In the figure at the right, two lines intersect to form four angles. If m∠7 = 9x and m∠8 = 11x, complete the paragraph proof to show that x = 9.

Given: Two intersecting lines with m∠7 = 9x and m∠8 = 11x

Prove: x = 9

Proof: ∠7 and ∠8 form a _____ angle so they are _____ angles. So, m∠7 + m∠8 = _____, by the definition of supplementary angles. By substitution, _____ + 11x = 180. So, x = _____ by the Division Property of Equality.

Construct an Argument Four towns lie on a straight road. Boyd is midway between Acton and Carson. Carson is midway between Boyd and Delta. Write a paragraph proof to show the distance from Acton to Boyd is the same as the distance from Carson to Delta.

Given: B is the midpoint of \(\overline{\text{AC}}\) and C is the midpoint of \(\overline{\text{BD}}\)

Prove: AB = CD

Proof: By the definition of midpoint, _____ = BC and _____ = CD. Therefore, AB = CD by _____.

Construct an Argument Complete the two-column proof to show that if ∠1 and ∠2 are supplementary and m∠1 = m∠2, then ∠1 and ∠2 are right angles.

Given: ∠1 and ∠2 are supplementary; m∠1 = m∠2

Prove: ∠1 and ∠2 are right angles

Statements                                   Reasons

a. ∠1 and ∠2 are

supplementary;

m∠1 = m∠2

b. m∠1 + m∠2 = 180°

c. m∠1 + m∠1 = 180°

d. 2(m∠1) = 180°

e. m∠1 = 90°

f. m∠2 = 90°                                   m∠1 = m∠2 (Given)

g. ∠1 and ∠2 are right angles