In right triangle $PQR$, if angle $Q$ is the right angle, side $PQ$ = $7$ and side $QR$ = $8$, what is the measure of angle $R$? Round to the nearest degree.

• A. $26°$
• B. $45°$
• C. $64°$
• D. $42°$

Answer: (D)

Verified step by step guidance

1. Identify the given elements in the right triangle PQR: angle Q is the right angle (90°), side PQ = 7, and side QR = 8. Since angle Q is right, sides PQ and QR are the legs of the triangle.
2. Recall that angle R is one of the acute angles, and we want to find its measure. To do this, we can use a trigonometric ratio involving the sides adjacent and opposite to angle R.
3. Determine which sides are opposite and adjacent to angle R. Side PQ is opposite angle R, and side QR is adjacent to angle R.
4. Use the tangent function, which relates the opposite side to the adjacent side for an angle in a right triangle: \(\tan(\angle R) = \frac{\text{opposite}}{\text{adjacent}} = \frac{PQ}{QR} = \frac{7}{8}\).
5. To find angle R, take the inverse tangent (arctangent) of the ratio: \(\angle R = \tan^{-1}\left(\frac{7}{8}\right)\). After calculating this value, round the result to the nearest degree.