congruent triangles - mendham borough schools · 46 solve for the measurements of the angles x and...
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Unit 4 Congruent Triangles.notebook
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CongruentTriangles
Geometry
AAS Congruence
Review of Triangle Congruence Proofs
Return to Table
Unit 4 Congruent Triangles.notebook
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Side
SideSide opposite
the sides of triangles
Adjacent Sides - two sides sharing a common vertex
leg
leg
leg
leg
In a right triangle, the hypotenuse is the side
If an isosceles triangle has 3 congruent sides, it
- the line segments that make up a polygon
Vertex (vertices)
Acute Triangle
Obtuse triangle
- 3 congruent angles
Scalene triangle - No congruent sides
Isosceles Triangle - At least two congruent sides
Unit 4 Congruent Triangles.notebook
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points. A triangle can be classified by its sides and angles.
Isosceles
Acute Equiangular
(also acute)
right
click
clickclick
click click
Measure and Classify the triangles by sides and angles
0 180
10170
20
160
30
15040
14050
13060
120
70
110
80
100
90
90
0°
100
80
110
70
120
60
130
50
140
40
150
30
160
20
17010
1800
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 2 3 4 50°
Unit 4 Congruent Triangles.notebook
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Measure and Classify the triangles by sides and angles
0 180
10170
20
160
30
15040
14050
13060
120
70
110
80
100
90
90
0°
100
80
110
70
120
60
130
50
140
40
150
30
160
20
17010
1800
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 2 3 4 50°
Side lengths: 3 cm, 4 cm, 5 cm
Equilateral
Right
Side lengths: 3 cm, 2 cm, 3 cm
Equilateral
Right
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Side lengths: 5 cm, 5 cm, 5 cm
Equilateral
Right
Equilateral
Right
Equilateral
Right
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Equilateral
Right
Side lengths: 3 cm, 4 cm, 5 cm
Equilateral
Right
8Side lengths: 3 cm, 3 cm, 3 cm
Equilateral
Right
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9 Classify the triangle by sides and angles
Equilateral
Right
10 Classify the triangle by sides and angles
Equilateral
Right
11 Classify the triangle by sides and angles
Equilateral
Right
Unit 4 Congruent Triangles.notebook
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12 An isosceles triangle
13is _______________ an isosceles triangle.
14 A triangle can have more than one obtuse angle.
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15 A triangle can have more than one right angle.
16 Each angle in an equiangular triangle measures 60°
17 An equilateral triangle is also an isosceles triangle
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Return to Table
The measures of the interior angles of a triangle sum to 180°
its three interior angles is 180°
Why is this true?
Find the measure of the missing angle
Theorem T1. The Triangle Sum Theorem says that the interior
Unit 4 Congruent Triangles.notebook
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18 What is the measurement of the missing angle?
m B =
19 What is the measurement of the missing angle?
m N =
20 What is the measure of the missing angle?
Unit 4 Congruent Triangles.notebook
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21what is the m A?
22what is the m F?
We can solve more "complicated" problems using the Triangle Sum Theorem.
Solve for x
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23 Solve for x in the diagram.
What is m Qm Rm S
24
Since T1. the Triangle Sum Theorem says the interior
Recall: two angles that add up to 90° are called
Unit 4 Congruent Triangles.notebook
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The measure of one acute angle of a right triangle is five times the measure of the other acute angle.
Find the measure of each acute angle.
Since this is a right triangle, we can use the Corollary to the Triangle Sum Theorem which says the two acute angles are
25 In a right triangle, the two acute angles sum to 90°
26 What is the measurement of the missing angle?
Unit 4 Congruent Triangles.notebook
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27 Solve for x
ChallengeClick to reveal
A
B C
28 Solve for x
ChallengeClick to reveal
D E
F
G
H
J
29 In the right triangle given, what is the measurement of each
Unit 4 Congruent Triangles.notebook
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30
31
32
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Exterior Angle Theorems
Exterior angles are adjacent to the interior angles.
Exterior angles and interior angles together form a straight line.
The sum of an exterior angle and an interior angle is 180 degrees.
Unit 4 Congruent Triangles.notebook
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Unit 4 Congruent Triangles.notebook
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Proof of the Exterior Angle Theorem
Q
Solve for x using the Exterior Angle Theorem
marked x°, is equal to the two nonadjacent interior angles.
What does x° + y° have to equal?180o
click
click
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click
click
click
click
click
click
click
33 Solve for the exterior angle, x.
Unit 4 Congruent Triangles.notebook
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34
B
35 Find the value of x using the Exterior Angles
Theorem?
A 34
B 17
C 60
D 86
36 Find the value of y using the Exterior Angles
Theorem?
A 34
B 17
C 60
D 86
Unit 4 Congruent Triangles.notebook
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37 Using the Exterior Angles Theorem, find the value
of x.
A 100
B 51
C 46
D 23
38 What is the value of Y?
A 80
B 40
C 51
D 100
39 Find the value of x.
A 40
B 37.5
C 20
D 10
Unit 4 Congruent Triangles.notebook
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40 PS bisects RST , what is the value of w?
A 100
B 110
C 115
D 125
41 Find the measure of angle 1.
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42
43 Find the measure of angle 3.
44 Find the measure of angle 4.
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45 Find the measure of angle 5.
Isosceles
Return to Table
at least
If an isosceles triangle has
- two congruent sides are called
the two angles adjacent to the base are the base angles
Unit 4 Congruent Triangles.notebook
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If two sides of a triangle are congruent, the angles opposite them are congruent.
Corollary to BAT (T3)
Find the values of x & y in the isosceles triangle below.
x = 44; Base Angles are Congruent
y + 44 + 44 = 180; Triangle Sum Th.y + 88 = 180y = 92
Find the values of x & y in the isosceles triangle below.
x = y; Base Angles are Congruent
x + y + 52 = 180; Triangle Sum Th.x + x + 52 = 180; Substitution2x + 52 = 1802x = 128x = 64
46 Solve for the measurements of the angles x and y
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47 Solve for x and y.
48 What are the measurements of the base angles?
49 The vertex angle of an isosceles triangle is 38°. What is a possible measurement for the base angles?
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Corollary to Converse of the BAT (T4)
50 What is the measurement of FD?
51 Classify the triangle by sides and angles
equilateral equiangular
obtuse
right
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52 Classify the triangle by sides and angles
equilateral equiangular
obtuse
right
53 Classify the triangle by sides and angles
equilateral equiangular
obtuse
right
54 Classify the triangle by sides and angles
equilateral equiangular
obtuse
right
Unit 4 Congruent Triangles.notebook
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Find the value of x and y
1. First, consider the top triangle. The 3 marks indicate this is an equilateral triangle
y°
4. Two adjacent angles whose non-shared sides form a straight angle
5. Two adjacent angles whose non-shared sides form a straight line are a linear pair.
7. Using the Base Angles Theorem (T3) and the Triangle Sum theorem (T1), we can
55
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56
57 Solve for x in the diagram.
3x - 17
28
&
Triangles
Return to Table
Unit 4 Congruent Triangles.notebook
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if they have the exact size and shape
The two triangles are congruent , write:
Answer
Unit 4 Congruent Triangles.notebook
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Part
Corresponding Sides Corresponding Angles
58 What is the corresponding part to J
JKL PQR=~
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59 What is the corresponding part to Q
JKL PQR=~
60 What is the corresponding part to QP
JKL PQR=~
61 Write a congruence statement for the two triangles
BVC XCZ=~
XCB BCX=~
VBC ZXC=~
CBV CZX=~
Unit 4 Congruent Triangles.notebook
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W
What else can be marked congruent?
62 Complete the congruence statement XYZ =~
XWZ
ZWX
WXZ
ZXW
T5. Third Angles TheoremIf two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent.
o
40o & m R = m U = 80o
degrees, thenClick to reveal
Find the value of x.
W
Theorem (T5), we know
2) The m B is easy to find withthe Triangle Sum Theorem (T1),
3) Substitute to find x
Unit 4 Congruent Triangles.notebook
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63 What is the measurement of J
64 Solve for x
65 Find the value of x.
Unit 4 Congruent Triangles.notebook
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Return to Table
From the Congruence and Triangles section, you learned that two triangles are congruent if the 3
3 corresponding pairs of angles are
However, we do not always need all 6 pieces of information to prove 2 triangles congruent.
Unit 4 Congruent Triangles.notebook
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If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
So, we have:
Unit 4 Congruent Triangles.notebook
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You need to be very careful that you get the corresponding congruent parts in the correct order
CAB is not congruent to HKJ
66 ~
67 ~
68 ~
Unit 4 Congruent Triangles.notebook
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Return to Table
Side-Angle-Side (SAS) CongruenceIf two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
Unit 4 Congruent Triangles.notebook
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69
triangle?
JKL, sides KL and JK
70~
Unit 4 Congruent Triangles.notebook
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71 ~
72~
~
~
~
~
73 What type of congruence exists between the two triangles?
Not congruent
Unit 4 Congruent Triangles.notebook
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74 What type of congruence exists between the two triangles?
Not congruent
75 What type of congruence exists between the two triangles?
Not congruent
76 What type of congruence exists between the two triangles?
Not congruent
Unit 4 Congruent Triangles.notebook
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77 What type of congruence exists between the two triangles?
Not congruent
78 What type of congruence exists between the two triangles?
Not congruent
Return to Table
Unit 4 Congruent Triangles.notebook
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Angle-Side-Angle (ASA) CongruenceIf two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
R
are congruent
Second: if it is not already marked, check and mark the diagram
Third: check your congruence postulates - what piece of
it need to be for your chosen congruence?
Unit 4 Congruent Triangles.notebook
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W
79 What is the included side for X and W?
W
80 What is the included side for X and Y
81
congruence between the two triangles?
Unit 4 Congruent Triangles.notebook
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82
congruence between the two triangles?
83
84 What type of congruence exists between the two triangles?
Not congruent
Unit 4 Congruent Triangles.notebook
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marking the diagrampulling the triangles apart (when needed)makes it much easier to understand the
85 What type of congruence exists between the two triangles?
Not congruent
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
click to reveal
click to reveal
click to reveal
86 What type of congruence exists between the two triangles?
Not congruent
click to reveal
Unit 4 Congruent Triangles.notebook
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87 What type of congruence exists between the two triangles?
Not congruent
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
click to reveal
click to reveal
click to reveal
88 What type of congruence exists between the two triangles?
Not congruent
At the intersection of two line you always have _____ angles.
89 What type of congruence exists between the two triangles?
SAS
ASA
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90 What type of congruence exists between the two triangles?
SAS
ASA
part.
AAS Congruence
Return to Table
Theorem (T7):
Angle-Angle-Side (AAS) CongruenceIf two angles and the nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent.
Unit 4 Congruent Triangles.notebook
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Why is AAS a Theorem ?
The Triangle Sum Theorem (T1) allows us to find the
So, by AAS,
congruence statement?
91
triangles?
Not Congruent
Unit 4 Congruent Triangles.notebook
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92
triangles?
Not Congruent
93
triangles?
Not Congruent
94
triangles?
Not Congruent
W
Unit 4 Congruent Triangles.notebook
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95
triangles?
Not Congruent
96
triangles?
Not Congruent
97
triangles?
Not Congruent
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98
triangles?
Not Congruent
Return to Table
Theorem (T8):
If the hypotenuse and a leg of one right triangle are equal to the corresponding hypotenuse and leg of another right triangle, then the two triangles are congruent.
Unit 4 Congruent Triangles.notebook
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c
HL Congruence theorem applies when the corresponding
case, the two right triangles are congruent.
Are the two triangles congruent?
99
triangles?
Not congruent
Given: QS = XZ RS = YZ
~~
Click to reveal
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If they are congruent what is the congruence statement?
100
triangles?
Not congruent
If they are congruent what is the congruence statement?
101
triangles?
Not congruent
If they are congruent what is the congruence statement?
102
triangles?
Not congruent
W
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If they are congruent what is the congruence statement?
103
triangles?
Not congruent
W
If they are congruent what is the congruence statement?
104
triangles?
Not congruent
If they are congruent what is the congruence statement?
105
triangles?
Not congruent
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If they are congruent what is the congruence statement?
106
triangles?
Not congruent
If they are congruent what is the congruence statement?
107
triangles?
Not congruent
If they are congruent what is the congruence statement?
What angles are congruent when parallel lines are cut by a transversal?
108
triangles?
Not congruent
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If they are congruent what is the congruence statement?
109
triangles?
Not congruent
If they are congruent what is the congruence statement?
110triangles?
Not congruent
Return to Table
Unit 4 Congruent Triangles.notebook
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1) Given
2) SSS Postulate
~~ ~1)
2) AFK = BGK~
Statements Reasons
~~ ~
HF = HJ~
Given
FG = JK~
Given
H is the midpoint of GK.
Given
GH = KH~
Def. of midpoint
FGH = JKH~
SSS
Solution (flow proof):
justified by the reasons on the right-side column. As we read down the table, we can see the thought process laid out.
Statements
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Statements
1. Given, AC bisects BCD
click ___________
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clickclick
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Statements
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H is the midpoint of GJ
click
click___________
click ___________
click ___________
click ___________
click ___________ click___________
Statements
A
B
C
D
click ___________
click ___________click ___________
___________
click ___________
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Statements
lines__ __
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Given: R is the midpoint of QS, PQR and TSR are right 's, PR = TR~
__
__
click
click ___________
click ___________
click ___________click ___________
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Statements Reasons
1)
2)
3)
4)
5)
1)
2)
3)
4)
5)
Given
SSS
Def. of midpoint
Def. of midpoint
~
~
~
~
E is the midpoint of AB and CD
Given
~
~
A
B
DC
E
Return to Table
orresponding ongruent ongruent
Unit 4 Congruent Triangles.notebook
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says that if two or more triangles are congruent by:
then all of their corresponding parts are also congruent.
C arts of C riangles are C
CPCTC
Sometimes, our goal is not to prove two triangles congruent, but to
that some other property is true.
or two angles are corresponding parts
2. Prove that the two triangles are congruent
3. State that the two parts are congruent, using as the reason:orresponding ongruent ongruent"
111 you
Unit 4 Congruent Triangles.notebook
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112 you
113 you
1 2
114 you
Unit 4 Congruent Triangles.notebook
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3. Given
4.
5.
6.
Statements
3. C is the midpoint of AD
4.
5.
6.
click ___________
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DB bisects ABC ABD = CBD~
click ___________
click ___________click ___________
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click ___________
_____
We are given that BCA = DCE, BC = CD, and B and D are right angles. Since all right angles are congruent, B = D. With the congruent angles and segments, we can conclude that ABC = EDC by ASA. Therefore, BA = DE by CPCTC.
~ ~~
~ ~
click_________________
________
___ _______ ________________________
Unit 4 Congruent Triangles.notebook
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7. If alt. int. 's =, then lines ||~
Statements
W
click ___________
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Triangle Coordinate
Proofs
Return to Table
A coordinate proof places a triangle or, any other geometric figure, into a coordinate plane.
- the geometric postulates, theorems, and properties, and
The only thing that changes from the proofs we have done
Formula to calculate side and segment lengths.
Unit 4 Congruent Triangles.notebook
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and is the point with coordinates
d =
Statements
A(4,1), B(5,6), and C(1,3) forms an isosceles right triangle
d =
Unit 4 Congruent Triangles.notebook
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Continued...
d =
After we plot the points, we can see that it forms a triangle.
Review of Triangle Congruence Proofs
Return to Table
of Contents
Unit 4 Congruent Triangles.notebook
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Objective:
Prove triangle congruence using triangle
congruence postulates and theorems
S
T
KL
R1
2
3
4
Given: Prove:
statements reasons