We are asked to find the measure of ∠ TQR. Let's show it on the given diagram.
We can see that QS⊥ RS and QT⊥ RT. We also know that both QS and QT equal 8. Therefore, Q is equidistant from the sides of ∠ TRS. Now, let's recall the Converse of the Angle Bisector Theorem.
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.
By this theorem, RQ is the bisector of ∠ TRS. From here, we can conclude that ∠ QRT and ∠ QRS are congruent.
With this information, we can write three congruence relations between △ SQR and △ TQR
∠ QRS & ≅ ∠ QRT
∠ RSQ & ≅ ∠ RTQ
SQ & ≅ TQ
By the Angle-Angle-Side Congruence Theorem, △ SQR and △ TQR are congruent.
△ SQR ≅ △ TQR
Since corresponding parts of congruent triangles are congruent, ∠ SQR and ∠ TQR are congruent. Therefore, they have the same measure.
m∠ TQR = 43 ^(∘)
