1. Geometry 1 Unit 4 Congruent Triangles
Casa Grande Union High School
Fall 2008

2. Geometry 1 Unit 4
4.1 Triangles and Angles

3. Classifying Triangles by Sides
Equilateral- all 3 sides are congruent
Isosceles- at least 2 sides are congruent
Scalene-
No sides congruent

4. Classifying Triangles by Angles
Acute Triangle- All angles
are less than 90°
Obtuse Triangle- 1 angle
greater than 90 °
Right Triangle-
1 angle measuring 90°

5. Classifying Triangles
Example 1
Name each triangle by its sides and angles
A B. C.

6. Parts of triangles Vertex (plural vertices) Adjacent sides
The points joining the sides of a triangle
Adjacent sides
Sides sharing a common vertex
Side AB is adjacent to side BC A B C

7. Interior angle Exterior angle Angle on the inside of a triangle
Angle outside the triangle that is formed by extending one side A B C
Interior angle
Exterior angle

8. Triangle Sum Theorem
The sum of the three interior angles of a triangle is 180º

9. Exterior Angle Theorem
The measure of the exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles.
Example: m∠1=m∠A+ m∠B B A C 1

10. Example 2
Find the measure of each angle.
2x + 10 x
x + 2

11. Example 3 Given that ∠ A is 50º and ∠B is 34º, what is the measure of
∠BCD?
What is the measure of ∠ACB? D A B C

12. Right triangle vocabulary
Legs
Sides that form the right angle
Hypotenuse
Side opposite the right angle
Legs
Hypotenuse

13. Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary. m∠A+ m∠B = 90°

14. Example 4
A. Given the following triangle, what is the length of the hypotenuse?
B. What are the length of the legs?
C. If one of the acute angle measures is 32°, what is the other acute angle’s measurement? 13 12 5

15. Legs Base Base Angles Vertex Angle
The two congruent sides of an isosceles triangle.
Base
The noncongruent side of an isosceles triangle.
Base Angles
The two angles that contain the base of an isosceles triangle.
Vertex Angle
The noncongruent angle in an isosceles triangle.

16. Isosceles triangle vocabulary
Legs
Base Angles
Vertex Angle
Base

17. Example 5 A B C
75º 15 7
A. Given the following isosceles triangle, what is the measurement of segment AC?
B. What is the measurement of angle A?

18. Example 6
Find the missing measures
80°
53°

19. Example 7 Given: ∆ABC with mC = 90° Prove: mA + mB = 90° Statement
Reason
1. mC = 90°
2. mA + mB + mC = 180°
3. mA + mB + 90° = 180°
4. mA + mB = 90°

20. 4.2 Congruence and Triangles
Geometry 1 Unit 4
4.2 Congruence and Triangles

21. Congruent Figures Congruent Figures
Figures are congruent if corresponding sides and angles have equal measures.
Corresponding Angles of Congruent Figures
When two figures are congruent, the angles that are in corresponding positions are congruent.
Corresponding Sides of Congruent Figures
When two figures are congruent, the sides that are in corresponding positions are congruent.

22. Congruent Figures For the triangles below, ∆ABC ≅ ∆PQR
The notation shows congruence and correspondence.
When writing congruence statements, be sure to list corresponding angles in the same order. A B C
Corresponding Angles
Corresponding Sides
A ≅ P
AB ≅ PQ
B ≅ Q
BC ≅ QR
C ≅ R
CA ≅ RP P Q R

23. Example 1
Complete the congruence statement for the two given triangles:
DEF
What side corresponds with DE?
What angle corresponds with E? S V T D E F

24. Example 2 In the diagram, ABCD ≅ KJHL Find the value of x.
Find the value of y. J H K L
(5y – 12)°
(4x – 3) cm A B D C
9 cm
6 cm
86°
91°
113°

25. Third Angles Theorem
If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the third angles are also congruent.

26. Example 3 (6y – 4)° Q R A 85° (10x + 5)° P 50° C B
Given ABC  PQR, find the values of x and y. A B C P Q R
85°
50°
(6y – 4)°
(10x + 5)°

27. Example 4
Decide whether the triangles are congruent. Justify your answer. F H E G J
58°

28. Example 5 Given: MN ≅ QP, MN || PQ, O is the midpoint of MQ and PN.
Prove: ∆MNO ≅ ∆QPO
Statements
Reasons 1. 2.
Alt. Interior Angles Theorem 3.
Vertical Angles Theorem
4.O is the midpoint of MQ and PN 5.
Def of Midpoint
6. ∆MNO ≅ ∆QPO P M O Q N

29. 4.3 Proving Triangles are Congruent: SSS and SAS
Geometry 1 Unit 4
4.3 Proving Triangles are Congruent: SSS and SAS

30. Warm-Up
Complete the following statement
BIG  B I G R A T

31. Definitions Included Angle Included Side
An angle that is between two given sides.
Included Side
A side that is between two given angles.

32. Example 1
Use the diagram. Name the included angle between the pair of given sides. K P J L

33. Triangle Congruence Shortcut
SSS
If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

34. Triangle Congruence Shortcuts
SAS
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

35. Example 2 U S T V STW   W Complete the congruence statement.
Name the congruence shortcut used. S T U V W
STW  

36. Example 3 H L M I N HIJ  LMN J
Determine if the following are congruent.
Name the congruence shortcut used. J H I L M N
HIJ  LMN
No. Triangle HIJ is congruent to MNL

37. Example 4 A B C O R X XBO   Complete the congruence statement.
Name the congruence shortcut used. B O X C A R
XBO  

38. Example 5 P T S Q Complete the congruence statement.
Name the congruence shortcut used.
SPQ   S P Q T

39. Constructing Congruent Triangles
Construct segment DE as a segment congruent to AB
Open your compass to the length of AC. Place the point of your compass on point D and strike an arc.
Open the compass to the width of BC. Place the point of your compass on E and strike an arc. Label the point where the arcs intersect as F. A C B

40. Example 6 Given: AB ≅ PB, MB ⊥ AP Prove: ∆MBA ≅ ∆MBP Statements
Reasons
1. MB ⊥ AP 2.
Perpendicular lines form right angles 3.
Right angles are congruent
4. AB ≅ PB
5. MB ≅ MB 6.

41. Example 7 Use SSS to show that ∆NPM ≅∆DFE N(-5, 1) P(-1, 6) M(-1, 1)

42. 4.4 Proving Triangles are Congruent: ASA and AAS
Geometry 1 Unit 4
4.4 Proving Triangles are Congruent: ASA and AAS

43. Triangle Congruence Shortcuts
ASA
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

44. Triangle Congruence Shortcuts
AAS
If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent.

45. Example 1 Q U A D QUA   Complete the congruence statement.
Name the congruence shortcut used. Q U A D
QUA  

46. Example 2 Complete the congruence statement.
Name the congruence shortcut used.
RMQ   M R Q N P

47. Example 3 A B C F E D ABC  FED
Determine if the following are congruent.
Name the congruence shortcut used. A B C F E D
ABC  FED

49. Example 4 Given: B ≅C, D ≅F; M is the midpoint of DF.
Prove: ∆BDM ≅∆CFM
Statements
Reasons 1. 2. 3.
Def of Midpoint 4.

50. 4.5 Using Congruent Triangles
Geometry 1 Unit 4
4.5 Using Congruent Triangles

51. Warm-up
State which postulate or theorem you can use to prove that the triangles are congruent.
Then, write the congruence statement. C G H S

52. Example 1 Given: NO is parallel to MP, MN is parallel to PO
Prove MN = OP
(Prove Δ MNO Δ OPM)
Mark the given information first
Then, mark the deduced information
Statements
Reasons
1.NO||MP, MN|| PO
1. Given 2. 3. 4. 5. 6.

53. Example 2 H J L K
Given: HJ || KL, JK || HL
Prove: LHJ ≅ JKL

54. Example 3 Given: MS || TR, MS ≅ TR Prove: A is the midpoint of MT.
Statements
Reasons 1. 2. 3. 4. 5. 6.

55. 4.6 Isosceles, Equilateral, and Right Triangles
Geometry 1 Unit 4
4.6 Isosceles, Equilateral, and Right Triangles

56. Warm-Up 1
Find the measure of each angle.
90°
30°
60° a b

57. Warm-Up 2
Find the measure of each angle.
110
150
90

58. Isosceles triangle theorems
Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are also congruent.
Converse of the Base Angles theorem
If two angles of a triangle are congruent, then the sides opposite them are also congruent

59. Example 1
35° x

60. Example 2
15° b a

61. Example 3
Find each missing measure
63°
10 cm m n p

62. Equilateral Triangles
If a triangle is equilateral, then it is equiangular.
If a triangle is equiangular, then it is equilateral.

63. Hypotenuse-Leg (HL)
If the hypotenuse and the leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent.

64. Example 4
Find the value of x
12 in
2x in

65. Example 5
Find the value of x and y. y x

66. Example 6
Find the value of x and y.
75° x° y°

67. 4.7 Triangles and Coordinate Proof
Geometry 1 Unit 4
4.7 Triangles and Coordinate Proof

68. Warm-up What is the midpoint formula? What is the distance formula?
What are some postulates and theorems you have learned about triangles this chapter?

69. Vocabulary Coordinate Proof
A proof involving placing geometric figures on a coordinate plane.
Uses the midpoint formula, distance formula, postulates and theorems to prove statements about the figure

70. Placing Figures in a Coordinate Plane
Complete the activity on p. 243 individually.
Compare your results to those of your partners.
What did you learn?

71. Example 1 A right triangle has legs of 3 units and 4 units.
Place the triangle on a coordinate grid. Label the vertices, then find the length of the hypotenuse. 3 4

72. Example 2 In the diagram, ΔABO ≅ ΔCBO.
Find the coordinates of point B.
C(10,0)
A(0,10)
O(0,0) B

73. Example 3 Write a plan to prove that OU bisects TOV. U(0,5) T(-3,5)

74. Example 4
Find the coordinates of P. P
N(h,0)
M(0,k)

75. Constructions review Duplicate the given triangle.
Write the steps that you used to construct the new triangle