Exercise 24 Page 508

Using the information in the diagram, we can relate a few of the angles. We already know that ∠ 4 ≅ ∠ 5. Additionally, we know that SE bisects ∠ RSF which means it cuts the angle in two congruent halves. If we let x represent the measures of ∠ 2 and ∠ 3, and we let y represent the measure of ∠ 4 and ∠ 5, we can draw the following figure.

Treating TR as a transversal to SF and RE, we can use the Corresponding Angles Theorem to find the measure of ∠ 1.

We also have that ∠ TSR is a straight angle. This means that the sum of the measures of the smaller angles along ∠ TSR add up to 180^(∘). Additionally, the measures of the angles in △ SER also add up to 180^(∘). Using this, we can write the following equations. &x+x+y=180 ⇔ 2x+y=180 &y+y+x=180 ⇔ 2y+x=180 Combining these equations, we get a system of equations which we can solve using the Elimination Method.

When we know the measure of x, we can substitute this into the first equation to find the measure of y.

Since we defined m∠ 1 as y, we can conclude that m∠ 1=60^(∘).