Exercise 37 Page 534

Let's analyze the given rhombus. Recall that in rhombus the diagonals are perpendicular. Therefore ∠ CEB is a right angle. This means that m∠ CEB = 90.

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. Therefore m ∠ ABD is equal to m ∠ DBC. Since m ∠ ABD=55, we know that m∠ DBC is also 55. m ∠ ABD = m ∠ DBC ⇒ m ∠ DBC = 55 Now, notice that ∠ DBC, ∠ ACB, and ∠ CEB are three angles in a triangle. By the Angle Sum Theorem, we can conclude that their measures add to 180. m ∠ DBC + m ∠ ACB + m∠ CEB = 180 We already know that m ∠ DBC = 55 and that m∠ CEB=90, so let's substitute these two values into our equation to find m ∠ ACB.

m ∠ DBC + m ∠ ACB + m∠ CEB = 180

SubstituteII

m ∠ DBC= 55, m∠ CEB= 90

55 + m ∠ ACB + 90 = 180

AddTerms

Add terms

m ∠ ACB + 145 = 180

SubEqn

LHS-145=RHS-145

m ∠ ACB = 35

The measure of ∠ ACB is 35.