Exercise 55 Page 146

Angles ∠ YDF and ∠ YDA are supplementary angles.

m∠ RDI = 29.5^(∘)

Practice makes perfect

We are given that DR bisects ∠ FDI. Let's note this information on the diagram.

From the previous exercise, we know that m ∠ YDA=59^(∘). Notice that angles ∠ YDA and ∠ FDI are a pair of opposite angles formed by two intersecting lines, AF and YI. This means they are vertical angles. The measures of vertical angles are congruent.

Therefore, m∠ FDI is also 59^(∘). m∠ FDR + m∠ RDI= m∠ FDI ⇓ m∠ FDR + m∠ RDI= 59^(∘) Because DR bisects ∠ FDI, we know that the measures of ∠ FDR and ∠ RDI are the also same. Let's use this fact to solve for m∠ RDI.

m∠ FDR + m∠ RDI= 59^(∘)

Substitute

m∠ FDR= m∠ RDI

m∠ RDI + m∠ RDI= 59^(∘)

AddTerms

Add terms

2 m∠ RDI= 59^(∘)

DivEqn

.LHS /2.=.RHS /2.

m∠ RDI= 29.5^(∘)

We found that the measure of ∠ RDI is 29.5^(∘).