Question

in the figure below, triangle qrt, triangle rsq, and triangle qst are scalene triangles.

if ( mangle qrs = 37^circ ) and ( mangle tqs = 34^circ ), what is ( mangle tqr )?

a. ( 90^circ )
b. ( 109^circ )
c. ( 93^circ )
d. ( 87^circ )

Explanation:

Step1: Find \( m\angle RQS \)

In right triangle \( RSQ \), \( \angle QSR = 90^\circ \) (given by the right angle symbol). The sum of angles in a triangle is \( 180^\circ \). So, \( m\angle RQS + m\angle QRS + m\angle QSR = 180^\circ \). We know \( m\angle QRS = 37^\circ \) and \( m\angle QSR = 90^\circ \). Substituting these values:
\( m\angle RQS + 37^\circ + 90^\circ = 180^\circ \)
\( m\angle RQS = 180^\circ - 37^\circ - 90^\circ = 53^\circ \).

Step2: Find \( m\angle TQR \)

We know \( m\angle TQS = 34^\circ \) and \( m\angle RQS = 53^\circ \). \( \angle TQR \) is the sum of \( \angle TQS \) and \( \angle RQS \). So:
\( m\angle TQR = m\angle TQS + m\angle RQS = 34^\circ + 53^\circ = 87^\circ \)? Wait, no—wait, wait, let's check again. Wait, no, \( \angle QSR \) is right, so \( \triangle RSQ \) has angles \( 90^\circ \), \( 37^\circ \), so \( \angle RQS = 180 - 90 - 37 = 53^\circ \). Then \( \angle TQR = \angle TQS + \angle RQS = 34 + 53 = 87^\circ \)? But wait, let's check the answer options. Option D is 87°, but wait, maybe I made a mistake. Wait, no—wait, \( \angle TQS = 34^\circ \), \( \angle RQS = 53^\circ \), so adding them gives 87°, which is option D. Wait, but let's recheck Step1. In \( \triangle RSQ \), right-angled at S, so angles: \( \angle QSR = 90^\circ \), \( \angle QRS = 37^\circ \), so \( \angle RQS = 180 - 90 - 37 = 53^\circ \). Then \( \angle TQR = \angle TQS + \angle RQS = 34 + 53 = 87^\circ \). Yes, that's correct.

Answer:

D. \( 87^\circ \)