Question

in the figure, $overline{bc}$ and $overline{ad}$ are line segments. what is the sum of $x$ and $y$? a. $108^{circ}$ b. $116^{circ}$ c. $118^{circ}$ d. $126^{circ}$

Explanation:

Step1: Find angle in left - triangle

In $\triangle ABO$, using angle - sum property of a triangle. Let the third angle in $\triangle ABO$ be $z$. Then $z = 180^{\circ}-(62^{\circ}+x)$. Also, $z$ and the $64^{\circ}$ angle are vertical angles, so $z = 64^{\circ}$. Then $180-(62 + x)=64$, $180-62 - x=64$, $118 - x=64$, $x = 54^{\circ}$.

Step2: Find angle in right - triangle

In $\triangle DOC$, using angle - sum property. Let the third angle in $\triangle DOC$ be $w$. Then $w = 180^{\circ}-(54^{\circ}+y)$. Also, $w$ and the $64^{\circ}$ angle are vertical angles, so $w = 64^{\circ}$. Then $180-(54 + y)=64$, $180-54 - y=64$, $126 - y=64$, $y = 62^{\circ}$.

Step3: Calculate $x + y$

$x + y=54^{\circ}+62^{\circ}=116^{\circ}$

Answer:

B. $116^{\circ}$