Transformation #1: ( x, y) ( x 7, y)

Transcription

1 Lesson 1A - Give it a Transformation! Name: Transformation #1: ( x, y) ( x 7, y) 6 y x a) Use Colored Pencil #1 to plot and label the following points. Connect them to form a triangle. A( -5,-2) B(-3,3) C(-1,-3) b) Get 3 new points by applying the transformation to the 3 original points listed above. Old New c) Use Colored Pencil #2 to plot and label the new points. Connect them to form a new figure. d) Describe how the original figure was changed by the transformation. Education Development Center, Inc., 2009

2 Transformation #2: ( x, y) ( x, y 6) y x a) Use Colored Pencil #1 to plot and label the following points. Connect them to form a parallelogram. A( -3,-3) B(-2,-5) C(5,-5) D(4,-3) b) Get 4 new points by applying the transformation to the 4 points listed above. Old New c) Use Colored Pencil #2 to plot and label the new points. Connect them to form a new figure. d) Describe how the original figure was changed by the transformation. Education Development Center, Inc., 2009

3 Transformation #3: ( x, y) (3 x,2 y) 6 y x a) Use Colored Pencil #1 to plot and label the following points. Connect them to form a triangle. A( -2,-2) B(2,-3) C(0,2) b) Get 3 new points by applying the transformation to the 3 points listed above. Old New c) Use Colored Pencil #2 to plot and label the new points. Connect them to form a new figure. d) Describe how the original figure was changed by the transformation. Education Development Center, Inc., 2009

4 Transformation #4: ( x, y) ( x, y) 6 y x a) Use Colored Pencil #1 to plot and label the following points. Connect them to form a trapezoid. A( -4,1) B(-2,3) C(2,2) D(4,1) b) Get 4 new points by applying the transformation to the 4 points listed above. Old New c) Use Colored Pencil #2 to plot and label the new points. Connect them to form a new figure. d) Describe how the original figure was changed by the transformation. Education Development Center, Inc., 2009

5 5. This one has a beautiful parabola 1. The nine marked points lie on a curve called a parabola. OLD NEW a) Record the coordinates of these points in the OLD column of the table. b) Apply the following transformation to all of the old points. Record the new coordinates in the NEW column. ( x, y) ( x 10,18 y) c) Plot the new points and draw a new parabola. Education Development Center, Inc., 2009

6 Problem #6: Transformation Rule: ( x, y) ( x 5, y 7) y x a) Plot and label the following points. Connect them to form a polygon. A( -5,4) B(-2,4) C(-2,5) D(-1,3) E(-2,1) F(-2,2) G(-5,2) b) Get 7 new points by applying the transformation to the 7 points listed above. Old New c) Plot your new set of points. Connect them to form a new figure. Education Development Center, Inc., 2009

7 Geometry N z2i0d1r4f jkpujthad ISgoZfztAwWavrzey klilucg.z \ WAIlslo FrbiCgghgtmsY \rbeyste`ravle]dd. Translations Name Date Period Graph the image of the figure using the transformation given. 1) translation: 5 units up 2) translation: 6 units left and 5 units up y y Y x E x G N A B P 3) translation: 7 units right and 3 units down y 4) translation: 3 units right and 1 unit up y X B S N Q x B x Y b D2z0G1f4O vkcudtean FS_oOf^tnwzaUrzeE zl^l^co.q c eakljlv Ar^ingVhhtxsy IrWeLsNesrsvheJdz.S p `MyaodYeQ Jw]iItHhG qi^nkfqifnsi^tsek BGueioWmZedtHrayQ. -1- Worksheet by Kuta Software LLC

8 5) translation: 4 units left S(1, -1), W(5, 1), J(3, -3) y 6) translation: 5 units left and 3 units down K(0, 0), I(0, 3), U(4, 1) y x x 7) translation: 2 units right and 6 units up G(-3, -2), S(1, -1), I(-1, -5) y 8) translation: 4 units right and 6 units up J(-3, -3), C(-2, -1), B(0, -1), I(1, -4) y x x r b2i0i1t4o [KTuKtmaL ws`opfat]wkakrkeo nlmlgck.z B FABlmln DrjiWghhwt^so vr]ezsfevrsvjeedh.f H DM[aqdie[ jwsiet]hn rimnzfpipn`ietqeb ZGtenobmCe_tArnyI. -2- Worksheet by Kuta Software LLC

9 Lesson 1C Reflections Part 1: Horizontal and Vertical Lines Name: Plot 5 points that have a y value equal to 6 Draw the line y=6 Plot 5 points that have an x value equal to 4 Draw the line x= Plot 5 points that have an x value equal to -2 Draw the line x=-2 Plot 5 points that have a y value equal to -1 Draw the line y=-1

10 5. Graph the horizontal and vertical lines given below. a) x 2 b) x 7 c) x 5 d) y 6 e) y 3 6. Reflect the following figure across the line y=1. 7. Reflect the following figure across the line x=2

11 Part 2: Reflections across horizontal and vertical lines 1. Apply the following transformation to the given figure. Label the vertices of the image. a) x, y 2 x, y A Original A(-4,5) Image C B(-3,1) B C(-1,3) 2. Apply the following transformations to the given figure. Label the vertices of the image. a) x, y x,4 y Original Image D(1,-1) E(5,-1) F(4,-3) D E G(2,-3) G F

12 3. Time to up our game and do some reflections! a) b) c) d)

13 Geometry Name a L2t0C1`5r hkfuit_ax \SfoWfItywradrzec GLiLYC^.f [ za_lllk crtivgzhwtesh IrOeasFeWrhvIeHdR. More Reflections Graph the image of the figure using the transformation given. Date Period 1) reflection across x = -1 y 2) reflection across x = 2 y x A V x M N W D M H 3) reflection across the x-axis M G y Z 4) reflection across y = -1 y N x x W H Q k u2^0n1n5n lkeubtfaa rscoqfpt]wtasreet NLuLCCB.N j LAslolv erbisgjhztusd CrdedsceerlvxeLdS.J e FMlasd`eZ WwniHthhb finn\fjionciet]ej vgneroumlewtyrmyk. -1- Worksheet by Kuta Software LLC

14 Write a rule to describe each transformation. 5) J T y 6) y U U' x Q Q' x J' T' T T' A) reflection across x = -3 B) reflection across y = 1 C) reflection across y = 2 D) reflection across x = -1 S A) reflection across y = -1 B) reflection across the y-axis C) reflection across y = -2 D) reflection across x = -2 S' 7) y 8) y U' U V' V J' K' K I' I B B' x x J A) reflection across y = -2 B) reflection across x = -2 C) reflection across x = -1 D) reflection across y = -1 A) reflection across y = 3 B) reflection across x = 3 C) reflection across the y-axis D) reflection across x = 1 X d2o0m1w5a KKvuCtlaN BSyoCfQtcwIamrked tlrlfcs.a d QAzlxlM ArmiRgQhgtksH orcebsaeprrvqe[dy.j m RM[aMdlez Gwti_tkhn OIDnOfriKnbirtMeL PGbeAoymReItWreyz. -2- Worksheet by Kuta Software LLC

15 Lesson 1E Everybody Rotate! 1. Before and After a) Name: Use patty paper to rotate the figure 90⁰ counterclockwise. Record the coordinates of the vertices for the original and the transformed figures. B' C Original Rotated A' B C' A b) E' F' Original Rotated D' D E F Rule for 90⁰ Counterclockwise Rotations: x, y y, x

16 2. Rotate it 90⁰ Counterclockwise three times! Rotate the given figure 3 times. Record the coordinates of each figure. Label each figure with the proper notations B D C A Original Rotated 90⁰ Rotated 180⁰ Rotated 270⁰ Rule for 90⁰ Counterclockwise Rotations: x, y y, x

17 3. One More Time Rotate the figure 3 times. T U S R Original Rotated 90⁰ Rotated 180⁰ Rotated 270⁰ Rule for 90⁰ Counterclockwise Rotations: x, y y, x

18 Rule for 90⁰ Counterclockwise Rotations: x, y y, x 4. Rotate 90⁰ counterclockwise. Use the rule!!! Record the coordinates of the original figure and its image. a) D C Original Rotated A B b) Original Rotated H F S

19 Lesson 1F - The Rule for 90⁰ Rotations Name: 1. One last time with tracing paper Rotate 90⁰ counterclockwise. Original Points Image Points Rule for 90⁰ Counterclockwise Rotations: x, y y, x 2. Use the rule!!! Original Points Image Points

20 Rule for 90⁰ Counterclockwise Rotations: x, y y, x 3. Here s another one. Original Points Image Points 4. Quadrilaterals are people too Original Points Image Points

21 Lesson 1G Practice those Transformations! Name: 1. Reflect triangle ABC over the line y=2. 2. The point P(4,2) is reflected over the x-axis and then translated right 5 units. What are the B coordinates of the image of point P? A C Answer: 3. Rotate the triangle below 180⁰ in the counterclockwise direction. Hint: Use the rule x, y ( x, y ). 4. Rotate the rectangle 90⁰F in the counterclockwise direction. Hint: Use the rule x, y ( y,x) T E F R S D G

22 5. The triangles on the right are congruent. Which of the following is a valid congruence statement for these triangles? D Q A. DPW FMQ P B. DPW FMQ C. WDP FMQ D. PDW FMQ E. PWD FMQ W F M Answer: 6. Identify the type of transformation that is illustrated in each figure (Hint: Each figure represents one of the following: translation, reflection, rotation, dilation, horizontal stretch, or vertical stretch) a) C C' b) B' C' A B B' A' B A' C D' A D Transformation: c) Transformation: d) Q C C' P' P R R' Q' Transformation: Transformation: e) E' F' f) U U' E F T V T' V' D G D' G' Transformation: Transformation:

23 Lesson 2A Writing Congruence Statements Name: 1. Take a good look at these figures a) In some ways, all of the objects above are the same. Explain. b) In some ways, they are different. Explain. c) What do we mean by congruent? d) Which squares are congruent? + = 1

24 2. How do we write it? How do we say it? a) Triangles Congruence statement: A J In words: B C R W b) Segments Congruence statement: S R In words: P Q c) Angles Congruence statement: E N 40 O 40 A In words: T d) Quadrilaterals Congruence statement: B D C M O N In words: P A 2

25 3 3. More Triangles Find 1 pair of congruent triangles and shade them with 2 different colored pencils. Write congruence statements. a) b) c) d) e) f) P D C B A V Z Y W X M E T I K P O N M S T E N R O H U T S R Q P

26 4. Segments and Angles Write a congruence statement for each pair of congruent segments and for each pair of congruent angles a) Congruent angles and segments: P A K R b) G Congruent angles and segments: F 65 5 H 5 65 I c) Congruent angles and segments: M 60 L O J 120 K N P 4

27 Writing Congruence Statements for Corresponding Parts If two triangles are congruent, then their corresponding sides and angles are congruent as well. Directions: The following pairs of triangles are congruent. Write congruence statements for the following triangles. List all the congruent angles and sides. 5) 6) a) Congruence statement: a) Congruence statement: b) Corresponding parts: b) Corresponding parts: 7) 8) a) Congruence statement: a) Congruence statement: b) Corresponding parts: b) Corresponding parts: 5

28 9) Write a congruence statement for the shapes below. List the congruent angles and sides. a) Congruence statement: b) Corresponding parts: 10) Write a congruence statement for the shapes below. List the congruent angles and sides. a) Congruence statement: b) Corresponding parts: 6

29 Lesson 3A Some Recipes for Triangles Name: Recipe #1: A Triangle with side lengths 10cm, 9cm, 4cm. Workspace: Draw a line segment that is 10cm long. Label the endpoints A and B. Draw a circle with center at A and radius 9cm. Draw a circle with center at B and radius 4cm. The circles intersect at two different points. Label one of these points C. Use a ruler to complete triangle ABC. Label each of the triangle s sides with its length. Questions: a) If somebody follows this recipe, how do you know that side AC will always end up being 9cm long? b) Could someone follow this recipe and get a triangle that is different than yours? 1

30 Recipe #2: A Triangle with side lengths 6cm, 8cm, 8cm. Draw a line segment that is 6cm long. Label the endpoints A and B. Draw a circle with center at A and radius 8cm. Draw a circle with center at B and radius 8cm. The circles intersect at two different points. Label one of these points C. Use a ruler to complete triangle ABC. Label each of the triangle s sides with its length. Questions: a) Is the shape of your triangle the same as your neighbor s triangle? b) Is the size of your triangle the same as your neighbor s triangle? 2

31 3. Try this. Draw a triangle that has these in the following order: 70⁰ angle, 10 cm side, 35⁰ angle 4. Draw a triangle with 30⁰, 60⁰, and 90⁰ angles a) Is the shape of your triangle the same as your neighbor s triangle? b) Is the size of your triangle the same as your neighbor s triangle? 3

32 5. Draw a triangle that has these in the following order: 10cm side, 120⁰ angle, 6 cm side. 6. Draw a triangle with a 40⁰ angle, a 7 cm side, and a 90⁰ angle in that order. 4

33 Lesson 3B Does It Imply Congruence? Name: 1. SSS (Side-Side-Side) Question: If 2 triangles have sides with the same lengths, do they have to be congruent? Example: Are all triangles with sides 12cm, 10cm, and 6 cm congruent? Sketch: Observe: All of our triangles have sides 12cm, 10cm, and 6cm. Are all of our triangles congruent? Conclusion: SSS Triangles are Congruent Triangles are not necessarily congruent 1

34 2. AAA (Angle-Angle-Angle) Question: If 2 triangles have angles with the same measurements, do they have to be congruent? Example: Are all triangles that have angle measurements 40, 60, 80 congruent? Sketch: Observe: All of our triangles have 40, 60, and 80 angles. Are all of our triangles congruent? Conclusion: AAA Triangles are Congruent Triangles are not necessarily congruent 2

35 3. ASA (Angle-Side-Angle) Sketch: Draw a triangle that has a 70 angle, an 8cm side, and a 30 angle (in that order) Observe: All of our triangles have a 70 angle, an 8cm side, and a 30 angle, with the 8 cm segment between the angles. Are all of our triangles congruent? Conclusion: ASA Triangles are Congruent Triangles are not necessarily congruent 3

36 4. SAS (Side-Angle-Side) Sketch: Draw a triangle that has a 14cm side, a sides.) 45, and a 9 cm side. (The angle is between these two Did we all get the congruent triangles? Conclusion: SAS Triangles are Congruent Triangles are not necessarily congruent 4

37 5. SSA (Side-Side-Angle) Sketch: Draw a triangle that has a 30 angle, an 8cm side, and a 5cm side. Show that there are really 2 triangles that could be drawn like this. Since 2 different triangles were drawn with and a 5cm side, what can we conclude? 30 angle, an 8cm side, Conclusion: SSA Triangles are Congruent Triangles are not necessarily congruent 5

38 Summary: These can be used to prove triangles are congruent: SSS - If the corresponding sides of triangles are congruent, then we can conclude that the triangles are congruent. ASA If two angles and the side between them are congruent, then the triangles are congruent. SAS If two sides and the angle between them are congruent, then the triangles are congruent. AAS If two sides and the angle between them are congruent, then the triangles are congruent. These are insufficient to conclude congruence: AAA, SSA 6

39 Lesson 3C Establishing Triangle Congruence Is there enough information to conclude that the triangles are congruent? Use SSS, SAS, ASA, or AAS to justify your answer!!! Name: 1) 2) 3) 4)

40 5) 6) 7) 8) 9) 10)

41 Unit 4, Lesson 4: What does the figure imply??? Based on the figures, which angles and segments have to be congruent? List the congruent angles and segments and give the reason why they must be congruent. a) Let s do one together These might help! These are the common reasons for concluding that segments or angles are congruent. A B Given in the figure. (Given) D List 4 congruencies (and why): C E Vertical angles are congruent. (V.A. are ) Alternate interior angles are congruent. (A.I. are ) An angle bisector divides an angle into two congruent angles. (Def n. Angle Bisector) Perpendicular lines form right angles (Def n Perp. Lines) All right angles are congruent (Right Angles are ) Corresponding angles are congruent. (Corr. Angles are ) A midpoint divides a segment into two congruent parts. (Def n. of Midpoint) The radii of a circle are congruent. (Radii are ) The base angles of an isosceles triangle are congruent. (Base Angles Th m). A median is a segment that connects a vertex and a midpoint. (Def n of Median) An altitude passes through a vertex and makes a right angle with one of the sides of a triangle. (Def n of Altitude) Corresponding parts of congruent triangles are congruent. (CPCTC) 1

42 Here we go!!! b) c) d) B F L T A M C G H K M is the midpoint of AC R P S P is the midpoint of RS List at least 3 congruencies: List at least 2 congruencies: List at least 3 congruencies: 2

43 e) f) T U M R P Q Q P A P is the midpoint of QA UD bisects QUA UD QA Give me 3 congruencies!!! D List at least 3 congruencies! 3

44 g) h) i) T R Y W L X M A E Y E W Z M is the center of the circle. List two pairs of congruent segments and one pair of congruent angles: WX is a median of triangle WYZ List at least 1 congruence.: Three congruencies please!!! 4

45 j) k) A R W M N B C S U List 4 congruencies: M is the midpoint of AB N is the midpoint of AC T Given: RST WUV There are six congruencies this time!!! V 5

46 l) There are at least 3 congruencies here A H Z O P m) Z Y n) D Name only one congruence! I G R T A YI is an altitude TA bisects DAR List 3 congruencies. 6

47 l) m) n) I Y A X K P W Z S O C WX is a median of triangle WYZ U KAP KUP IO is the perpendicular bisector of SC. (so point O is the midpoint ) List at least 1 congruence: Five congruencies please!!! Two congruencies are required!!! 7

48 Last two figures: o) M P p) R P Q S U This is tricky!!! There are two congruencies. S There are 4 congruencies for you to find! 8

49 Unit 4 - Lesson 5: Triangle Congruence Proofs Don t just say it Prove it!!! Proof #1: L Given: G is the midpoint of AL. A L Prove: AGN LGE N G E A b) Write a two column proof. 1

50 Proof #2: C B Given: M is the midpoint of AB AD CB M Prove: ADM BCM A D b) Write a two column proof. 2

51 Proof #3: Given: EI bisects KET K I T K T Prove: KIE TIE E b) Write a two column proof. 3

52 Proof #4: E Given: G is the center of the circle Prove: EGF KGH H G F K b) Write a two column proof. Are You still there??? Writing proofs takes practice 4

53 Proof #5: Proof #6: Given: DT RT DA RA D Given: AC bisects BAD and BCD A Prove: DAT RAT T A Prove: CBA CDA B D R C 5

54 Proof #7: Proof #8: Given: OM bisects LMN LOM NOM L M Given: Y is the midpoint of BD A E Prove: LOM NOM O AB BD and ED BD AB ED B C D Prove: ABC EDC N 6

55 7

56 Lesson 6 - Four CPCTC Proofs Name: Proof #1: Proof #2: Given: P S O is the midpoint of PS R O S Given: E is the midpoint of MY XM XY M E Y Prove: R Q Prove: MXE YXE P Q X

57 Proof #3: R Proof #4: Given: REA REP AE PE Given: DA CB AB DC A B Prove: AR PR E Prove: AB DC D C A P

58 CPCTC Cut and Paste #1 Given: Name: A BD bisects ABC and ADC Prove: AD CD B D C Statement Reason

59 3 4 BD BD 1 2 BAD BCD BD bisects ABC and ADC AD CD Definition of angle bisector CPCTC A.S.A Definition of angle bisector Reflexive Property Given

60 CPCTC Cut and Paste #2 Name: Given: CDA and CDB are right angles C D is the midpoint of AB Prove: CA CB A D B Statement Reason

61 CDA CDB S.A.S CD CD AD BD Given CA CB Definition of Midpoint CDA and CDB are right angles ACD BCD Reflexive Property CPCTC Given D is the midpoint of AB Right angles are congruent

62 Lesson 6D Properties of Parallelograms Name: I Can Draw a Parallelogram! Use the grid to draw 4 different parallelograms. 1

63 2. Measure the sides and angles of the parallelogram below. Check the properties that seem to be true for all parallelograms. B C A D Sides and Angles Both Pairs of opposite angles congruent. Exactly 1 pair of opposite angles are congruent Diagonals The diagonals are perpendicular to each other. The diagonals are congruent. Both Pairs of opposite sides are congruent. The diagonals bisect each other. Exactly 1 pair of opposite sides are congruent. Adjacent Angles are supplementary. 2

64 3. Prove it!!! For parallelograms, opposite angles are congruent. Given: Quadrilateral ABCD is a parallelogram AB DC and AD BC A B Prove: A C D C b) Write a two column proof. 3

65 4. For parallelograms, opposite sides are congruent. Given: Quadrilateral ABCD is a parallelogram AB DC and AD BC A B Prove: AB DC D C b) Write a two column proof. 4

66 5. The diagonals of a parallelogram bisect each other. Given: Quadrilateral ABCD is a parallelogram AB DC and AD BC A P B Prove: AP CP D C b) Write a two column proof. 5

67 6. Apply the Properties! a) Find x: b) c) 6

68 Lesson 6E: Properties of Kites Name: 1. What is a kite? Vocabulary Word 2. Which properties do kites have? Put a check mark by the properties that seem to be true for kites. i) Both pairs of opposite angles are congruent. ii) Exactly one pair of opposite angles are congruent. iii) Adjacent angles are supplementary. iv) Both pairs of opposite sides are congruent. v) Exactly one pair of opposite sides are congruent. vi) Diagonals are perpendicular. vii) Both diagonals bisect each other. viii) One diagonal bisects the other. ix) The diagonals are congruent 1

69 2. Draw Some! Use the grid below to draw 4 different kites. 2

70 3. Prove one of the properties!!! For kites, one pair of opposite angles are congruent. B Given: Quadrilateral ABCD is a kite with AB BC and AD CD A C Prove: A C D b) Write a two column proof. c) In the figure above, why is ABD CBD??? (This means that diagonal BD bisects ABC ) 3

71 4. Does our drawing method really work? How do we know that we get a kite? B Given: AC BD AM CM A M C Prove: Quadrilateral ABCD is kite. D Statements Reasons 1. AC BD AM CM AMB CMB BM BM ABM CMD AB CB AMD CMD MD MD AMD CMD AD CD Quadrilateral ABCD is a kite. 11.!!!CPCTC Rocks!!! 4

72 Lesson 7A: Constructions and Proofs Name: Part 1: Angle Bisectors 1. Use a compass and straightedge to construct the angle bisectors of the angles shown below. Pictorial summary of steps: 1. Start with the original angle. 2. Draw an arc and mark where it intersects the angle. 3. Use the two new points to draw 2 more arcs. 4. Draw a ray through where the two arcs intersect. 1

73 2. Question: How is the construction of angle bisectors related to congruent triangles? a) Construct the angle bisector for the angle below. b) Explain how congruent triangles can be used to explain why the construction works. 2

74 3. Prove it!!! Given: AB AC BP CP P Prove: AP bisects BAC B C A 3

75 Part 2: Perpendicular Bisectors 4. Construct the perpendicular bisectors of the segments below. I C B S T Pictorial summary of steps: E P Q P Q Q P M Q P 4

76 5. Another Question: How can we prove that this construction works??? a) Construct the perpendicular bisector for the segment below. b) Explain how two different pairs of congruent triangles will be used to prove that this construction works. 5

77 6. Let s prove it! Fill in the missing reasons. Given: AC CB BD DA C Prove: P is the midpoint AB and DC AB A P B D Statements Reasons 1. AC BC DA DB CD CD CAD CBD ACP BCP CP CP ACP BCP AP BP P is the midpoint of AP APC BPC mapc mbpc mapc mapc Substitution 13. mapc Arithmetic 14. APC is a right angle DC AB 15. 6

78 Lesson 7B Two Classic Proofs Name: Give us A Formal Proof Given: A A triangle with vertices and angles labeled as shown in 3 the diagram. Prove: m1 m2 m B 1 2 C Statement Reason 1. A triangle with vertices and angles labeled as shown in the diagram. 1. Given

79 2. There is something that is always true about vertical angles. What is it??? 40??? Oh yeah? Prove it!!! a b d c Given: Two intersecting lines with the angles labeled as shown in the figure. Prove: a=c Statement Reason 1. Two lines intersect each other to form four angles as shown in the figure. 1. Given 2. ab bc ab b c abbbc b a c 6. Reason Bank: Transitive Property of Equality Given Subtraction Property of Equality Simplification Angles that form a linear pair have measures that add to 180⁰

80 Lesson 7C Euclid Cut and Paste Name: Given: A A triangle with vertices and angles labeled as shown in 3 the diagram. Prove: m1 m2 m B 1 2 C Statement Reason

81 Angles that form a straight line add to 180⁰ Draw a line through point A that is parallel to BC. (Label the new angle on the left 4 and the new angle on the right 5 ) m1 m3 m2 180 m4 m 1 A triangle with vertices and angles labeled as shown in the diagram. Alternate Interior Angles are Congruent Substitution The Parallel Postulate Alternate Interior Angles are Congruent m4 m3 m5 180 m5 m 2 Given

82 Lesson 7D Vertical Angles Cut and Paste You think it s true? Prove it!!! Name: a b d c Given: Two intersecting lines with the angles labeled as shown in the figure. Prove: a=c Statement Reason

83 Angles that form a linear pair have measures that add to 180⁰ Transitive Property of Equality bc 180 abbbc b a c Two lines intersect each other to form four angles as shown in the figure. Angles that form a linear pair have measures that add to 180⁰ Simplification Given ab 180 Subtraction Property of Equality ab b c

84 Our Final Task Name: 1. The following proof demonstrates that if corresponding angles are congruent, then same-side interior angles are supplementary. Use the answers from the reason bank to supply the missing reasons in the proof. Given: Prove: 4 8 p and 6 are supplementary q Statement Reason m4 m m6 m m6 m and 6 are supplementary 5. Reason Bank: Definition of Supplementary Angles Substitution Given Definition of Congruence Angles Angles that form a linear pair have measures that sum to 180⁰

85 2. The following proof demonstrates that if corresponding angles are congruent, then same-side interior angles are supplementary. Use the answers from the reason bank to supply the missing reasons in the proof. Given: Prove: 1 5 p q Statement Reason Reason Bank: Vertical Angles are Congruent Transitive Property of Congruence Given

86 17 Reasons that you can use in your proofs. (These are the common ones ) 1. Given. (Everybody s favorite!!!) 2. Vertical angles are congruent. 3. Corresponding angles are congruent. (They have to be formed by a transversal that crosses parallel lines.) 4. Alternate interior angles are congruent. (They have to be formed by a transversal that crosses parallel lines.) 5. Definition of Midpoint. (A midpoint divides a segment into two congruent parts.) 6. Definition of Angle Bisector. (An angle bisector divides an angle into two congruent angles.) 7. Definition of Perpendicular lines. (Perpendicular lines form right angles.) 8. All right angles are congruent. 9. The radii of a circle are congruent. 10. Definition of a Median. (A median is a segment that connects a vertex and a midpoint.) 11. Definition of an Altitude. (An altitude passes through a vertex and makes a right angle with one of the sides of a triangle.) 12. The base angles of an isosceles triangle are congruent. (Base Angles Theorem) 13. If the base angles of a triangle are congruent, then the triangle is isosceles (Converse of Base Angles Theorem) 14. The sides of an equilateral triangle are congruent. 15. SSS, SAS AAS, ASA 16. CPCTC (Corresponding Parts of Congruent Triangles are Congruent). 17. Reflexive Property (Every segment is congruent to itself)