1. Warm-Up: Perspective drawing
To draw parallel lines use perspective drawing.
Draw your own perspective drawing to describe your real world .

2. 4.1 Congruence and Transformations
Objective: Draw, identify, and describe transformations in the coordinate plane.
Use the properties of rigid motions to determine whether figures are congruent and prove figures congruent.

3. Discover Transformations
Graph each set of points, color code each set.
Determine the transformation: Translation, Dilation, Rotation or Reflection
Analyze any patterns
Examine if the shapes are congruent. Explain.

4. Practice:
P. 219 # 1, 2, 3- 6, even.
Determine whether the polygons are congruent. Support your answer by describing a transformation.

5. Discuss & Draw Triangles
1. Draw 3 real world examples of triangles. A. B. C.

6. Practice:
P. 219 # 1, 2, 3- 6, even.
Determine whether the polygons are congruent. Support your answer by describing a transformation.

7. By their ANGLES and SIDES
Classify Triangles
By their ANGLES and SIDES

8. Real World Example

9. Classifying Triangles
Polygon: 3-sided close figure
Name 3 sides:
AB, BC, & AC
Name 3 angles:
< A or < CAB
< B or < ABC
< C or < BCA
Name 3 vertices:
A, B, C

10. Classifying triangles
Angle Sum Addition
Equal to 180º
(<1 + <2 + <3 = 180º)

11. Classify Triangles by “Angles”

12. Acute triangle
ALL Angles are acute.

13. Obtuse triangle
ONE Angle is obtuse.

14. Right Triangle
ONE right angle

15. Equiangular triangle
ACUTE Triangle
ALL Angles are congruent

16. Classify Triangles By “SIDES”

17. Scalene triangle
NO Sides are congruent

18. Isosceles triangles
2 sides are congruent

19. Equilateral triangle
ALL Sides are congruent

20. Real World Example

21. Practice Worksheet Part 1 # 1-8 Classification
Part 2  #1-12 Missing Measure
Practice p. 227 #1-17

22. Warm-up: 1. M: (x, y)  ( x + 5, y – 4)
G(4, -1), H(7, 3), I(7, -1)
Name coordinates of image
Graph:
Transformation:
Congruent or not? Explain.
2. Classify by sides and angles; find x. X°

23. Real world example

24. Classify Triangles
Practice

25. Identify Triangles A. Equilateral B. Isosceles C. Scalene D. Right
Answer: C

26. Identify Triangles A. Equilateral B. Isosceles C. Scalene D. Right
Answer: A

27. Identify Triangles A. Equilateral and acute B. Isosceles and right
C. Obtuse and Scalene
D. Right and Equilateral
Answer: B

28. Identify Triangles A. Equilateral and right B. Isosceles and obtuse
C. Scalene and Right
D. Scalene and obtuse
Answer: D

29. Identify Triangles A. Acute B. Obtuse C. Equiangular D. Right
Answer: B

30. Identify Triangles A. Equilateral B. Acute C. Scalene D. Right
Answer: D

31. Identify Triangles A. Equilateral B. Isosceles C. Scalene D. Obtuse
Answer: C

32. Identify Triangles A. Equilateral B. Isosceles C. Acute D. Right
Answer: C

33. Identify Triangles A. Equilateral & Obtuse B. Isosceles & right
C. Acute & Equiangular
D. Right & scalene
Answer: D

34. Identify Triangles A. Equilateral B. Obtuse C. Acute D. Isosceles
Answer: B

35. Practice: Name The triangle by sides and angles
1. 45, 90, 45
Answer: Right, Isosceles
2. 100, 30, 50
Answer: Obtuse, Scalene
3. 60, 60, 60
Answer: Acute, Equilateral, Equiangular
4. 33, 77, 70
Answer: Acute, Scalene

36. Practice
Check p. 227 #1-17

37. Measuring Angles in triangles: Trio Activity
Return questions and answers & check work

38. Warm-Up:
1. Find x.
65º º xº
80º
2. Find x.
50º º xº
62º º

39. Practice: Name The triangle
1. 35, 60, 85
Answer: Acute, Scalene
2. 110, 25, 45
Answer: Obtuse, Scalene
3. 45, 90, 45
Answer: Right, Isosceles
4. 60, 60, 60
Answer: Acute, Equilateral, Equiangular

40. Check: Trio Activity

41. corollary interior exterior interior angle exterior angle
Vocabulary
corollary
interior
exterior
interior angle
exterior angle
remote interior angle

42. 4.3 Angle Relationships in Triangle

43. Angle Relationships in Triangles
A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

44. Angle Relationships in Triangles
The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure.
4 is an exterior angle.
The remote interior angles of 4 are 1 and 2.
Exterior
Interior
3 is an interior angle.

45. Angle Relationships in Triangles

46. Angle Relationships in Triangles

47. Angle Relationships in Triangles Practice:
Pairs:
P. 235 # 4, 5, 6,
Draw figures
Substitute measures
Analyze given info
Compute measures

48. Quiz: Classify triangles and Measure Angles
No talking during Quiz
Take your time
Good LUCK!
14. M: (x, y) (y, -x)
L (3, 1)
M (3, 4)
N (5, 4)
O (5, 1)
Questions 14 & 15:
a) Name coordinates of image,
b) Name Transformation,
c) Determine if figures are congruent, explain.
15. M: (x, y) (x-1, y+1)
N (1, -2)
O (0, 4)
P (2, 4)

49. cont  Quiz: Classify triangle and Measure Angles
No talking during Quiz
Take your time
Good LUCK!
14. M: (x, y) (y, -x)
L (3, 1)
M (3, 4)
N (5, 4)
O (5, 1)
Questions 14 & 15:
a) Name coordinates of image,
b) Name Transformation,
c) Determine if figures are congruent, explain.
15. M: (x, y) (x-1, y+1)
N (1, -2)
O (0, 4)
P (2, 4)

50. Warm-Up:
1. After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ. 2.

51. check Quiz: Classify triangles and Measure Angles

52. Congruent triangles ABC = NOP
Sketch, label, & mark all congruent parts.
4.4
Stand Up: Form Student Triangles
Triangle have to be congruent
Each student represent one unit
Cannot Talk
Side 1 vs Side 2
JKL = XYZ
ABC = NOP

53. Congruent triangles
Define: Triangles that are the same size and shape.
Each triangle have 6 corresponding parts.
3 corresponding angles
corresponding sides

54. SKETCH, LABEL, & MARK all congruent parts.
ABC = JKL
B K
A C J L
Angles Sides

55. Practice Activity Worksheet Page 242 #1-10 Page 246
Congruent Triangles forms Origami

56. Warm-up: 1. ∆ABC  ∆JKL and AB = 2x + 12. JK = 4x – 50. Find x and AB.
Given that polygon MNOP  polygon QRST, identify the congruent corresponding part.
2. NO  ____ T  ____

57. Complete & check P. 242 #1-10 Activity Page 246
Congruent Triangles forms Origami

58. Prove triangles are Congruent
List postulates

59. objectives
Apply SSS, SAS, ASA, and AAS to construct triangles and solve problems.
Prove triangles congruent by using SSS, SAS, ASA, and AAS.
Use CPCTC to prove parts of triangles are congruent.

60. VOCABULARY
triangle rigidity
included angle
included side
CPCTC

61. Prove Triangles Are Congruent
In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent.
The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape.

62. Prove Triangles Are Congruent Triangle Congruence: SSS
For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate.

63. Triangle Congruence: SSS
Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.
Remember!

64. Triangle Congruence: SSS
It is given that AC  DC and that AB  DB. By the Reflexive Property of Congruence, BC  BC. Therefore ∆ABC  ∆DBC by SSS
Use SSS to explain why ∆ABC  ∆DBC.

65. Triangle Congruence: SSS
Use SSS to explain why ∆ABC  ∆CDA.
It is given that AB  CD and BC  DA.
By the Reflexive Property of Congruence, AC  CA.
So ∆ABC  ∆CDA by SSS.

66. Triangle Congruence: SAS
An included angle is an angle formed by two adjacent sides of a polygon.
B is the included angle between sides AB and BC.

67. Triangle Congruence: SAS
It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent.

68. Prove Triangles Are Congruent Triangle Congruence: SAS

69. Triangle Congruence: SAS
The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.
Caution

70. Triangle Congruence: SAS
The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ  ∆VWZ.
It is given that XZ  VZ and that YZ  WZ. By the Vertical s Theorem. XZY  VZW. Therefore ∆XYZ  ∆VWZ by SAS.

71. Triangle Congruence: SAS
Use SAS to explain why ∆ABC  ∆DBC.
It is given that BA  BD and ABC  DBC. By the Reflexive Property of , BC  BC. So ∆ABC  ∆DBC by SAS.

72. Triangle Congruence: SAS
The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angles, you can construct one and only one triangle.

73. Triangle Congruence: SAS
Show that the triangles are congruent for the given value of the variable.
∆STU  ∆VWX, when y = 4.
ST  VW, TU  WX, and T  W.
∆STU  ∆VWX by SAS.

74. Triangle Congruence: SSS & SAS
Which postulate, if any, can be used to prove the triangles congruent?

75. Triangle Congruence: ASA
An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

76. Prove Triangles Are Congruent Triangle Congruence: ASA

77. Triangle Congruence: ASA
Determine if you can use ASA to prove NKL  LMN. Explain.
By the Alternate Interior Angles Theorem. KLN  MNL. NL  LN by the Reflexive Property. No other congruence relationships can be determined, so ASA cannot be applied

78. Prove Triangles Are Congruent Triangle Congruence: AAS
You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle- Angle-Side (AAS).

79. Triangle Congruence: sss, sas, asa, and AAS
Identify the postulate or theorem that proves the triangles congruent.

80. Prove Triangles Are Congruent Triangle Congruence: CPCTC
CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent.

81. Triangle Congruence: CPCTC
SSS, SAS, ASA, and AAS use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent.
Remember!

82. Triangle Congruence
Practice 3.5 – 3.7
SSS
SAS
ASA
AAS
CPCTC

83. Warm-Up
Isosceles powerpoint

84. Practice:Triangle Congruence-sss, sas, asa, AAS, and CPCTC
Analyze the congruent parts
Match congruent triangles
Write the congruent statements
Identify the postulate or theorem that proves the triangles congruent

85. Check: Triangle Congruence
Practice 3.5 – 3.7
SSS
SAS
ASA
AAS
CPCTC

86. Analyze isosceles triangles
Identify congruent parts of triangles
Online powerpoint

87. Isosceles and Equilateral Triangles
Practice
Worksheet
P. 288 #1-10

88. Warm-up:
1. If XYZ is congruent to PRS, then write the 6 congruence statements.
2. Name theorem Find x & y 32
x y

89. Isosceles and Equilateral Triangles
Check Practice
Worksheet
P. 288 #1-10

90. Quiz: Take your Time Label all parts NO talking DO YOUR BEST!
When finish, complete study guide.
TEST: Monday

91. Warm-Up: solve
1. In XYZ, m< X = 3x – 31, m< Y = 5x+8, and m< Z = 4x -13, find each angle.
A ACD is equilateral.
m< B = 30º. Find m< BAC.
B C D

92. Practice Chapter 4 Study Guide Ask Questions
Label congruent parts on triangles
Analyze figures

93. Focused on finding measurements.
Practice before Test
Focused on finding measurements.

94. Find the angle measures.yº
answer: x= 40º y= 100º
xº º

95. Find the angle measures.
answer: x = 6 and y =16 y
7xº º

96. Find the angle measures.
answer: x =12 and y = 60º
5xº yº

97. Find the angle measures.
72º
45º xº
Answer: = 117

98. Find the angle measures.54
3xº
Answer: x = 18

99. Find the angle measures.xº
answer: x = 80
52º º

100. Find the angle measures.
8xº
7xº
Answer: 15x = 90…. X = 6

101. Find the angle measures.
answer: x = 120º xº

102. Find the angle measures.
40º xº
Answer: x = 110º

103. Find x & the angle measures.
B C
Answer: x = <A = 70 <B = 90 <C = 20

104. Chap 4 test: Identifying congruent triangles
NO talking
Take your time
LABEL, LABEL, LABEL
Show all your work
Read, read, & reread each question
GOOD LUCK!
#32 Construction
For full credit,
Classify triangle
List the angle measurements
Use protractor and straightedge
For additional credit, construct a 2nd triangle using compass and straightedge