8th grade Ch5 L2

Use the figure to complete the paragraph proof. (Refer to the diagram with points C, G, F, E, D) Given: m∠1 = m∠2, ∠1 and ∠2 are supplementary. Prove: ∠1 and ∠2 are right angles. Proof: m∠1 + m∠2 = ______ since they are supplementary angles. Since m∠1 = m∠2, then m∠1 + m∠1 = 180° by ______. Solving the equation gives m∠1 = ______. Since m∠1 = m∠2, then m∠2 is also ______. Therefore, ∠1 and ∠2 are right angles.

• 180°; substitution; 90°; 90°

• 90°; addition; 45°; 45°

• 180°; addition; 60°; 60°

• 90°; substitution; 180°; 180°

Refer to the figure above. Complete the two-column proof to show that if EG = 3x – 1, ED = 2x + 4, and EG = ED, then x = 5.

• Given

• Substitution

• Subtract 2x from both sides

• Add 1 to both sides

Deductive reasoning is used in algebra and geometry proofs to:

• arrive at conclusions based on accepted facts and logical steps

• make guesses without evidence

• use only experimental data

• ignore established theorems

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.

• linear pair; supplementary; 180; 9x; 9

• vertical; complementary; 90; 11x; 11

• adjacent; congruent; 360; 9x; 20

• opposite; supplementary; 180; 11x; 10

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 AC and C is the midpoint of BD. Prove: AB = CD. Proof: By the definition of midpoint, __________ = BC and __________ = CD. Therefore, AB = CD by __________.

• AB; CD; substitution

• BC; AB; addition

• AC; BD; subtraction

• AB; BC; multiplication

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. Prove: ∠1 and ∠2 are right angles. a. ∠1 and ∠2 are supplementary; m∠1 = m∠2

• Given

• Definition of Supplementary Angles

• Vertical Angles Theorem

• Addition Property of Equality

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. b. m∠1 + m∠2 = 180°

• Definition of supplementary angles

• Vertical angles theorem

• Transitive property of equality

• Definition of congruent angles

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. c. m∠1 + m∠1 = 180°

• Substitution

• Addition Property of Equality

• Definition of Supplementary Angles

• Transitive Property

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. d. 2(m∠1) = 180°

• Combine like terms

• Apply the definition of supplementary angles

• Subtract m∠1 from both sides

• Use the transitive property

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.

• Division Property of Equality

• Addition Property of Equality

• Transitive Property of Equality

• Substitution Property