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Geometry and Trigonometry Difficulty: Hard

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 $w$ ?

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Explanation

The correct answer is $StartFraction 101 Over 2 EndFraction$ . In the figure, lines $q$ , $r$ , and $s$ form a triangle. One interior angle of this triangle is vertical to the angle marked $a °$ ; therefore, the interior angle also has measure $a °$ . It's given that $a equals 43$ . Therefore, the interior angle of the triangle has measure $43 degree$ . A second interior angle of the triangle forms a straight line, $q$ , with the angle marked $b °$ . Therefore, the sum of the measures of these two angles is $180 degree$ . It's given that $b equals 122$ . Therefore, the angle marked $b °$ has measure $122 degree$ and the second interior angle of the triangle has measure $left parenthesis 180 minus 122 right parenthesis degree$ , or $58 degree$ . The sum of the interior angles of a triangle is $180 degree$ . Therefore, the measure of the third interior angle of the triangle is $(180 - 43 - 58) °$ , or $79 degree$ . It's given that parallel lines $q$ and $t$ are intersected by line $r$ . It follows that the triangle's interior angle with measure $79 degree$ 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 $w °$ , the sum of $79 degree$ , $w °$ , and $w °$ is $180 degree$ . It follows that $79 + w + w = 180$ , or $79 + 2 w = 180$ . Subtracting $79$ from both sides of this equation yields $2 w equals 101$ . Dividing both sides of this equation by $2$ yields $w = \frac{101}{2}$ . Note that 101/2 and 50.5 are examples of ways to enter a correct answer.