Exercise 23 Page 394

Let's find the measures of the numbered angles one at a time.

Measure of ∠ 1, ∠ 2, ∠ 3 and ∠ 4

Before we begin, let's name the vertices of our kite. We will call it quadrilateral ABCD.

If a quadrilateral is a kite, then its diagonals are perpendicular. Therefore m ∠ 1 = 90, m ∠ 2 = 90, m ∠ 3 = 90 and m ∠ 4 = 90.

Measure of ∠ 5 and ∠ 6

Notice that △ BAD and △ BCD are congruent by the Side-Side-Side (SSS) Congruence Theorem. Since corresponding parts of congruent triangles are congruent, ∠ ABD ≅ ∠ CBD and ∠ ADB ≅ ∠ CDB. Because these angles are congruent, we know that their measures are equal.

m ∠ 5 = 46 m ∠ 6 = 34

Measure of ∠ 7

Notice that ∠ 3, ∠ 6 and ∠ 7 are three angles in a triangle. By the Triangle Angle-Sum Theorem, we know that their measures add to 180. m ∠ 3 + m ∠ 6 + m ∠ 7 = 180 We already know that m ∠ 3 = 90 and m ∠ 6 = 34. Let's substitute these values and solve the equation to find m ∠ 7.

m ∠ 3 + m ∠ 6 + m ∠ 7 = 180

SubstituteII

m ∠ 3= 90, m ∠ 6= 34

90 + 34 + m ∠ 7 = 180

AddTerms

Add terms

124 + m ∠ 7 = 180

SubEqn

LHS-124=RHS-124

m ∠ 7 = 56

Measure of ∠ 8

Notice that ∠ 4, ∠ 5 and ∠ 8 are three angles in a triangle. By the Triangle Angle-Sum Theorem, we know that their measures add to 180. m ∠ 4 + m ∠ 5 + m ∠ 8 = 180 We already know that m ∠ 4 = 90 and m ∠ 5 = 46. Let's substitute these values and solve the equation to find m ∠ 8.

m ∠ 4 + m ∠ 5 + m ∠ 8 = 180

SubstituteII

m ∠ 4= 90, m ∠ 5= 46

90 + 46 + m ∠ 8 = 180

AddTerms

Add terms

136 + m ∠ 8 = 180

SubEqn

LHS-136=RHS-136

m ∠ 8 = 44

Measure of ∠ 10

Notice that ∠ 2, ∠ 10, and the angle of measure 46 are three angles in a triangle. By the Triangle Angle-Sum Theorem, we know that their measures add to 180. m ∠ 2 + 46 + m ∠ 10 = 180 We already know that m ∠ 2 = 90. Let's substitute this value and solve the equation to find m ∠ 10.

m ∠ 2 + 46 + m ∠ 10 = 180

Substitute

m ∠ 2= 90

90 + 46 + m ∠ 10 = 180

AddTerms

Add terms

136 + m ∠ 10 = 180

SubEqn

LHS-136=RHS-136

m ∠ 10 = 44

Measure of ∠ 9

Notice that ∠ 1, ∠ 9, and the angle of measure 34 are three angles in a triangle. By the Triangle Angle-Sum Theorem, we know that their measures add to 180. m ∠ 1 + m ∠ 9 + 34 = 180 We already know that m ∠ 1 = 90. Let's substitute this value and solve the equation to find m ∠ 9.

m ∠ 1 + m ∠ 9 + 34 = 180

Substitute

m ∠ 1= 90

90 + m ∠ 9 + 34 = 180

AddTerms

Add terms

124 + m ∠ 9 = 180

SubEqn

LHS-124=RHS-124

m ∠ 9 = 56