Counterclockwise from top right, the lines are labeled s, r, q, and t. Line s intersects line r above line

Question

Counterclockwise from top right, the lines are labeled s, r, q, and t.
Line s intersects line r above line q.
Line s and line r each intersect line q and line t.
At the intersection of line s and line r, 1 angle is labeled counterclockwise from top as follows:
- Top: a/li>
At the intersection of line s and line q, 1 angle is labeled counterclockwise from top-left as follows:
- Top left: b/li>
At the intersection of line r and line t:
- The line segment divides the bottom-left angle into two equal angles, each labeled w/li>
A note indies the figure is not drawn to scale.

In the figure, parallel lines $q$ and $t$ are intersected by lines $r$ and $s$ . If $a equals 43$ and $b equals 122$ , what is the value of $w$ ?

Accepted Answer

The correct answer is 101 2 . In the figure, lines q , r , and s form a triangle. One interior angle of this triangle is vertical to the angle marked a &#176; ; therefore, the interior angle also has measure a &#176; . It's given that a = 43 . Therefore, the interior angle of the triangle has measure 43 &#176; . A second interior angle of the triangle forms a straight line, q , with the angle marked b &#176; . Therefore, the sum of the measures of these two angles is 180 &#176; . It's given that b = 122 . Therefore, the angle marked b &#176; has measure 122 &#176; and the second interior angle of the triangle has measure 180 - 122 &#176; , or 58 &#176; . The sum of the interior angles of a triangle is 180 &#176; . Therefore, the measure of the third interior angle of the triangle is 180 - 43 - 58 &#176; , or 79 &#176; . It's given that parallel lines q and t are intersected by line r . It follows that the triangle's interior angle with measure 79 &#176; is congruent to the same side interior angle between lines q and t formed by lines t and r . Since this angle is supplementary to the two angles marked w &#176; , the sum of 79 &#176; , w &#176; , and w &#176; is 180 &#176; . It follows that 79 + w + w = 180 , or 79 + 2 w = 180 . Subtracting 79 from both sides of this equation yields 2 w = 101 . Dividing both sides of this equation by 2 yields w = 101 2 . Note that 101/2 and 50.5 are examples of ways to enter a correct answer.

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