math

1. Math 2 Warm Up
In the Geometry Textbook,
p. 177 #1-13
p. 178 #4, 6-9
Turn in after we check!!

2. Math 2 Warm Up

3. Math 2 Warm Up

4. Unit 2: Congruence & Similarity
“Congruent Figures”
Objective: To recognize congruent figures and their
corresponding congruent parts.
congruent figures: two or more figures (segments,
angles, triangles, etc.) that have the
“same shape” and the “same size”.
symbol for congruent: ≅
congruent polygons: two polygons are congruent if
all the pairs of corresponding sides and
all the pairs of corresponding angles are
congruent.

5. Examples of Congruent Polygons
"Slide", "Flip", "Turn"...Translate, Reflection, Rotate
If two figures are congruent, then one figure can be mapped
onto the other one by a one or series of “rigid motions”!

6. How do I know if sides or angles are congruent?
1. If figures are drawn to scale, then measure the corresponding
angles and measure the corresponding sides.
2. If figures are not drawn to scale, by special markings.
Side Markings (“ticks”)
Angle Markings (“hoops”)

7. Example: Given: ∆REM ≅ ∆FEM
List the corresponding congruent parts.
“Reflexive Property of Congruence”
"If two figures share the same side or the same angle, then the
shared sides or shared angles are congruent to each other."
R
E
F
M

8. Example: Given: ∆RTV ≅ ∆LTC
List the corresponding congruent parts.
V
R
T
C
L

9. Example: Given: ∆RTV ≅ ∆CTL
List the corresponding congruent parts.
V
R
T
C
L

10. Example: Given: ΔPQR ≅ ΔSTU,
m∠S = 36°, m∠Q = 27°,
PR = 7 cm, and TS = 18 cm
1. Find the m∠T.
2. Find the m∠U.
3. Find the SU.
4. Find the QP.

11. Proving Triangles Congruent
Example: Prove: ∆EAB ≅ ∆CDB .

12. Proving Triangles Congruent
Example: Prove ∆PQR ≅ ∆PSR.
“Third Angle Theorem”
"If two angles of one triangle are congruent two angles of
another triangle, then the third angles are congruent."

13. Proving Triangles Congruent
Example: Prove ∆ABC ≅ ∆EBC .
“Right Angle Theorem”
“All right angles are congruent."

14. Given: AE ≅ DC, EB ≅ CB, B is the midpoint of AD, ∠E≅ ∠C
Prove: ∆ABE ≅ ∆DBC

15. Example: Given: ∆ABC ≅ ∆QTJ
List the corresponding congruent parts.
A
B
C
J T
Q

16. ΔABC ≅ ΔHFC
A
B
C H
F
1. List the corresponding congruent sides.
2. List the corresponding congruent angles.

18. End of Day 1

19. Objective: To prove two triangles congruent
by using the sides of the triangles.
NCTM Illuminations:
Triangle Congruence
Unit 2: Congruency & Similarity
“Proving Triangles Congruent: Sides”

20. Side-Side-Side

21. Can you prove the two triangles congruent?

22. Side-Angle-Side

23. Can you prove the two triangles congruent?

24. Example 1
Given: M is the midpoint of 𝐗𝐘, AX ≅ 𝐀𝐘
Prove: ∆AMX ≅ ∆AMY
A
X Y
M

25. Example 2
Given: RS bisects ∠GSH, SG ≅ 𝐒𝐇
Prove: ∆RSG ≅ ∆RSH R
G H
S

26. Example 3
Given: O is the midpoint of 𝐏𝐑 and 𝐄𝐖
Prove: ∆POW ≅ ∆ROE
P
R
W
E
O

27. Example 4
Given: 𝐑𝐒 ǁ 𝐓𝐊, 𝐑𝐒 ≅ 𝐓𝐊
Prove: ∆RSK ≅ ∆TKS
R
K T
S

28. Example 5*
Given: 𝐀𝐁 is the perpendicular bisector of 𝐗𝐘
Prove: ∆AXB ≅ ∆AYB
A
X Y
B

30. End of Day 2

32. Unit 2: Congruency & Similarity
“Proving Triangles Congruent: Angles”
Objective: To prove two triangles congruent by
using the angles of the triangles.
NCTM Illuminations:
Triangle Congruence

33. Angle-Side-Angle

34. Angle-Angle-Side

35. Can you prove the two triangles congruent?
1. Given: HI ≅ 𝐐𝐑,
∠G ≅ ∠P, ∠H ≅ ∠Q
2. Given: ∠LPA ≅ ∠YAP,
∠LAP ≅ ∠YPA
L
A
Y
P
G
H
I
P
R
Q

36. Can you prove the two triangles congruent?
3. Given: ∠B ≅ ∠C,
AX bisects ∠BAC
4. Given: TR ǁ AL
Y
R
A
T
L
A
B C
X

37. Example 5
Given: P is the midpoint of AB , ∠A ≅ ∠B
Prove: ∆APX ≅ ∆BPY
P
B
X
Y
A

38. Example 6
Given: PR bisects ∠SPQ, ∠S ≅ ∠Q
Prove: ∆SRP ≅ ∆QRP
R
P
Q
S

39. Example 7
Given: XQ ǁ TR, X𝑹 bisects QT
Prove: ∆XMQ ≅ ∆RMT
T
Q
R
X
M

40. Can you prove the two triangles congruent?
3. Given: ∠B ≅ ∠C,
AX bisects ∠BAC
4. Given: ∠YRT ≅ ∠YAL,
∠RTY ≅ ∠YLA
Y
R
A
T
L
A
B C
X

42. End of Day 3

43. Unit 2: Congruency & Similarity
“Proving Right Triangles Congruent”
Objective: To prove two triangles congruent
by using the sides of the triangles.
Leg
Leg

44. Hypotenuse-Leg
Theorem 4-6 Hypotenuse – Leg (HL)
If the hypotenuse and a leg of one right triangle are
congruent to the hypotenuse and a leg of another right
triangle, then the triangles are congruent

45. Given: △PQR and △XYZ are right triangles, with right angles Q and Y respectively.
𝑃𝑅 ≅ 𝑋𝑍, and 𝑃𝑄 ≅𝑋𝑌
Prove: △PQR ≅ △XYZ
P
R
X
Y Z

46. A
C
E
D
Given: CA ≅ ED
AD is the perpendicular bisector of CE
Prove: △CBA ≅ △EBD
B

47. W Z
J K
Given: WJ ≅ KZ
∠W and ∠K are right angles
Prove: △JWZ ≅ △ZKJ

48. Corresponding Parts of Congruent Triangles are Congruent
“C.P.C.T.C.”
We have used SSS, SAS, ASA, AAS, and HL to prove triangles
are congruent. We also discussed the definition of congruent
shapes (all corresponding parts of those shapes are also
congruent). We will use the abbreviation CPCTC to say that
Corresponding Parts of Congruent Triangles are Congruent.
1st Prove the triangles are congruent
2nd Use CPCTC for your reason the
parts are congruent

49. Given: ∠EDG ≅ ∠EDF
∠DEG and ∠DEF are right angles
Prove: EF ≅ EG
F
D
E
G

50. S
T
P
O
Given: 𝑆𝑃 ≅ 𝑃𝑂
∠SPT ≅ ∠OPT
Prove: ∠S ≅ ∠O

51. P R
M G
Given: 𝑃𝑅 ∥ 𝑀𝐺 and 𝑀𝑃 ∥ 𝐺𝑅
Prove: 𝑃𝑅 ≅ 𝑀𝐺 and 𝑀𝑃 ≅ 𝐺𝑅

52. End of Day 4

53. Unit 2: Congruency & Similarity
“Isosceles Triangles”
Objective: To identify and apply properties of
isosceles triangles.
“legs” – are the two congruent sides.
“base” – is the third non-congruent
side.
“vertex angle” – is the angle formed
by the legs.
“base angles” – are the angles formed
using the base as a side.

54. Isosceles Triangle Theorem
“If two sides of a triangle are congruent, then the
angles opposite those sides are congruent.”
If THEN

55. Example 1
Given: YX ≅ ZX and m∠YXZ = 30°
Find the m∠ZYX and m∠YZX.
X
Z
Y

56. Example 2
Given: MO ≅ 𝐍𝐎 and m∠NMO = 50°
Find the m∠MNO and m∠MON.
O
M
N

57. Converse Isosceles Triangle Theorem
“If two angles of a triangle are congruent, then the
sides opposite those angles are congruent.”
If THEN

58. Example 3
Given: ∠R ≅ ∠T, RS = 5x – 8,
ST = 2x + 7, RT = 4x + 2
Find x, RS, ST, and RT.
S
R
T

59. Example 4
List any pair of segments that you would know are
congruent.

60. Example 5
List any pair of segments that you would know are
congruent.

61. Isosceles Bisector Theorem
“The bisector of the vertex angle of an isosceles triangle
is the perpendicular bisector of the base.”
If THEN

62. Example 6
Find the value of x, y, and z.
Z°
23°

63. Example 7
Find the value of x, y, and z.
Z°

64. Equilateral Triangle Theorem
“If all three sides of a triangle are congruent, then all
three angles of the triangle are congruent.”
If THEN

65. Converse Equilateral Triangle Theorem
“If all three angles of a triangle are congruent, then all
three sides of the triangle are congruent.”
If THEN

66. Example 8
Find the value of w, x, y, and z..
X
y
Z
W

67. Example 9
Find the measures of ...
m∠BCA = ______ m∠BCD = ______
m∠DCE = ______ m∠BAG = ______
m∠DEF = ______ m∠GAH = ______

68. Example 10
Find the measures of ...
m∠ABE = ______
m∠BAE = ______
m∠BDF = ______
m∠BDC = ______
m∠EFC = ______
m∠DFG = ______
m∠BEF = ______
m∠AEG = ______
m∠EBC = ______

69. End of day 5

70. Unit 2: Congruence & Similarity
“Similarity in Right Triangles”
Objective: To find and use the relationships in similar
right triangles.
geometric mean of two positive numbers a and b, is
the positive number x such that
𝒂
𝒙
=
𝒙
𝒃
.
Find the geometric mean of…
1. 3 and 12
2. 5 and 7

74. “Right Triangle Similarity”
a b
c
x
y z
Which segments are geometric means?
What proportions can be set up?

75. Example 1
Solve for x, y, and z.

76. Example 2
Solve for x, y, and z.

77. Example 3
Solve for x, y, and z.

78. Example 4
Solve for x.

79. Example 5
Solve for x.

80. Side Splitter Theorem
if a line is parallel to a side of a triangle and intersect the
other two sides, then this line divides those
two sides proportionally.
A
D E
B C
AD AE
DB EC
=
www.youtube.com/watch?v=6C2xHEGRTyl
For proof:

81. 12
8
9
x
Solve for x.

82. X
X + 4
5
7
A
B
C
D
E
What is the length of side BC?

83. Triangle Angle Bisector Theorem
If a ray bisects an angle of a triangle, then it divides
the opposite side into two segments that are
proportional to the two other sides of the triangle.
D
C
B
A
BC BA
CD DA
=
https://www.youtube.com/watch?v=XLYUveKSCtY
For proof:

84. A
D
C
B
15
20
8 x
Solve for x.

85. A
D
B
C
X-4
25
25
20
Solve for x.

86. Math 2 Assignment
In the Geometry textbook:
pp. 442-443
#1-7 odd, 15-21, 34, 36, 39, 51

87. End of Day 6

88. Unit 2: Congruence & Similarity
“Similar Figures”
Objective: To identify similar polygons, prove two
triangles similar and use similar figures to find
missing measurements.
Two polygons are similar if,
1) their corresponding angles are congruent
and
2) their corresponding sides are proportional
(same ratio).
 The ratio of the lengths of corresponding sides is the
similarity ratio.

89. Understanding Similarity
Determine whether the triangles are similar.
If they are, write a similarity statement and give the
similarity ratio.

90. Practice!

91. Using Similarity

92. Example
Given: LMNO ~QRST
Find the value of x .

93. Angle-Angle Similarity

94. Side-Angle-Side Similarity
If
Then

95. Side-Side-Side Similarity
If
Then

96. Practice!

97. Example
Use similar triangles to solve for x - if possible.
1.
2.

98. Example
Use similar triangles to solve for x - if possible.
3.
4.

99. Example
Use similar triangles to solve for x - if possible.
5.

100. Applying Similar Triangles
In sunlight, a flagpole casts a 15 ft shadow. At the same
time of day a 6 ft person casts a 4 ft shadow. Use similar
triangles to find the height of the flag pole?

101. Applying Similar Triangles
Brianna places a mirror 24 feet from the base of a tree.
When she stands 3 feet from the mirror, she can see the
top of the tree reflected in it. If her eyes are 5 feet
above the ground, how tall is the tree?

102. Example
Given: ∆ABC ~∆XYZ
Complete each statement.
a)
b)

103. Example
Given: ∆ABC ~∆XYZ
Find the value of x .

104. End of Day 7