Exercise 26 Page 312

Consider the diagram below. In order to find the value of x we need to find the relationships between the angles and use the corresponding theorems and postulates. We will add an additional angle measure y.

Let's find the relationship between the angle that measures 54 and the angle with the measure of y. These angles are supplementary angles, so the sum of their measure is 180 by the Supplement Theorem. y+54=180Now, let's solve it for y.

y+54=180

SubEqn

LHS-54=RHS-54

y=126

The value of y is 126. Now, we need to find the relationship between the angles with measures 126 and 2x.

The angles with measures 126 and (2x) lie on opposite sides of the transversal and they are interior angles formed by parallel lines. These are called alternate interior angles. Therefore, to find the value of x, we can use the Alternate Interior Angles Theorem.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

According to this theorem, the angles are congruent, so their measures are the same. 126= 2x We can solve this equation for x.

126=2x

DivEqn

.LHS /2.=.RHS /2.

63=x

RearrangeEqn

Rearrange equation

x=63

The value of x is 63 by the Supplement Theorem and the Alternate Interior Angles Theorem.