Congruent Triangles - Mendham Borough Schools · 46 Solve for the measurements of the angles x and...

Unit 4 Congruent Triangles.notebook

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CongruentTriangles

Geometry

AAS Congruence

Review of Triangle Congruence Proofs

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2

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


3

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°


4

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


Equilateral

Right


5


Equilateral

Right

Equilateral

Right

Equilateral

Right


6

Equilateral

Right


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


Equilateral

Right


Equilateral

Right


8

12 An isosceles triangle

13is _______________ an isosceles triangle.

14 A triangle can have more than one obtuse angle.


9

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


10

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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


11

18 What is the measurement of the missing angle?

m B =


m N =

20 What is the measure of the missing angle?


12

21what is the m A?

22what is the m F?

We can solve more "complicated" problems using the Triangle Sum Theorem.

Solve for x


13

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


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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°



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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


16

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.


18


19

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


20

click

click

click

click

click

click

click

33 Solve for the exterior angle, x.


21

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


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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


23

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.


24

42




25


Isosceles

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at least

If an isosceles triangle has

- two congruent sides are called

the two angles adjacent to the base are the base angles


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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?


28

Corollary to Converse of the BAT (T4)

50 What is the measurement of FD?


equilateral equiangular

obtuse

right


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obtuse

right



obtuse

right



obtuse

right


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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


31

56

57 Solve for x in the diagram.

3x - 17

28

&

Triangles

Return to Table


32

if they have the exact size and shape

The two triangles are congruent , write:

Answer


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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=~


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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


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63 What is the measurement of J

64 Solve for x

65 Find the value of x.


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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.


38

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

So, we have:


39

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 ~


40

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.


41

69

triangle?

JKL, sides KL and JK

70~


42

71 ~

72~

~

~

~

~

73 What type of congruence exists between the two triangles?

Not congruent


43


Not congruent


Not congruent


Not congruent


44


Not congruent


Not congruent

Return to Table


45

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?


46

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?


47

82

congruence between the two triangles?

83


Not congruent


48

marking the diagrampulling the triangles apart (when needed)makes it much easier to understand the


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


Not congruent

click to reveal


49


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


Not congruent

At the intersection of two line you always have _____ angles.


SAS

ASA


50


SAS

ASA

part.

AAS Congruence

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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.


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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


52

92

triangles?

Not Congruent

93

triangles?

Not Congruent

94

triangles?

Not Congruent

W


53

95

triangles?

Not Congruent

96

triangles?

Not Congruent

97

triangles?

Not Congruent


54

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.


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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


101

triangles?

Not congruent


102

triangles?

Not congruent

W


57


103

triangles?

Not congruent

W


104

triangles?

Not congruent


105

triangles?

Not congruent


58


106

triangles?

Not congruent


107

triangles?

Not congruent


What angles are congruent when parallel lines are cut by a transversal?

108

triangles?

Not congruent


59


109

triangles?

Not congruent


110triangles?

Not congruent

Return to Table


60

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 ___________

click ___________

click ___________

click ___________

clickclick

click ___________

click ___________click ___________

click ___________

Statements

click ___________

click ___________

click ___________ click ___________

click ___________


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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 ___________

click ___________

click ___________

Statements

lines__ __

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________


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Given: R is the midpoint of QS, PQR and TSR are right 's, PR = TR~

__

__

click

click ___________

click ___________

click ___________click ___________

click ___________

click ___________

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


64

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


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112 you

113 you

1 2

114 you


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3. Given

4.

5.

6.

Statements

3. C is the midpoint of AD

4.

5.

6.

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

DB bisects ABC ABD = CBD~

click ___________

click ___________click ___________

click ___________

click ___________

click ___________

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_________________

________

___ _______ ________________________


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7. If alt. int. 's =, then lines ||~

Statements

W

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

Triangle Coordinate

Proofs

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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.


68

and is the point with coordinates

d =

Statements

A(4,1), B(5,6), and C(1,3) forms an isosceles right triangle

d =


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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


70

Objective:

Prove triangle congruence using triangle

congruence postulates and theorems

S

T

KL

R1

2

3

4

Given: Prove:

statements reasons

Congruent Triangles - Mendham Borough Schools · 46 Solve for the measurements of the angles x and...

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