Exercise 53 Page 444

We can find m∠ 4 in two steps, starting by finding m∠ 1 and then m∠ 4.

Finding m∠ 1

Let's focus on quadrilateral ABDC first. The markers on the diagram indicate that this quadrilateral has a right angle at A, at B, and at C.

This means that ABDC is a rectangle, so it also has a right angle at B. Let's indicate this with a marker and focus on triangle △ ABD.

It is given that in this triangle m∠ B= 90 and m∠ D= 64. We can use the Triangle Angle Sum Theorem to find m∠ A= m∠ 1.

m∠ A+ m∠ B+ m∠ D=180

SubstituteValues

Substitute values

m∠ A+ 90+ 64=180

▼

Solve for m∠ A

AddTerms

Add terms

m∠ A+154=180

SubEqn

LHS-154=RHS-154

m∠ A=26

The measure of ∠ 1 is 26.

Finding m∠ 4

Let's turn our attention now to the angle at A. Put the measure of ∠ 1 we just found on the diagram.

We can see that ∠ 1 and ∠ 4 together form a right angle, so their measure add up to 90^(∘). Since we know the measure of ∠ 1, this observation allows us to find the measure of ∠ 4.

m∠ 1+ m∠ 4=90

Substitute

m∠ 1= 26

26+ m∠ 4=90

SubEqn

LHS-26=RHS-26

m∠ 4=64

The measure of ∠ 4 is 64^(∘).