Question

solving for angle measures of right triangles
determining an unknown angle measure of a right triangle
what is the measure of ∠r in △pqr? round to the nearest degree.
42°
64°
45°

Explanation:

Step1: Identify sides relative to ∠R

In right triangle \( \triangle PQR \) with right angle at \( P \), \( PR = 12.7 \) (adjacent to \( \angle R \)) and \( QR = 14.1 \) (hypotenuse). We use the cosine function: \( \cos(\angle R)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{PR}{QR} \).

Step2: Calculate cosine of ∠R

Substitute values: \( \cos(\angle R)=\frac{12.7}{14.1}\approx0.9007 \).

Step3: Find ∠R using inverse cosine

\( \angle R=\cos^{-1}(0.9007) \). Using a calculator, \( \cos^{-1}(0.9007)\approx25.8^\circ \)? Wait, no, wait—wait, maybe I mixed up adjacent and opposite. Wait, \( PR = 12.7 \), \( PQ \)? Wait, no, the right angle is at \( P \), so \( PR \) and \( PQ \) are legs, \( QR \) is hypotenuse. Wait, maybe \( PR = 12.7 \) (opposite to \( \angle Q \)) and adjacent to \( \angle R \)? Wait, no, let's re-express. Let's define: in \( \triangle PQR \), right-angled at \( P \), so \( \angle P = 90^\circ \), sides: \( PR = 12.7 \), \( PQ \) (unknown), \( QR = 14.1 \) (hypotenuse). To find \( \angle R \), the adjacent side to \( \angle R \) is \( PR = 12.7 \), hypotenuse \( QR = 14.1 \). So \( \cos(\angle R)=\frac{PR}{QR}=\frac{12.7}{14.1}\approx0.9007 \). Then \( \angle R=\cos^{-1}(0.9007)\approx25.8^\circ \)? But the options are 42, 64, 45. Wait, maybe I mixed up opposite and adjacent. Wait, maybe \( PR \) is opposite to \( \angle Q \), and \( PQ \) is adjacent? Wait, no, let's check the triangle. Point \( P \) is right angle, so \( PR \) and \( PQ \) are legs, \( QR \) is hypotenuse. So \( \angle R \) is at \( R \), so the sides: opposite to \( \angle R \) is \( PQ \), adjacent is \( PR \), hypotenuse \( QR \). Wait, maybe the length 12.7 is \( PQ \) (opposite to \( \angle R \)) and 14.1 is \( QR \) (hypotenuse). Then \( \sin(\angle R)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{PQ}{QR}=\frac{12.7}{14.1}\approx0.9007 \). Then \( \angle R=\sin^{-1}(0.9007)\approx64^\circ \). Ah, that makes sense. So I had adjacent and opposite mixed. So correct: \( \sin(\angle R)=\frac{12.7}{14.1}\approx0.9007 \), so \( \angle R=\sin^{-1}(0.9007)\approx64^\circ \).

Answer:

\( 64^\circ \) (corresponding to the option with \( 64^\circ \))