Chapter 2: Introduction To Reasoning and Proofs - 2.6 Exercise Set - PDF

## Chapter 2: Introduction To Reasoning and Proofs ### **2.6 Exercise Set** **Proof** Fill in each blank with a reason that justifies the statement. **1.** Statements | Reasons ---|--- 3x - 10 = 65 | Given 3x = 75 | a. Addition Property of Equality x = 25 | b. Division Property of Equality **...

## Chapter 2: Introduction To Reasoning and Proofs ### **2.6 Exercise Set** **Proof** Fill in each blank with a reason that justifies the statement. **1.** Statements | Reasons ---|--- 3x - 10 = 65 | Given 3x = 75 | a. Addition Property of Equality x = 25 | b. Division Property of Equality **2.** Statements | Reasons ---|--- 11x + 14 = 80 | Given 11x = 66 | a. Subtraction Property of Equality x = 6 | b. Division Property of Equality **3.** Statements | Reasons ---|--- 9x - 4 = 7x + 44 | Given 2x - 4 = 44 | a. Subtraction Property of Equality 2x = 48 | b. Addition Property of Equality x = 24 | c. Division Property of Equality **4.** Statements | Reasons ---|--- 3x + 6 - 2x + 31 | Given 5x + 6 = 31 | a. Combine like terms 5x = 25 | b. Subtraction Property of Equality x = 5 | c. Division Property of Equality **5.** Statements | Reasons ---|--- 1/2(2x - 5) = 10 | Given 2x - 5 = 20 | a. Multiplication Property of Equality 2x = 25 | b. Addition Property of Equality x = 30 | c. Division Property of Equality **6.** Statements | Reasons ---|--- 5(x + 3) = -4 | Given 5x + 15 = -4 | a. Distributive Property 5x = -19 | b. Subtraction Property of Equality x = -19/5 | c. Division Property of Equality **7.** Statements | Reasons ---|--- 15 + 7(3x + 4) = -20 | Given 15 + 21x + 28 = -20 | a. Distributive Property 21x + 43 = -20 | b. Combine like terms 21x = -63 | c. Subtraction Property of Equality x = -3 | d. Division Property of Equality **8.** Statements | Reasons ---|--- -20 + 2(11 - 2x) = 26 | Given -20 + 22 - 4x = 26 | a. Distributive Property 2 - 4x = 26 | b. Combine like terms -4x = 24 | c. Subtraction Property of Equality x = -6 | d. Division Property of Equality **9.** Given: ∠CDE and ∠EDF are supplementary. Statements | Reasons ---|--- m∠CDE + m∠EDF = 180° | a. Definition of supplementary angles x° + (3x + 20)° = 180° | b. Substitution Property 4x° + 20° = 180° | c. Combine like terms 4x° = 160° | d. Subtraction Property of Equality x° = 40° | e. Division Property of Equality **10.** Given: XY = 42 Statements | Reasons ---|--- XZ + ZY = XY | a. Segment Addition Postulate 3(n + 4) + 3n = 42 | b. Substitution Property 3n + 12 + 3n = 42 | c. Distributive Property 6n + 12 = 42 | d. Combine like terms 6n = 30 | e. Subtraction Property of Equality n = 5 | f. Division Property of Equality *** **Name the property of equality or congruence that justifies going from the first statement to the second statement.** **11.** 2x + 1 = 7 | 2x = 6 | Subtraction Property of Equality **12.** 5x = 20 | x = 4 | Division Property of Equality **13.** ST = QR | QR = ST | Symmetric Property of Equality **14.** AB = BC | AB + BC = 12 | Addition Property of Equality **Matching** Match each statement with the appropriate property that it describes. **15.** If ∠M = ∠N, then ∠N = ∠M | Symmetric Property of Equality **16.** If CD = PQ and PQ = RF, then CD = RF | Transitive Property of Equality **17.** If RT = 5, then 9(RT) = 45 | Multiplication Property of Equality **18.** If 3x - 12 = 15, then 3x = 27 | Addition Property of Equality **19.** If m∠A = 27° and m∠C = m∠A + 10°, then m∠C = 37° | Substitution Property of Equality **20.** If -2(x - 7) = 5, then -2x + 14 = 5 | Distributive Property of Equality *** **Fill in the Blank** Use the given property to complete each statement. **21.** Symmetric Property of Congruence If ∠H = ∠K, then **∠K = ∠H**. **22.** Symmetric Property of Equality If AB = YU, then **YU = AB**. **23.** Distributive Property 3(x-1) = **3x - 3** **24.** Reflexive Property of Congruence ∠POR = **∠POR** **25.** Transitive Property of Congruence If ∠XYZ = ∠AOB and ∠AOB = ∠WYT, then **∠XYZ = ∠WYT**. **26.** Substitution Property If LM = 7 and EF + LM = NP, then **EF + 7 = NP**. *** **Proof: ** For Exercises 27 and 28, fill in the missing statements or reasons for the two-column proof. **27.** Given: C is the midpoint of AD. Statements | Reasons ---|--- 1. C is the midpoint of AD. | 1. Given 2. AC = CD | 2. Definition of Midpoint 3. AC = CD | 3. Segment are equal in length 4. 4x = 2x + 12 | 4. Substitution Property 5. 2x = 12 | 5. Subtraction Property of Equality 6. x = 6 | 6. Division Property of Equality **28.** Given: AC is the angle bisector of ∠DAG. Statements | Reasons ---|--- 1. AC is the bisector of ∠DAG. | 1. Given 2. ∠DAC = ∠CAG | 2. Definition of Angle Bisector 3. m∠DAC = m∠ CAG | 3. Angles have equal measure 4. 9x = 6x + 9 | 4. Substitution 5. 3x = 9 | 5. Subtraction Property of Equality 6. x = 3 | 6. Division Property of Equality *** **Consider the following relationships among people. Tell whether each relationship is "reflexive," "symmetric," "transitive," or "none of these." Explain.** - **has the same birthday as:** Reflexive, Symmetric, Transitive - **lives in a different state than:** Symmetric - **is taller than:** None - **is shorter than:** None **39.** **Find the Error** The statements below "show" that 1 = 2. Describe the error. Statements | Reasons ---|--- a = b | Given ab = b² | Multiplication Property of Equality aba²b² - a² | Subtraction Property of Equality a(b - a) = (b + a)(b - a) | Distributive Property a = b + a | Division Property of Equality a = a + a | Substitution Property 1 = 2 | Simplify * **Error:** You cannot divide both sides of an equation by (b - a), because when a = b, the denominator becomes 0, and you can't divide by zero. **40. Find the Error** The statements below "show" that 0 = 3. Describe the error. Statements | Reasons ---|--- x = -3 | Given -3x = 9 | Multiplication Property of Equality -3xx² = 9x² | Multiplication Property of Equality -x(3 + x) = (3 - x)(3 + x) | Distributive Property -x = 3 - x | Division Property of Equality 0 = 3 | Addition Property of Equality * **Error:** When you multiply both sides by x², both sides should be multiplied by -x² to preserve equivalence. The error is that the value of -x² is always positive, causing the equation to not be balanced. *** **Review and Preview** Use the diagram at the right. Find each measure. - **41.** m∠AOC = 85° - **42.** m∠DOB = 60° - **43.** m∠AOD = 130° - **44.** m∠BOE = 45° Find the value of each variable. - **45.** x = 12 - **46.** x = 5 **47. Name four points on line *t*.** *Points G, A, B, and C are on line *t*. **48. Are points G, A, and B collinear?** *Yes, points G, A, and B are collinear. **49. Are points F, I, and H collinear?** *Yes, points F, I, and H are collinear. **50. Name the line on which point E lies.** *Point E lies on line *r*. **51. Name line *t* in three other ways.** *Line *t* can also be named as line AC, line GB, or line GC. **52. Name the point at which lines *t* and *r* intersect.** *Lines *t* and *r* intersect at point A. *** ### **2.7 Proving Theorems About Angles** **Objective 1:** Prove and Use Theorems About Angles. **Vocabulary:** - paragraph proof **Theorem 2.7-1: 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.* *** **2.7 Exercise Set** **Mixed Practice** Use the theorems in this section to find the measures of the angles. **18.** Proof Use a 2-column proof and the given figure: Given: m∠1 = m∠2 Prove: m∠3 = m∠4 Statements | Reasons ---|--- 1. m∠1 = m∠2 | 1. Given 2. m∠1 + m∠3 = 180° | 2. Definition of Supplementary Angles 3. m∠2 + m∠4 = 180° | 3. Definition of Supplementary Angles 4. m∠1 + m∠3 = m∠2 + m∠4 | 4. Transitive Property of Equality 5. m∠3 = m∠4 | 5. Subtraction Property of Equality **Name two pairs of congruent angles in each figure. Justify your answers by naming the theorem.** - **19.** ∠1 ≅ ∠2, ∠3 ≅ ∠4 (Vertical Angles Theorem) - **20.** ∠1 ≅ ∠3,∠2 ≅ ∠4 (Corresponding Angles Theorem) **21. Find the value of each variable and the measure of each angle.** x = 5; (2x)° = 10°; (3x)° = 15°; (5x)° = 25° **22. Find the value of each variable and the measure of each angle.** x = 12; (x - 4y)° = 4°; (2x - 5y)° = 19°; (4y - 8)° = 8° **23.** In the photograph, the legs of the table are constructed so that ∠3 = ∠4. What theorem can you use to justify the statement that ∠3 = ∠4? *Vertical Angles Theorem* **24.** Explain why this statement is true: If m∠ABC + m∠XYZ = 180° and ∠ABC = ∠XYZ, then ∠ABC and ∠XYZ are right angles. *Since ∠ABC and ∠XYZ are equal and supplementary, they must each measure 90° because the sum of two supplementary angles is 180°. Therefore, they are both right angles.* **25.** Proof Prove Theorem 2.7-2: Supplements of the same angle (or of equal angles) are equal in measure. Given: ∠A and ∠B are supplementary. ∠C and ∠D are supplementary. m∠A = m∠C. Prove: m∠B = m∠D. Statements | Reasons ---|--- 1. ∠A and ∠B are supplementary |1. Given 2. ∠C and ∠D are supplementary | 2. Given 3. m∠A + m∠B = 180° | 3. Definition of Supplementary Angles 4. m∠C + m∠D = 180° | 4. Definition of Supplementary Angles 5. m∠A = m∠C | 5. Given 6. m∠A + m∠B = m∠C + m∠D | 6. Transitive Property of Equality 7. m∠B = m∠D | 7. Subtraction Property of Equality **26.** Proof: Prove Theorem 2.7-6 Two equal supplementary angles are right angles. Given: m∠1 = m∠2 m∠1 + m∠ 2 = 180° Prove: m∠1 and m∠2 are right angles. Statements | Reasons ---|--- 1. m∠1 = m∠2 | 1. Given 2. m∠ 1 + m∠2 = 180° | 2. Given 3. 2 * m∠1 = 180° | 3. Substitution Property 4. m∠1 = 90° | 4. Division Property of Equality 5. Since m∠1 = m∠2 and m∠1 = 90°, then m∠2 = 90° | 5. Substitution Property 6. Therefore, m∠1 and m∠2 are right angles. | 6. Definition of Right Angle **27.** Multiple Steps a) What is the measure of the angle formed by Main St. and 116th St.? *The measure is 145°* b) What is the measure of the angle formed by Park St. and 116th St.? *The measure is 35°.* **28.** In the game of miniature golf, the ball bounces off the wall at the same angle it hit 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 2.7-1, what are the values of x and y? *x = 40°, y = 40°, using the Equal Complements Theorem* **29. Find the value of each variable and the measure of each angle.** *x = 15, y = 5; (4x - 35)° = 25°; (x + 10)° = 25°; (6y - 10)° = 20° **30. Find the value of each variable and the measure of each angle.** *x = 14, y = 8; (3x+ 8)° = 46°; (5x-20)° = 50°; (5x+4y)° = 110° **31.** A is twice as large as its complement, B. *∠A = 60°, ∠B = 30°* **32.** A is half as large as its complement, B. * ∠A = 30°, ∠B = 60°* **33.** ∠A is twice as large as its supplement, ∠B. *∠A = 120°, ∠B = 60°* **34.** ∠A is half as large as twice its supplement, ∠B. * ∠A = 60°, ∠B = 120°* **35. Find the value of each variable and the measure of each angle.** *x = 7: 2x° = 14°; (y-x)° = 7°; (x+y)° = 21° **36. Find the value of each variable and the measure of each angle.** *x = 10, y = 20; (x + y + 5)° = 35°; 2y = 40°; (x - y)° = -10°; (x + y + 10)° = 40°; 2x° = 20° **37. Find the value of each variable and the measure of each angle.** *x = 12, y = 8; (x + y + 5)° = 25°; 2x° = 24°; (x - y)° = 4°; (x + y)° = 20°; 2y° = 16° **38. Find the value of each variable and the measure of each angle.* *x = 15, y = 5; (x + y + 5)° = 25°; 2y° = 10°; 2x° = 30°; (x + y)° = 20° **39. Give an example of vertical angles in your home or classroom. ** *Possible examples: the two angles formed by opposing walls, the two angles formed by opposing sides at a table corner.* **40. How are angles that form a right angle related?** *Angles that form a right angle are complementary angles. They add up to 90 degrees.* **41. Find the coordinates of a point B so that ∠BOA and ∠AOX are adjacent complementary angles.** *The complementary angle to ∠AOX would be ∠BOX. To add up to 90 degrees, a point B would need to lie on the y axis at a distance of 4 units above the x-axis so that ∠BOX = 90°. Therefore, point B would be (0, 4). **42. Find the coordinates of a point C so that OC is a side of a different angle that is adjacent and complementary to ∠AOX** *The coordinates of point C would be (0, 4) to make ∠COA complementary to ∠AOX. This would result in a right angle made by ∠COA and ∠AOX combined, making them complementary angles.* **43. 3x + 7 = 19 | 3x= 12 | Subtraction Property of Equality** **44. 4x = 20 | x = 5 | Division Property of Equality** **45. ∠1 and ∠3 = ∠2 | Substitution Property** **46. m∠Q = m∠ R | Substitution Property** **47. Name four points on line *t*.** *Points G, A, B, and C are on line *t*. **48. Are points G, A, and B collinear?** *Yes, points G, A, and B are collinear. **49. Are points F, I, and H collinear?** *Yes, points F, I, and H are collinear. **50. Name the line on which point E lies.** *Point E lies on line *r*. **51. Name line *t* in three other ways.** *Alternatively, line *t* could be named as line AC, line GB, or line GC. **52. Name the point at which lines *t* and *r* intersect.** *Lines *t* and *r* intersect at point A.