Slide Formalizing Geometric Proofs Copyright © 2014 Pearson Education, Inc.

Slide 2 Review Algebra Properties of Equality Let a, b, and c be any real numbers. Addition Property If a = b, then a + c = b + c. Subtraction Property If a = b, then a – c = b – c. Multiplication Property If a = b, then a  c = b  c. Division Property If a = b and c ≠ 0, then Reflexive Property a = a Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c. Substitution Property If a = b, then b can replace a in any expression. Copyright © 2014 Pearson Education, Inc.

Slide 3 The Distributive Property Use multiplication to distribute a to each term of the sum or difference within the parentheses. Copyright © 2014 Pearson Education, Inc.

Slide 4 Copyright © 2014 Pearson Education, Inc.

Slide 5 Example Fill in each blank with the reason to justify the statement StatementsReasons a. CG = MN MN = CG Given ? b. m ∠ P = m ∠ Q m ∠ Q = m ∠ B m ∠ P = m ∠ B Given ? c. m ∠ 1 + m ∠ 2 + m ∠ 3 = 50° m ∠ 1 + m ∠ 2 = 43° 43° + m ∠ 3 = 50° Given ? Stating Properties Copyright © 2014 Pearson Education, Inc. Symmetric Property of Equality Transitive Property of Equality Substitution Property

Slide 6 Writing a Two-Column Proof A proof is an argument that uses logic to establish the truth of a statement. There are many formats for proofs, but for now, we will use a two-column proof. A two-column proof lists numbered statements on the left and corresponding numbered reasons or justifications on the right. These statements show the logical order of the proof. Copyright © 2014 Pearson Education, Inc.

Slide 7 Example Write a two-column proof. Given: m ∠ 1 = m ∠ 3 Prove: m ∠ BAG = m ∠ EAC Solution Make sure the figure is marked with the given info. Study what you want to prove and form a plan. For example, since m ∠ 1 = m ∠ 3, we can add m ∠ 2 to both m ∠ 1 and m ∠ 3. The resulting angles are the angles we are interested in and will have equal measure. Writing a Two-Column Proof Copyright © 2014 Pearson Education, Inc.

Slide 8 Example StatementsReasons 1. m ∠ 1 = m ∠ 3 1. Given Writing a Two-Column Proof Copyright © 2014 Pearson Education, Inc. 2. Reflexive Property of Equality 3. Addition Property of Equality 4. Angle Addition Postulate 5. Substitution Property

Slide 9 Corollary 1.7 Equal Complements Theorem Complements of the same angle (or of equal angles) are equal in measure. If m ∠ 1 + m ∠ 2 = 90° and m ∠ 3 + m ∠ 2 = 90°, then m ∠ 1 = m ∠ 3. Copyright © 2014 Pearson Education, Inc.

Slide 10 Example Given: ∠ A and ∠ B are complementary. ∠ C and ∠ D are complementary. m ∠ A = m ∠ C Prove: m ∠ B = m ∠ D Proving the Equal Complements Cor. 1.7 Copyright © 2014 Pearson Education, Inc.

Slide 11 Example StatementsReasons 1. ∠ A and ∠ B are complements. ∠ C and ∠ D are complements. 1. Given Proving the Equal Complements Cor. 1.7 Copyright © 2014 Pearson Education, Inc. 2. Given 3. Definition of complementary angles. 4. Transitive Property 5. Substitution Property 6. Subtraction Property of Equality

Slide 12 Corollary 1.9 Equal Supplements Theorem Complements of the same angle (or of equal angles) are equal in measure. If m ∠ 1 + m ∠ 2 = 180° and m ∠ 3 + m ∠ 2 = 180°, then m ∠ 1 = m ∠ 3. Copyright © 2014 Pearson Education, Inc.

Slide 13 Theorem 1.10 If two lines intersect forming adjacent angles 1 and 2, then the angles are supplementary. Copyright © 2014 Pearson Education, Inc.

Slide 14 Example Given: ∠ 1 and ∠ 2 form a linear pair. Prove: ∠ 1 and ∠ 2 are supplementary angles. Proving Theorem 1.10 Copyright © 2014 Pearson Education, Inc.

Slide 15 Example Solution Proving Theorem 1.10 Copyright © 2014 Pearson Education, Inc.

Slide 16 Theorem 1.11Vertical Angles Theorem Definitions Two non adjacent angles formed by intersecting lines are called vertical angles. Two angles that are equal in measure are congruent. Thm 1.11 Vertical angles are congruent. Copyright © 2014 Pearson Education, Inc.

Slide 17 Example Given: ∠ 1 and ∠ 3 are vertical angles. Prove: ∠ 1 ≅ ∠ 3 Proving the Vertical Angles Theorem Copyright © 2014 Pearson Education, Inc.

Slide 18 Example StatementsReasons 1. ∠ 1 and ∠ 3 are vertical angles 1. Given Proving the Vertical Angles Theorem Copyright © 2014 Pearson Education, Inc. 2. Definition of linear pair angles 3. Linear Pair Theorem. 4. Equal Supplements Theorem 2. ∠ 1 and ∠ 2 form a linear pair. ∠ 3 and ∠ 2 form a linear pair. 3. ∠ 1 and ∠ 2 are supplementary ∠ 3 and ∠ 2 are supplementary

Slide 19 Right Angles Congruent Theorem Thm 1.1 All right angles are congruent. Copyright © 2014 Pearson Education, Inc.

Slide 20 Example Given: ∠ 1 and ∠ 2 are right angles. Prove: ∠ 1 ≅ ∠ 2 Solution Proof: ∠ 1 and ∠ 2 being right angles is given. By the definition of right angles, m ∠ 1 = 90° and m ∠ 2 = 90°. By the Transitive Property, since both angles equal 90°, it is true that m ∠ 1 = m ∠ 2. Then ∠ 1 ≅ ∠ 2 by the definition of congruent angles. Proving the Right Angles Congruent Theorem Copyright © 2014 Pearson Education, Inc.

Slide 21 Theorem 1.5 Equal Supplementary Angles Theorem Two equal supplementary angles are right angles. If m ∠ 1 = m ∠ 2 and m ∠ 1 + m ∠ 2 = 180°, then ∠ 1 and ∠ 2 are right angles. Copyright © 2014 Pearson Education, Inc.