Exercise 3 Page 508

Recall that diagonals in a rectangle are congruent and bisect each other.

33.5

Practice makes perfect

We are asked to find the measure of ∠TQR using the given information. Let's notice that ∠PTQ and ∠QTR are supplementary angles, so the sum of their measures is 180. This means that m∠QTR=180-67= 113.

Since diagonals in a rectangle have the same lengths and bisect each other, TQ=TR. This means that the triangle TQR is an isosceles triangle and m∠TQR=m∠TRQ.

Using the Triangle Sum Theorem we can find the measure of ∠TQR. Let's recall that according to this theorem, the sum of the measures of angles in a triangle is always 180. We will use the fact we found above that m∠TQR=m∠TRQ.

m∠QTR+m∠TQR+m∠TRQ=180

SubstituteII

m∠QTR= 113, m∠TRQ= m∠TQR

113+m∠TQR+ m∠TQR=180

AddTerms

Add terms

113+2m∠TQR=180

SubEqn

LHS-113=RHS-113

2m∠TQR=67

DivEqn

.LHS /2.=.RHS /2.