Content Standards G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective

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1 Content Standards G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Mathematical Practices 2 Reason abstractly and quantitatively. 6 Attend to precision.

2 You measured and classified angles. Identify and classify triangles by angle measures. Identify and classify triangles by side measures.

3 acute triangle equiangular triangle obtuse triangle right triangle equilateral triangle isosceles triangle scalene triangle

5 Classify Triangles by Angles A. Classify the triangle as acute, equiangular, obtuse, or right.

6 Classify Triangles by Angles B. Classify the triangle as acute, equiangular, obtuse, or right.

7 A. ARCHITECTURE The frame of this window design is made up of many triangles. Classify ΔACD. A. acute B. equiangular C. obtuse D. right

8 B. ARCHITECTURE The frame of this window design is made up of many triangles. Classify ΔADE. A. acute B. equiangular C. obtuse D. right

9 Classify Triangles by Angles Within Figures Classify ΔXYZ as acute, equiangular, obtuse, or right. Explain your reasoning.

10 Classify ΔACD as acute, equiangular, obtuse, or right. A. acute B. equiangular C. obtuse D. right

12 ARCHITECTURE The triangle truss shown is modeled for steel construction. Classify ΔJMN, ΔJKO, and ΔOLN as equilateral, isosceles, or scalene. Classify Triangles by Sides

13 ARCHITECTURE The frame of this window design is made up of many triangles. Classify ΔABC. A. isosceles B. equilateral C. scalene D. right

14 Classify Triangles by Sides Within Figures If point Y is the midpoint of VX, and WY = 3.0 units, classify ΔVWY as equilateral, isosceles, or scalene. Explain your reasoning.

15 If point C is the midpoint of BD, classify ΔABC as equilateral, isosceles, or scalene. A. equilateral B. isosceles C. scalene

16 Finding Missing Values ALGEBRA Find the measures of the sides of isosceles triangle KLM with base KL.

17 ALGEBRA Find x and the measure of each side of equilateral triangle ABC if AB = 6x 8, BC = 7 + x, and AC = 13 x. A. x = 10; all sides are 3. B. x = 6; all sides are 13. C. x = 3; all sides are 10. D. x = 3; all sides are 16.

18 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.

19 You classified triangles by their side or angle measures. Apply the Triangle Angle-Sum Theorem. Apply the Exterior Angle Theorem.

20 auxiliary line exterior angle remote interior angles flow proof corollary

23 SOFTBALL The diagram shows the path of the softball in a drill developed by four players. Find the measure of each numbered angle. Use the Triangle Angle-Sum Theorem

24 Find the measure of 3. A. 95 B. 75 C. 57 D. 85

27 GARDENING Find the measure of FLW in the fenced flower garden shown. Use the Exterior Angle Theorem

28 The piece of quilt fabric is in the shape of a right triangle. Find the measure of ACD. A. 30 B. 40 C. 50 D. 130

30 Find the measure of each numbered angle. Find Angle Measures in Right Triangles

31 Find m 3. A. 50 B. 45 C. 85 D. 130

32 Content Standards G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices 6 Attend to precision. 3 Construct viable arguments and critique the reasoning of others.

33 You identified and used congruent angles. Name and use corresponding parts of congruent polygons. Prove triangles congruent using the definition of congruence.

34 congruent congruent polygons corresponding parts

36 Identify Corresponding Congruent Parts Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.

37 The support beams on the fence form congruent triangles. In the figure ΔABC ΔDEF, which of the following congruence statements correctly identifies corresponding angles or sides? A. B. C. D.

38 Use Corresponding Parts of Congruent Triangles In the diagram, ΔITP ΔNGO. Find the values of x and y.

39 In the diagram, ΔFHJ ΔHFG. Find the values of x and y. A. x = 4.5, y = 2.75 B. x = 2.75, y = 4.5 C. x = 1.8, y = 19 D. x = 4.5, y = 5.5

41 ARCHITECTURE A drawing of a tower s roof is composed of congruent triangles all converging at a point at the top. If IJK IKJ and m IJK = 72, find m JIH. Use the Third Angles Theorem

42 TILES A drawing of a tile contains a series of triangles, rectangles, squares, and a circle. If ΔKLM ΔNJL, KLM KML, and m KML = 47.5, find m LNJ. A. 85 B. 45 C D. 95

43 Prove That Two Triangles are Congruent Write a two-column proof. Prove: ΔLMN ΔPON

44 Proof: Prove That Two Triangles are Congruent

45 Find the missing information in the following proof. Prove: ΔQNP ΔOPN Proof: Statements Reasons Given Reflexive Property of Congruence 3. Q O, NPQ PNO3. Given 4. QNP ONP 4.? 5. ΔQNP ΔOPN 5. Definition of Congruent Polygons

46 A. CPCTC B. Vertical Angles Theorem C. Third Angles Theorem D. Definition of Congruent Angles

48 Content Standards G.CO.10 Prove theorems about triangles. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 1 Make sense of problems and persevere in solving them.

49 You proved triangles congruent using the definition of congruence. Use the SSS Postulate to test for triangle congruence. Use the SAS Postulate to test for triangle congruence.

50 included angle

52 Use SSS to Prove Triangles Congruent Write a flow proof. Given: QU AD, QD AU Prove: ΔQUD ΔADU

53 Answer: Use SSS to Prove Triangles Congruent

54 Which information is missing from the flowproof? Given: AC AB D is the midpoint of BC. Prove: ΔADC ΔADB A. AC AC B. AB AB C. AD AD D. CB BC

55 SSS on the Coordinate Plane EXTENDED RESPONSE Triangle DVW has vertices D( 5, 1), V( 1, 2), and W( 7, 4). Triangle LPM has vertices L(1, 5), P(2, 1), and M(4, 7). a. Graph both triangles on the same coordinate plane. b. Use your graph to make a conjecture as to whether the triangles are congruent. Explain your reasoning. c. Write a logical argument that uses coordinate geometry to support the conjecture you made in part b.

56 SSS on the Coordinate Plane

57 Determine whether ΔABC ΔDEF for A( 5, 5), B(0, 3), C( 4, 1), D(6, 3), E(1, 1), and F(5, 1). A. yes B. no C. cannot be determined

59 Use SAS to Prove Triangles are Congruent ENTOMOLOGY The wings of one type of moth form two triangles. Write a two-column proof to prove that ΔFEG ΔHIG if EI FH, and G is the midpoint of both EI and FH.

60 Prove: ΔFEG ΔHIG Use SAS to Prove Triangles are Congruent Given: EI FH; G is the midpoint of both EI and FH.

61 The two-column proof is shown to prove that ΔABG ΔCGB if ABG CGB and AB CG. Choose the best reason to fill in the blank. Proof: Statements Reasons 1. Given 2.? Property 3. ΔABG ΔCGB 3. SSS A. Reflexive B. Symmetric C. Transitive D. Substitution

62 Use SAS or SSS in Proofs Write a paragraph proof. Prove: Q S

63 Answer: Use SAS or SSS in Proofs

64 Choose the correct reason to complete the following flow proof. A. Segment Addition Postulate B. Symmetric Property C. Midpoint Theorem D. Substitution

65 Content Standards G.CO.10 Prove theorems about triangles. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 5 Use appropriate tools strategically.

66 You proved triangles congruent using SSS and SAS. Use the ASA Postulate to test for congruence. Use the AAS Theorem to test for congruence.

67 included side

69 Use ASA to Prove Triangles Congruent Write a two-column proof.

70 Use ASA to Prove Triangles Congruent

71 Fill in the blank in the following paragraph proof. A. SSS B. SAS C. ASA D. AAS

73 Use AAS to Prove Triangles Congruent Write a paragraph proof.

74 Complete the following flow proof. A. SSS B. SAS C. ASA D. AAS

75 Apply Triangle Congruence MANUFACTURING Barbara designs a paper template for a certain envelope. She designs the top and bottom flaps to be isosceles triangles that have congruent bases and base angles. If EV = 8 cm and the height of the isosceles triangle is 3 cm, find PO.

76 Apply Triangle Congruence

77 The curtain decorating the window forms 2 triangles at the top. B is the midpoint of AC. AE = 13 inches and CD = 13 inches. BE and BD each use the same amount of material, 17 inches. Which method would you use to prove ΔABE ΔCBD? A. SSS B. SAS C. ASA D. AAS

79 Content Standards G.CO.10 Prove theorems about triangles. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Mathematical Practices 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others.

80 You identified isosceles and equilateral triangles. Use properties of isosceles triangles. Use properties of equilateral triangles.

81 legs of an isosceles triangle vertex angle base angles

83 Congruent Segments and Angles A. Name two unmarked congruent angles.

84 Congruent Segments and Angles B. Name two unmarked congruent segments.

85 A. Which statement correctly names two congruent angles? A. PJM PMJ B. JMK JKM C. KJP JKP D. PML PLK

86 B. Which statement correctly names two congruent segments? A. JP PL B. PM PJ C. JK MK D. PM PK

89 A. Find m R. Find Missing Measures

90 B. Find PR. Find Missing Measures

91 A. Find m T. A. 30 B. 45 C. 60 D. 65

92 B. Find TS. A. 1.5 B. 3.5 C. 4 D. 7

93 Find Missing Values ALGEBRA Find the value of each variable.

94 Find the value of each variable. A. x = 20, y = 8 B. x = 20, y = 7 C. x = 30, y = 8 D. x = 30, y = 7

95 Apply Triangle Congruence Given: HEXAGO is a regular polygon. ΔONG is equilateral, N is the midpoint of GE, and EX OG. Prove: ΔENX is equilateral.

96 Proof: Apply Triangle Congruence

97 Given: HEXAGO is a regular hexagon. NHE HEN NAG AGN Prove: HN EN AN GN Proof: Statements 1. HEXAGO is a regular hexagon. 2. NHE HEN NAG AGN 3. HE EX XA AG GO OH 4. ΔHNE ΔANG Reasons 1. Given 2. Given 3. Definition of regular hexagon 4. ASA

98 Proof: Statements 5. HN AN, EN NG 6. HN EN, AN GN 7. HN EN AN GN Reasons 5.? 6. Converse of Isosceles Triangle Theorem 7. Substitution A. Definition of isosceles triangle B. Midpoint Theorem C. CPCTC D. Transitive Property

99 Content Standards G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Mathematical Practices 1 Make sense of problems and persevere in solving them. 7 Look for and make use of structure.

100 You proved whether two triangles were congruent. Identify reflections, translations, and rotations. Verify congruence after a congruence transformation.

101 transformation preimage image congruence transformation isometry reflection translation rotation

102

103 Identify Congruence Transformations A. Identify the type of congruence transformation shown as a reflection, translation, or rotation.

104 Identify Congruence Transformations B. Identify the type of congruence transformation shown as a reflection, translation, or rotation.

105 Identify Congruence Transformations C. Identify the type of congruence transformation shown as a reflection, translation, or rotation.

106 A. Identify the type of congruence transformation shown as a reflection, translation, or rotation. A. reflection B. translation C. rotation D. none of these

107 B. Identify the type of congruence transformation shown as a reflection, translation, or rotation. A. reflection B. translation C. rotation D. none of these

108 C. Identify the type of congruence transformation shown as a reflection, translation, or rotation. A. reflection B. translation C. rotation D. none of these

109 Identify a Real-World Transformation BRIDGES Identify the type of congruence transformation shown by the image of the bridge in the river as a reflection, translation, or rotation.

110 GAME Identify the type of congruence transformation shown by the image of the chess piece as a reflection, translation, or rotation. A. reflection B. translation C. rotation D. none of these

111 Verify Congruence after a Transformation Triangle PQR with vertices P(4, 2), Q(3, 3), and R(5, 2) is a transformation of ΔJKL with vertices J( 2, 0), K( 3, 5), and L( 1, 4). Graph the original figure and its image. Identify the transformation and verify that it is a congruence transformation.

112 Verify Congruence after a Transformation

113 Triangle ABC with vertices A( 1, 4), B ( 4, 1), and C( 1, 1) is a transformation of ΔXYZ with vertices X ( 1, 4), Y( 4, 1), and Z( 1, 1). Graph the original figure and its image. Identify the transformation and verify that it is a congruence transformation. A. B. C. D.

114 Content Standards G.CO.10 Prove theorems about triangles. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 2 Reason abstractly and quantitatively.

115 You used coordinate geometry to prove triangle congruence. Position and label triangles for use in coordinate proofs. Write coordinate proofs.

116 coordinate proof

117 Position and Label a Triangle Position and label right triangle XYZ with leg d units long on the coordinate plane.

118 Which picture on the following slide would be the best way to position and label equilateral triangle ABC with side w units long on the coordinate plane?

119 A. B. C. D.

120

121 Identify Missing Coordinates Name the missing coordinates of isosceles right triangle QRS.

122 Name the missing coordinates of isosceles right ΔABC. A. A(d, 0); C(0, 0) B. A(0, f); C(0, 0) C. A(0, d); C(0, 0) D. A(0, 0); C(0, d)

123 Write a Coordinate Proof Write a coordinate proof to prove that the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base is perpendicular to the base.

124 Finish the following coordinate proof to prove that the segment drawn from the right angle to the midpoint of the hypotenuse of an isosceles right triangle is perpendicular to the hypotenuse.

their slopes are opposite. B. the sum of their slopes is zero.

125 Proof: The coordinates of the midpoint D are The slope of is or 1. The slope of or 1, therefore because.? A. their slopes are opposite. B. the sum of their slopes is zero. C. the product of their slopes is 1. D. the difference of their slopes is 2.

126 Classify Triangles DRAFTING Write a coordinate proof to prove that the outside of this drafter s tool is shaped like a right triangle. The length of one side is 10 inches and the length of another side is 5.75 inches.

127 Classify Triangles

128 FLAGS Tracy wants to write a coordinate proof to prove this flag is shaped like an isosceles triangle. The altitude is 16 inches and the base is 10 inches.

129 What ordered pair should she use for point C? A. (10, 10) B. (10, 5) C. (16, 10) D. (16, 5)

Chapter 4 Unit 6 SPRING GEOMETRY Name Hour

Chapter 4 Unit 6 SPRING GEOMETRY Name Hour CONGRUENT TRIANGLES Chapter 4 Unit 6 SPRING 2019 GEOMETRY Name Hour Geometry Classifying Triangles 4.1 Objectives: Triangles can be classified by their and/or their. 1) classify triangles by their angle