Question

find the value of each variable in the figure on the right.
x = 82 (simplify your answer.)
y = (simplify your answer.)

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. In the given triangle, we have two known angles 29° and 69°. Let the third - interior angle be \(z\). So, \(z+29^{\circ}+69^{\circ}=180^{\circ}\).

Step2: Calculate the third - interior angle \(z\)

\[

$$\begin{align*} z&=180^{\circ}-(29^{\circ} + 69^{\circ})\\ z&=180^{\circ}-98^{\circ}\\ z& = 82^{\circ} \end{align*}$$

\]
Since \(x\) and \(z\) are vertical angles, \(x = 82^{\circ}\) (vertical angles are equal).

Step3: Use the linear - pair property

\(x\) and \(y\) form a linear pair. A linear pair of angles is supplementary, i.e., \(x + y=180^{\circ}\).

Step4: Solve for \(y\)

\[

$$\begin{align*} y&=180^{\circ}-x\\ y&=180^{\circ}-82^{\circ}\\ y& = 98^{\circ} \end{align*}$$

\]

Answer:

98