ \sin \theta =\frac{5}{8}\\ \sin \theta =0.625\\ \theta ={\sin }^{-1}\left(0.625\right)\\ \theta ={38.7}^{\circ }\end{array}$

Hence, $\angle D=\theta ={38.7}^{\circ }$

In the given parallelogram $\angle D$ is opposite to $\angle B$ and $\angle A$ is opposite to $\angle C$

So, $\angle D=\angle B\text{ and }\angle A=\angle C$ because opposite angles of a parallelogram are equal

Now,

  $\begin{array}{l}\angle A+\angle B+\angle C+\angle D={360}^{\circ }\left[{\text{Sum of all angles of a parallelogram is 360}}^{\circ }\right]\\ 2\left(\angle C+\angle D\right)={360}^{\circ }\\ \angle C+\angle D={180}^{\circ }\\ \angle C={180}^{\circ }-\angle D\\ \angle C={180}^{\circ }-{38.7}^{\circ }\\ \angle C={141.3}^{\circ }\end{array}$

Since the opposite angles of a parallelogram is equal. Therefore, $\angle A\text{ and }\angle C={141.3}^{\circ }$ and $\angle B\text{ and }\angle D={38.7}^{\circ }$