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Geometry and Trigonometry / Lines, angles, and triangles Difficulty: Easy

Right triangles $P Q R$ and $S T U$ are similar, where $P$ corresponds to $S$ . If the measure of angle $Q$ is $18 degree$ , what is the measure of angle $S$ ?

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Explanation

Choice B is correct. In similar triangles, corresponding angles are congruent. It’s given that right triangles $P Q R$ and $S T U$ are similar, where angle $P$ corresponds to angle $S$ . It follows that angle $P$ is congruent to angle $S$ . In the triangles shown, angle $R$ and angle $U$ are both marked as right angles, so angle $R$ and angle $U$ are corresponding angles. It follows that angle $Q$ and angle $T$ are corresponding angles, and thus, angle $Q$ is congruent to angle $T$ . It’s given that the measure of angle $Q$ is $18 degree$ , so the measure of angle $T$ is also $18 degree$ . Angle $U$ is a right angle, so the measure of angle $U$ is $90 degree$ . The sum of the measures of the interior angles of a triangle is $180 degree$ . Thus, the sum of the measures of the interior angles of triangle $S T U$ is $180$ degrees. Let $s$ represent the measure, in degrees, of angle $S$ . It follows that $s + 18 + 90 = 180$ , or $s + 108 = 180$ . Subtracting $108$ from both sides of this equation yields $s equals 72$ . Therefore, if the measure of angle $Q$ is $18$ degrees, then the measure of angle $S$ is $72$ degrees.

Choice A is incorrect. This is the measure of angle $T$ .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the sum of the measures of angle $S$ and angle $U$ .