Exercise 9 Page 425

Let's take a look at the given figure. Notice that all four angles in the quadrilateral are right angles, which makes it a rectangle.

Since we already know that the given figure is a rectangle, it means that its opposite sides are parallel. Notice that the diagonal of the rectangle lies on the transversal between two parallel sides. By the Alternate Interior Angles Theorem, the measure of angle x^(∘) is congruent to 36^(∘).

Now, the diagonals divide the figure into four isosceles triangles.

Let's consider the lower triangle with one of the angle measures equal to 36^(∘). Since it is an isosceles triangle, the other base angle is also 36^(∘).

The remaining angle in the considered triangle and angle y^(∘) are vertical angles and are therefore congruent.

To find y^(∘), we are going to apply the Triangle Angle Sum Theorem, which states that the measures of angles in a triangle add up to 180^(∘).

36^(∘)+36^(∘)+y^(∘)=180

AddTerms

Add terms

72^(∘)+y^(∘)=180^(∘)

SubEqn

LHS-72^(∘)=RHS-72^(∘)

y^(∘)=108^(∘)

We found the values of x^(∘) and y^(∘). To find z^(∘), notice that the angles of measure y^(∘)=108^(∘) and z^(∘) are supplementary. This means that their measures also add up to 180^(∘). 108^(∘)+z^(∘)=180^(∘) ⇔ z^(∘)=72^(∘) We found that x^(∘)=36^(∘), y^(∘)=108^(∘), and z^(∘)=72^(∘).