Content Standards G.CO.10 Prove theorems about triangles. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to

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1 Content Standards G.CO.10 Prove theorems about triangles. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others.

2 You used segment and angle bisectors. Identify and use perpendicular bisectors in triangles. Identify and use angle bisectors in triangles.

3 perpendicular bisector concurrent lines point of concurrency circumcenter incenter

5 Use the Perpendicular Bisector Theorems A. Find BC.

6 Use the Perpendicular Bisector Theorems B. Find XY.

7 Use the Perpendicular Bisector Theorems C. Find PQ.

8 A. Find NO. A. 4.6 B. 9.2 C D. 36.8

9 B. Find TU. A. 2 B. 4 C. 8 D. 16

10 C. Find EH. A. 8 B. 12 C. 16 D. 20

13 Use the Circumcenter Theorem GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden?

14 BILLIARDS A triangle used to rack pool balls is shown. Would the circumcenter be found inside the triangle? A. No, the circumcenter of an acute triangle is found in the exterior of the triangle. B. Yes, circumcenter of an acute triangle is found in the interior of the triangle.

16 Use the Angle Bisector Theorems A. Find DB.

17 Use the Angle Bisector Theorems B. Find m WYZ.

18 Use the Angle Bisector Theorems C. Find QS.

19 A. Find the measure of SR. A. 22 B. 5.5 C. 11 D. 2.25

20 B. Find the measure of HFI. A. 28 B. 30 C. 15 D. 30

21 C. Find the measure of UV. A. 7 B. 14 C. 19 D. 25

23 Use the Incenter Theorem A. Find ST if S is the incenter of ΔMNP.

24 Use the Incenter Theorem B. Find m SPU if S is the incenter of ΔMNP.

25 A. Find the measure of GF if D is the incenter of ΔACF. A. 12 B. 144 C. 8 D. 65

26 B. Find the measure of BCD if D is the incenter of ΔACF. A. 58 B. 116 C. 52 D. 26

27 Content Standards G.CO.10 Prove theorems about triangles. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 6 Attend to precision. 3 Construct viable arguments and critique the reasoning of others.

28 You identified and used perpendicular and angle bisectors in triangles. Identify and use medians in triangles. Identify and use altitudes in triangles.

29 median centroid altitude orthocenter

31 Use the Centroid Theorem In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.

32 In ΔLNP, R is the centroid and LO = 30. Find LR and RO. A. LR = 15; RO = 15 B. LR = 20; RO = 10 C. LR = 17; RO = 13 D. LR = 18; RO = 12

33 Use the Centroid Theorem In ΔABC, CG = 4. Find GE.

34 In ΔJLN, JP = 16. Find PM. A. 4 B. 6 C. 16 D. 8

35 Find the Centroid on a Coordinate Plane SCULPTURE An artist is designing a sculpture that balances a triangle on top of a pole. In the artist s design on the coordinate plane, the vertices are located at (1, 4), (3, 0), and (3, 8). What are the coordinates of the point where the artist should place the pole under the triangle so that it will balance?

36 BASEBALL A fan of a local baseball team is designing a triangular sign for the upcoming game. In his design on the coordinate plane, the vertices are located at ( 3, 2), ( 1, 2), and ( 1, 6). What are the coordinates of the point where the fan should place the pole under the triangle so that it will balance? A. B. C. ( 1, 2) D. (0, 4)

38 Find the Orthocenter on a Coordinate Plane COORDINATE GEOMETRY The vertices of ΔHIJ are H(1, 2), I( 3, 3), and J( 5, 1). Find the coordinates of the orthocenter of ΔHIJ.

39 COORDINATE GEOMETRY The vertices of ΔABC are A( 2, 2), B(4, 4), and C(1, 2). Find the coordinates of the orthocenter of ΔABC. A. (1, 0) B. (0, 1) C. ( 1, 1) D. (0, 0)

41 Content Standards G.CO.10 Prove theorems about triangles. Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others.

42 You found the relationship between the angle measures of a triangle. Recognize and apply properties of inequalities to the measures of the angles of a triangle. Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle.

46 Use the Exterior Angle Inequality Theorem

47 Use the Exterior Angle Inequality Theorem

48 A. B. C. D.

49 A. B. C. D.

51 Order Triangle Angle Measures List the angles of ΔABC in order from smallest to largest.

52 List the angles of ΔTVX in order from smallest to largest. A. X, T, V B. X, V, T C. V, T, X D. T, V, X

53 Order Triangle Side Lengths List the sides of ΔABC in order from shortest to longest.

54 List the sides of ΔRST in order from shortest to longest. A. RS, RT, ST B. RT, RS, ST C. ST, RS, RT D. RS, ST, RT

55 Angle-Side Relationships HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?

56 KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing? A. B. C. D. A and D B and F C and E A and B

57 Content Standards G.CO.10 Prove theorems about triangles. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 2 Reason abstractly and quantitatively.

58 You wrote paragraph, two-column, and flow proofs. Write indirect algebraic proofs. Write indirect geometric proofs.

59 indirect reasoning indirect proof proof by contradiction

61 State the Assumption for Starting an Indirect Proof A. State the assumption you would make to start an indirect proof for the statement is not a perpendicular bisector.

62 State the Assumption for Starting an Indirect Proof B. State the assumption you would make to start an indirect proof for the statement 3x = 4y + 1.

63 State the Assumption for Starting an Indirect Proof

64 A. B. C. D.

65 A. B. C. D.

66 A. B. MLH PLH C. D.

67 Write an Indirect Algebraic Proof Write an indirect proof to show that if 2x + 11 < 7, then x > 2. Given: 2x + 11 < 7 Prove: x > 2

68 Which is the correct order of steps for the following indirect proof? Given: x + 5 > 18 Prove: x > 13 I. In both cases, the assumption leads to a contradiction. Therefore, the assumption x 13 is false, so the original conclusion that x > 13 is true. II. Assume x 13. III. When x < 13, x + 5 = 18 and when x < 13, x + 5 < 18.

69 A. I, II, III B. I, III, II C. II, III, I D. III, II, I

70 Indirect Algebraic Proof EDUCATION Marta signed up for three classes at a community college for a little under $156. There was an administration fee of $15, and the class costs are equal. How can you show that each class cost less than $47?

71 SHOPPING David bought four new sweaters for a little under $135. The tax was $7, but the sweater costs varied. Can David show that at least one of the sweaters cost less than $32? A. Yes, he can show by indirect proof that assuming that every sweater costs $32 or more leads to a contradiction. B. No, assuming every sweater costs $32 or more does not lead to a contradiction.

72 Indirect Proofs in Number Theory Write an indirect proof to show that if x is a prime number not equal to 3, then x is not an integer. 3

73 You can express an even integer as 2k for some integer k. How can you express an odd integer? A. 2k + 1 B. 3k C. k + 1 D. k + 3

74 Geometry Proof Write an indirect proof. Given: ΔJKL with side lengths 5, 7, and 8 as shown. Prove: m K < m L

75 Which statement shows that the assumption leads to a contradiction for this indirect proof? Given: ΔABC with side lengths 8, 10, and 12 as shown. Prove: m C > m A

76 A. Assume m C m A + m B. By angle-side relationships, AB > BC + AC. Substituting, or This is a false statement. B. Assume m C m A. By angleside relationships, AB BC. Substituting, This is a false statement.

77 Content Standards G.CO.10 Prove theorems about triangles. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively.

78 You recognized and applied properties of inequalities to the relationships between the angles and sides of a triangle. Use the Triangle Inequality Theorem to identify possible triangles. Prove triangle relationships using the Triangle Inequality Theorem.

80 Identify Possible Triangles Given Side Lengths A. Is it possible to form a triangle with side lengths of 6 1, 6 1, and 14 1? If not, explain why not

81 Identify Possible Triangles Given Side Lengths B. Is it possible to form a triangle with side lengths of 6.8, 7.2, 5.1? If not, explain why not.

82 A. yes B. no

83 B. Is it possible to form a triangle given the side lengths 4.8, 12.2, and 15.1? A. yes B. no

84 In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR? A 7 B 9 C 11 D 13 Find Possible Side Lengths

85 In ΔXYZ, XY = 6, and YZ = 9. Which measure cannot be XZ? A. 4 B. 9 C. 12 D. 16

86 Proof Using Triangle Inequality Theorem TRAVEL The towns of Jefferson, Kingston, and Newbury are shown in the map below. Prove that driving first from Jefferson to Kingston and then Kingston to Newbury is a greater distance than driving from Jefferson to Newbury.

87 Jacinda is trying to run errands around town. She thinks it is a longer trip to drive to the cleaners and then to the grocery store, than to the grocery store alone. Determine whether Jacinda is right or wrong. A. Jacinda is correct, HC + CG > HG. B. Jacinda is not correct, HC + CG < HG.

88 Content Standards G.CO.10 Prove theorems about triangles. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 1 Make sense of problems and persevere in solving them.

89 You used inequalities to make comparisons in one triangle. Apply the Hinge Theorem or its converse to make comparisons in two triangles. Prove triangle relationships using the Hinge Theorem or its converse.

91 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD.

92 Use the Hinge Theorem and Its Converse B. Compare the measures m ABD and m BDC.

93 A. Compare the lengths of FG and GH. A. FG > GH B. FG < GH C. FG = GH D. not enough information

94 B. Compare m JKM and m KML. A. m JKM > m KML B. m JKM < m KML C. m JKM = m KML D. not enough information

96 Use the Hinge Theorem HEALTH Doctors use a straight-leg-raising test to determine the amount of pain felt in a person s back. The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Nitan can tolerate the doctor raising his right leg 35 and his left leg 65 from the table. Which leg can Nitan raise higher above the table?

97 Meena and Rita are both flying kites in a field near their houses. Both are using strings that are 10 meters long. Meena s kite string is at an angle of 75 with the ground. Rita s kite string is at an angle of 65 with the ground. If they are both standing at the same elevation, which kite is higher in the air? A. Meena s kite B. Rita s kite

98 Apply Algebra to the Relationships in Triangles ALGEBRA Find the range of possible values for a.

99 Find the range of possible values of n. A. 6 < n < 25 B. C. n > 6 D. 6 < n < 18.3

100 Prove Triangle Relationships Using Hinge Theorem Write a two-column proof. Given: JK = HL; JH KL m JKH + m HKL < m JHK + m KHL Prove: JH < KL

101 Which reason correctly completes the following proof? Given: Prove: AC > DC

102 Statements Reasons Given Reflexive Property 3. m ABC = m ABD + m DBC 3. Angle Addition Postulate 4. m ABC > m DBC 4. Definition of Inequality 5. AC > DC 5.?

103 A. Substitution B. Isosceles Triangle Theorem C. Hinge Theorem D. none of the above

104 Given: Prove Relationships Using Converse of Hinge Theorem Prove:

105 Prove Relationships Using Converse of Hinge Theorem

106 Which reason correctly completes the following proof? Given: X is the midpoint of ΔMCX is isosceles. CB > CM Prove:

107 Statements 1. X is the midpoint of MB; ΔMCX is isosceles Reasons 1. Given Definition of midpoint Reflexive Property 4. CB > CM 4. Given 5. m CXB > m CXM 5.? Definition of isosceles triangle Isosceles Triangle Theorem 8. m CXB > m CMX 8. Substitution

108 A. Converse of Hinge Theorem B. Definition of Inequality C. Substitution D. none of the above

Five-Minute Check (over Lesson 5 5) CCSS Then/Now Theorems: Inequalities in Two Triangles Example 1: Use the Hinge Theorem and its Converse Proof:

Five-Minute Check (over Lesson 5 5) CCSS Then/Now Theorems: Inequalities in Two Triangles Example 1: Use the Hinge Theorem and its Converse Proof: Hinge Theorem Example 2: Real-World Example: Use the Hinge