Chapter 5.6, Problem 46E

Solution

To find: To state whether the statement is true of false.

The given statement is true.

Given information:

In the given statement the side of the parallelogram are $a$ and $b$ and an included angle of $\theta$ . The area of the parallelogram is given as $ab\sin \theta$ .

Formula used:

Area of a triangle for sides $a$ and $b$ and an included angle of $\theta$ is given as:

  $A_{\text{triangle }}=\frac{1}{2}ab\sin \theta$

Area of parallelogram is given as:

  $A_{\text{parallelogram }}=2\times A_{\text{triangle }}$

Calculation:

A parallelogram can be divided into two congruent triangles along any of the two diagonals.

That is why here, the sides of the triangle will be $a$, $b$ and included angle will be $\theta$ .

Write the area of the triangle for sides $a$, $b$ and included angle of $\theta$

  $A_{\text{triangle }}=\frac{1}{2}ab\sin \theta$

Substitute $\frac{1}{2}ab\sin \theta$ for $A_{\text{triangle }}$ in $A_{\text{parallelogram }}=2\times A_{\text{triangle }}$ to calculate area of the parallelogram

  $\begin{array}{c}{A}_{\text{parallelogram }}=2\times A_{\text{triangle }}\\ =2\times \frac{1}{2}ab\sin \theta \\ =ab\sin \theta \end{array}$

Thus, area of the parallelogram is $ab\sin \theta$ .

Therefore, the given statement is true.