Question

if ∠tqr is a right angle, m∠rqs = 10°, and m∠tqs = (8x)°, find each angle measure below.
diagram: t (top), q (bottom right), r (left of q, horizontal), s (left of r, horizontal); ∠rqs = 10°, ∠tqs = (8x)°, right angle at q
x=
( mangle tqr = )
( mangle tqs = )

Explanation:

Step1: Analyze angle relationship

Since \( \angle TQR \) is a right angle (\( 90^\circ \)) and \( \angle TQS=\angle TQR - \angle RQS \), we have \( (8x)^\circ=90^\circ - 10^\circ \).

Step2: Solve for x

Simplify the right - hand side: \( 90 - 10 = 80 \), so the equation becomes \( 8x=80 \). Divide both sides by 8: \( x=\frac{80}{8}=10 \).

Step3: Find \( m\angle TQR \)

Given that \( \angle TQR \) is a right angle, so \( m\angle TQR = 90^\circ \).

Step4: Find \( m\angle TQS \)

Substitute \( x = 10 \) into \( m\angle TQS=(8x)^\circ \), we get \( m\angle TQS=8\times10 = 80^\circ \).

Answer:

\( x = 10 \)
\( m\angle TQR=90^\circ \)
\( m\angle TQS = 80^\circ \)