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Geometry, Student Text and Homework Helper Page 1 of 2 Skip Directly to Table of Contents | Skip Directly to Main Content Change text size Show/Hide TOC Page Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles Congruent 2-6 Proving Angles Congruent Teks Focus TEKS (6)(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. Additional TEKS (1)(E), (1)(G), (4)(A) Vocabulary • Paragraph proof – A paragraph proof gives statements and reasons or justifications of a proof, written as sentences in a paragraph. • Theorem – a conjecture or statement that you prove true • Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated. • Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING You can use given information, definitions, properties, postulates, and previously proven theorems as reasons in a proof. take note Key Concept Paragraph Proof The proof in Problem 3 is two-column, but there are many ways to display a proof. A paragraph proof is written as sentences in a paragraph. Below is a paragraph proof from Problem 3. Each statement in the Problem 3 proof is red in the paragraph proof. Proof Given: ∠1 ≅ ∠4 Prove: ∠2 ≅ ∠3 Proof: ∠1 ≅ ∠4 is given. ∠4 ≅ ∠2 because vertical angles are congruent. By the Transitive Property of Congruence, ∠1 ≅ ∠2. ∠1 ≅ ∠3 because vertical angles are congruent. By the Transitive Property of Congruence, ∠2 ≅ ∠3. take note Theorem 2-1 Vertical Angles Theorem Vertical angles are congruent. ∠1 ≅ ∠3 and ∠2 ≅ ∠4 d For a proof of Theorem 2-1, see Problem 1. PearsonTEXAS.com Page 71 https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Copyright 2016 © Pearson Education, Inc. or its affiliate(s). All rights reserved. Page 2 of 2 Privacy Policy | Terms of Use | Rights and Permissions https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Page 1 of 1 Skip Directly to Table of Contents | Skip Directly to Main Content Change text size Show/Hide TOC Page Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles Congruent take note Theorem 2-2 Congruent Supplements Theorem Theorem If … ∠1 and ∠3 are supplements and ∠2 and ∠3 are supplements If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. Then … ∠1 ≅ ∠2 For a proof of Theorem 2-2, see Problem 5. Theorem 2-3 Congruent Complements Theorem Theorem If … ∠1 and ∠2 are complements and ∠3 and ∠2 are complements Then … ∠1 ≅ ∠3 If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. You will prove Theorem 2-3 in Exercise 4. Theorem 2-4 Theorem If … ∠1 and ∠2 are right angles Then … ∠1 ≅ ∠2 All right angles are congruent. You will prove Theorem 2-4 in Exercise 23. Theorem 2-5 Theorem If … ∠1 ≅ ∠2, and ∠1 and ∠2 are supplements If two angles are congruent and supplementary, then each is a right angle. Then … m∠1 = m∠2 = 90 You will prove Theorem 2-5 in Exercise 11. Page 72 Copyright 2016 © Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy | Terms of Use | Rights and Permissions https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Page 1 of 2 Skip Directly to Table of Contents | Skip Directly to Main Content Change text size Show/Hide TOC Page Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles Congruent Problem 1 TEKS Process Standards (1)(D) Proof Proving the Vertical Angles Theorem Given: ∠1 and ∠3 are vertical angles. Prove: ∠1 ≅ ∠3 Statements 1) ∠1 and ∠3 are vertical angles. Reasons 1) Given 2) ∠ s that form a linear pair are supplementary. 2) ∠1 and ∠2 are supplementary. ∠2 and ∠3 are supplementary. ∠1 and ∠2 form a linear pair. ∠2 and ∠3 form a linear pair. 3) m∠1 + m∠2 = 180 3) The sum of the measures of supplementary ∠ s is 180. m∠2 + m∠3 = 180 4) m∠1 + m∠2 = m∠2 + m∠3 4) Transitive Property of Equality 5) m∠1 = m∠3 5) Subtraction Property of Equality 6) ∠1 ≅ ∠3 s with the same measure are ≅. 6) ∠ Proof Verifying the Vertical Angles Theorem Using Line Segments Plan How can you use the relationship between ∠1, ∠2, and ∠3 in your proof? ∠1 and ∠3 both form linear pairs with ∠2. Since angles that form a linear pair are supplementary, you can show that ∠1 and ∠2 are supplements, and ∠2 and ∠3 are supplements. The Vertical Angles Theorem works for vertical angles formed by line segments as well as lines. Suppose line ¯¯¯¯ ¯¯¯¯ segments AB and CD intersect at point E. Write a paragraph proof to show that ∠AEC ≅ ∠BED. Both ∠AEC and ∠BED are supplementary to ∠CEB, because ∠AEC and ∠CEB form a linear pair, and ∠CEB and ∠BED form a linear pair. By the definition of supplementary angles, m∠AEC + m∠CEB = 180 and m∠CEB + m∠BED =180. Then m∠AEC + m∠CEB = m∠CEB + m∠BED by the Transitive Property of Equality. Subtract m∠CEB from each side. By the Subtraction Property of Equality, m∠AEC = m∠BED. Angles with the same measure are congruent, so ∠AEC = ∠BED. Problem 2 Applying the Vertical Angles Theorem What is the value of x? Plan How do you get started? Think The two labeled angles are vertical angles, so set them equal. Write 2x + 21 = 4x 21 = 2x Solve for x by subtracting 2x from each side and then dividing by 2. 21 =x 2 Look for a relationship in the diagram that allows you to write an equation with the variable. PearsonTEXAS.com Page 73 https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Copyright 2016 © Pearson Education, Inc. or its affiliate(s). All rights reserved. Page 2 of 2 Privacy Policy | Terms of Use | Rights and Permissions https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Page 1 of 1 Skip Directly to Table of Contents | Skip Directly to Main Content Change text size Show/Hide TOC Page Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles Congruent Problem 3 Proof Writing a Proof Using the Vertical Angles Theorem Given: ∠1 ≅ ∠4 Prove: ∠2 ≅ ∠3 Statements Reasons 1) ∠1 ≅ ∠4 1) Given 2) ∠4 ≅ ∠2 2) Vertical angles are ≅. 3) ∠1 ≅ ∠2 3) Transitive Property of Congruence 4) ∠1 ≅ ∠3 4) Vertical angles are ≅. 5) ∠2 ≅ ∠3 5) Transitive Property of Congruence Think Why does the Transitive Property work for statements 3 and 5? In each case, an angle is congruent to two other angles, so the two angles are congruent to each other. Problem 4 Distinguishing Between Mathematical Concepts For each statement, determine whether it is an undefined term, a definition, a postulate, a conjecture, or a theorem. I. A segment is a part of a line that consists of two endpoints and all points between them. II. If ∠1 and ∠3 are supplements and ∠2 and ∠3 are supplements, then ∠1 ≅ ∠2. III. A plane contains infinitely many lines. Think IV. If ∠1 and ∠2 are supplementary, then one of the angles is obtuse. Why are some terms undefined? V. If two distinct planes R and W intersect, they intersect in exactly one line. Statement I is the definition of the term segment. Statement II is a theorem, specifically the Congruent Supplements Theorem. Statement III describes a plane, but plane is an undefined term. A counterexample can show that Statement IV is incorrect, so it is a conjecture. It is an accepted fact that two planes intersect in exactly one line, so Statement V is a postulate. The terms point, line, and plane are not defined because their definitions would require terms that also need defining. Problem 5 TEKS Process Standard (1)(G) Proof Writing a Paragraph Proof Plan How can you use the given information? ∠1 and ∠3 are supplementary. ∠2 and ∠3 are supplementary. Prove: ∠1 ≅ ∠2 ∠1 and ∠3 are supplementary because it is given. So m∠1 + m∠3 = 180 by the definition of supplementary angles. ∠2 and ∠3 are supplementary because it is given, so m∠2 + m∠3 = 180 by the same definition. By the Transitive Proof: Property of Equality, m∠1 + m∠3 = m∠2 + m∠3. Subtract m∠3 from each side. By the Subtraction Property of Equality, m∠1 = m∠2. Angles with the same measure are congruent, so ∠1 ≅ ∠2. Given: Both ∠1 and ∠2 are supplementary to ∠3. Use their relationship with ∠3 to relate ∠1 and ∠2 to each other. Page 74 Copyright 2016 © Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy | Terms of Use | Rights and Permissions https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Page 1 of 2 Skip Directly to Table of Contents | Skip Directly to Main Content Change text size Show/Hide TOC Page Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles Congruent PRACTICE and APPLICATION EXERCISES Find the value of each variable and the measure of each labeled angle. Scan page for a Virtual Nerd™ tutorial video. 1. 2. For additional Proof 3. Justify Mathematical Arguments (1)(G) Complete the following proof by filling in the blanks. support when completing your homework, go to PearsonTEXAS.com . Given: ∠1 ≅ ∠3 Prove: ∠6 ≅ ∠4 Statements Reasons 1) ∠1 ≅ ∠3 1) Given 2) ∠3 ≅ ∠6 2) a. _?_ 3) b. _?_ 3) Transitive Property of Congruence 4) ∠1 ≅ ∠4 4) c. _?_ 5) ∠6 ≅ ∠4 5) d. _?_ Proof 4. Fill in the blanks to complete this proof of the Congruent Complements Theorem (Theorem 2−3). If two angles are complements of the same angle, then the two angles are congruent. ∠1 and ∠2 are complementary. ∠3 and ∠2 are complementary. Prove: ∠1 ≅ ∠3 ∠1 and ∠2 are complementary and ∠3 and ∠2 are complementary because it is given. By the definition of complementary angles, m∠1 + m∠2 = a. _?_ and m∠3 + m∠2 = b. _?_. Then m∠1 + m∠2 = m∠3 + m∠2 by the Proof: Transitive Property of Equality. Subtract m∠2 from each side. By the Subtraction Property of Equality, you get m∠1= c. _?_. Angles with the same measure are d. _?_, so∠1 ≅ ∠3. 5. Apply Mathematics (1)(A) What is the measure of the angle formed by Park St. and 116th St.? Given: 6. Apply Mathematics (1)(A) Give an example of vertical angles in your home or classroom. d Proof 7. Use Multiple Representations to Communicate Mathematical Proof Ideas (1)(D) Write a paragraph proof for the Vertical Angles Theorem (Theorem 2−1). Include a sketch of intersecting lines and label each angle. https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Page 2 of 2 PearsonTEXAS.com Page 75 Copyright 2016 © Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy | Terms of Use | Rights and Permissions https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Page 1 of 1 Skip Directly to Table of Contents | Skip Directly to Main Content Change text size Show/Hide TOC Page Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles Congruent 8. Explain Mathematical Ideas (1)(G) In the figure at the right, m∠2 = 21 and m∠5 = 138. Find m∠1. Show your work. 9. Two lines that intersect form four angles. If one of the angles has a measure of 55, what are the measures of the remaining angles? 10. Apply Mathematics (1)(A) In the game of miniature golf, the ball bounces off the wall at the same angle it hits the wall. (This is the angle formed by the path of the ball and the line perpendicular to the wall at the point of contact.) In the diagram, the ball hits the wall at a 40° angle. Using Theorem 2−3, what are the values of x and y? d Proof 11. Justify Mathematical Arguments (1)(G) Fill in the blanks to complete this proof of Theorem 2−5. If two angles are congruent and supplementary, then each is a right angle. Given: ∠W and ∠V are congruent and supplementary. Prove: ∠W and ∠V are right angles. ∠W and ∠V are congruent because a. _?_. Because congruent angles have the same measure, m∠W = b. _?_. ∠W and ∠V are supplementary because it is given. By the definition of supplementary angles, m∠W + m∠V = c. _?_. Substituting m∠W for m∠V, you get m∠W +m∠W = 180, or 2m∠W = 180. By the Proof: d. _?_ Property of Equality, m∠W = 90. Since m∠W = m∠V, m∠V = 90 by the Transitive Property of Equality. Both angles are e. _?_ angles by the definition of right angle. Page 76 Copyright 2016 © Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy | Terms of Use | Rights and Permissions https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Page 1 of 1 Skip Directly to Table of Contents | Skip Directly to Main Content Change text size Show/Hide TOC Page Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles Congruent 12. Apply Mathematics (1)(A) In the photograph below, the legs of the table are constructed so that ∠1 ≅ ∠2. What theorem can you use to justify the statement that ∠3 ≅ ∠4? d 13. Explain Mathematical Ideas (1)(G) Explain why this statement is true: If m∠ABC + m∠XYZ = 180 and ∠ABC = ∠XYZ, then ∠ABC and ∠XYZ are right angles. Find the measure of each angle. 14. ∠A is twice as large as its complement, ∠B. 15. ∠A is half as large as its complement, ∠B. 16. ∠A is twice as large as its supplement, ∠B. 17. ∠A is half as large as twice its supplement, ∠B. Proof 18. Write a proof for this form of Theorem 2-2. If two angles are supplements of congruent angles, then the two angles are congruent. ∠1 and ∠2 are supplementary. Given: ∠3 and ∠4 are supplementary. ∠2 ≅ ∠4 Prove: ∠1 ≅ ∠3 Proof 19. Justify Mathematical Arguments (1)(G) Two lines intersect and one of the angles formed is a right angle. Prove that all four angles are right angles using the Vertical Angles Theorem. Determine whether the statement is an undefined term, a definition, a postulate, a conjecture, or a theorem. Explain your reasoning. 20. If an angle is obtuse, then it measures 100°. 21. If ∠1 and ∠2 are complements and ∠3 and ∠2 are complements, then ∠1 ≅ ∠3. ¯¯¯¯ ¯¯¯¯ 22. If two distinct lines AB and CD intersect, then they intersect in exactly one point. PearsonTEXAS.com Page 77 Copyright 2016 © Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy | Terms of Use | Rights and Permissions https://media.pearsoncmg.com/curriculum/math/hs2016_tx/html_books/tx_gm/homework... 10/13/2016 Geometry, Student Text and Homework Helper Page 1 of 1 Skip Directly to Table of Contents | Skip Directly to Main Content Change text size Show/Hide TOC Page Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles Congruent Proof 23. Justify Mathematical Arguments (1)(G) Fill in the blanks to complete this proof of Theorem 2-4. All right angles are congruent. Given: ∠X and ∠Y are right angles. Prove: ∠X ≅ ∠Y ∠X and a. _?_ are right angles because it is given. By the definition of b. _?_, m∠X = 90 and m∠Y = 90. By the Transitive Property of Equality, m∠X = c. _? Proof: _. Because angles of equal measure are congruent, d. _?_. 24. Analyze Mathematical Relationships (1)(F) Explain the relationship between undefined terms and terms with definitions. Include an example to illustrate your explanation. Find the value of each variable and the measure of each angle. 25. 26. 27. 28. Justify Mathematical Arguments (1)(G) Sketch a pair of intersecting line segments. Label each of the four resulting angles. Write a paragraph proof to show that one pair of vertical angles in your sketch have equal measures. 29. Given: ∠7 ≅ ∠8 Prove: ∠5 ≅ ∠6 TEXAS Test Practice 30. ∠1 and ∠2 are vertical angles. If m∠1 = 63 and m∠2 = 4x − 9, what is the value of x? 31. What is the area in square centimeters of a triangle with a base of 5 cm and a height of 8 cm? 32. What is the measure of an angle with a supplement that is four times its complement? 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