Congruent Trianglescontent.njctl.org/.../congruent-triangles-2014-06-03-3-slides-per-page-w-answers.pdf · Scalene triangle - No congruent sides Isosceles Triangle - 2 congruent sides

This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.

Click to go to website:www.njctl.org

New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative®

Slide 1 / 209

www.njctl.org

CongruentTriangles

Geometry

2014-06-03

Slide 2 / 209

Table of ContentsClassifying TrianglesInterior Angle Theorems

Isosceles Triangle Theorem

Congruence & TrianglesSSS CongruenceSAS CongruenceASA CongruenceAAS CongruenceHL Congruence

CPCTCTriangle Coordinate Proofs

Triangle Congruence Proofs

Exterior Angle Theorems

Slide 3 / 209

http://www.njctl.orghttp://www.njctl.orghttp://www.njctl.orghttp://www.njctl.orghttp://www.njctl.orgpage1svgpage6svgpage54svgpage70svgpage84svgpage89svgpage94svgpage115svgpage132svgpage128svgpage156svgpage155svgpage16svg

Return to Tableof Contents

ClassifyingTriangles

Slide 4 / 209

Parts of a triangle

Side

Side

Vertex

Vertex VertexA B

C

Side opposite

andare adjacent sides

interior

Vertex (vertices) - points joining the sides of triangles

Adjacent Sides - two sides sharing a common vertex

Slide 5 / 209

Parts of a triangle (cont'd)

hypotenuseleg

leg

leg

leg

base

In a right triangle, the hypotenuse is the side opposite the right angle. The legs are the 2 sides that form the right angle.

In an isosceles triangle, the base is the side that is not congruent to the other two sides (legs).

If an isosceles triangle has 3 congruent sides, it is an equilateral triangle.

Slide 6 / 209

page2svg

Definitions

Triangle - three-sided polygon

Polygon - a closed plane figure composed of line segments

Sides - the line segments that make up a polygonVertex (vertices) - the endpoints of the sidesAcute Triangle - all angles < 90°

Obtuse triangle - one angle is between, 90° < angle < 180°

Right Triangle - one 90° angle

Equiangular Triangle - 3 congruent angles

Equilateral Triangle - 3 congruent sides

Scalene triangle - No congruent sides

Isosceles Triangle - 2 congruent sides

Slide 7 / 209

A triangle is formed by line segments joining three noncollinear points. A triangle can be classified by its sides and angles.

Equilateral

3 congruent sides

Isosceles

2 congruent sides

Scalene

No congruent sides

Classification by Sides

Classification by Angles

3 acuteangles

Acute Equiangular

3 congruentangles

Right

1 right angle

Obtuse

1 obtuse angle

(also acute)

Slide 8 / 209

Classify the triangles by sides and angles

equilateralequiangular

scaleneacute

isoscelesacute

isoscelesobtuse

isoscelesright

click

clickclick

click click

Slide 9 / 209

Measure and Classify the triangles by sides and anglesExample

isosceles, right isosceles, acuteClick for AnswerClick for Answer Click for AnswerClick for Answer scalene, obtuseClick for AnswerClick for Answer

Slide 10 / 209

Measure and Classify the triangles by sides and anglesExample

scalene, obtuse scalene, acuteClick for AnswerClick for Answerequilateral, acute/equiangularClick for AnswerClick for Answer Click for AnswerClick for Answer

Slide 11 / 209

1 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

Slide 12 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

[This object is a pull tab]

Ans

wer

C Scalene

Bonus: F Right32+42 = 529+16 = 25

Slide 12 (Answer) / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

Slide 13 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

B Isosceles

Bonus: D Acute

3 cm 3 cm

2 cm



A EquilateralB IsoscelesC Scalene

D AcuteE EquiangularF RightG Obtuse

Ans

wer

Slide 14 / 209


A EquilateralB IsoscelesC Scalene

D AcuteE EquiangularF RightG Obtuse


Ans

wer

A Equilateral

Bonus:E Equiangular (all equilateral triangles are equiangular)D Acute (all angles are 60o )


4 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90°

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

Slide 15 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

F Right

Bonus: C Scalene (all angles are different, so all sides are different)



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

Slide 16 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

G Obtuse

Bonus: C Scalene (all angles are different, so all sides are different)



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

Slide 17 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

D AcuteE Equiangular

Bonus:A Equilateral (if a triangle is equiangular, then it's equilateral)


7 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cmAngle measures: 37°, 53°, 90°

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

Slide 18 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer

C ScaleneF Right



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

Slide 19 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse


Ans

wer A Equilateral

D AcuteE Equiangular


9 Classify the triangle by sides and angles

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

A B120°

C

Ans

wer

Slide 20 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

A B120°

C


Ans

wer

C ScaleneG Obtuse



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

L

MN

Ans

wer

Slide 21 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

L

MN


Ans

wer

B IsoscelesF Right



A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse H

J

K45°

85°

50° Ans

wer

Slide 22 / 209


A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse H

J

K45°

85°

50°


Ans

wer

C ScaleneD Acute


12 An isosceles triangle is _______________ an equilateral triangle.

A Sometimes

B Always

C Never

Ans

wer

Slide 23 / 209

12 An isosceles triangle is _______________ an equilateral triangle.

A Sometimes

B Always

C Never


Ans

wer

A Sometimes


13 An obtuse triangle is _______________ an isosceles triangle.

A Sometimes

B Always

C Never

Ans

wer

Slide 24 / 209

13 An obtuse triangle is _______________ an isosceles triangle.

A Sometimes

B Always

C Never


Ans

wer

A Sometimes


14 A triangle can have more than one obtuse angle.

True

False

Ans

wer

Slide 25 / 209

14 A triangle can have more than one obtuse angle.

True

False


Ans

wer

False


15 A triangle can have more than one right angle.

True

False

Ans

wer

Slide 26 / 209

15 A triangle can have more than one right angle.

True

False


Ans

wer

False


16 Each angle in an equiangular triangle measures 60°

True

False

Ans

wer

Slide 27 / 209

16 Each angle in an equiangular triangle measures 60°

True

False


Ans

wer

True


17 An equilateral triangle is also an isosceles triangle

True

False

Ans

wer

Slide 28 / 209

17 An equilateral triangle is also an isosceles triangle

True

False


Ans

wer

False


InteriorAngle

Theorems


Slide 29 / 209

T1. Triangle Sum TheoremThe measures of the interior angles of a triangle sum to 180°

A B

C

If you have a triangle, then you know the sum of its three interior angles is 180°

Why is this true? Click here to go to the lab titled, "Triangle Sum Theorem"

Slide 30 / 209

Example: Triangle Sum TheoremFind the measure of the missing angle

J

K L

32°

20°

Theorem T1. The Triangle Sum Theorem says that the interior angles of must sum to 180°.

and substituting the information from the diagram

32° + + 20° = 180°

So,

+ 52° = 180°= 128° Check: 128+32+20=180

Slide 31 / 209

page2svghttps://njctl.org/courses/math/geometry/congruent-triangles/triangle-sum-theorem-lab/

A B

C

52°

53°

18 What is the measurement of the missing angle?

m∠B =

Ans

wer

Slide 32 / 209

A B

C

52°

53°


m∠B =


Ans

wer 52 + 53 + m


57°L

M

N

m∠N =


Ans

wer 90 + 57 + m

(draw a diagram)

21 In ABC, if m B is 84° and m C is 36°, what is the m A?

Ans

wer

Slide 35 / 209

(draw a diagram)

21 In ABC, if m B is 84° and m C is 36°, what is the m A?


Ans

wer

triangle ABC = 180°So, 84 + 36 + = 180°

=180° - 120°= 60°


(draw a diagram)

22 In DEF, if m D is 63° and m E is 12°, what is the m F?

Ans

wer

Slide 36 / 209

(draw a diagram)

22 In DEF, if m D is 63° and m E is 12°, what is the m F?


Ans

wer triangle DEF = 180°

So, 63 + 12 + m F = 180°m F =180° - 75°m F = 105°


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

Solve for x

55°

(12x+8)°

(8x-3)°P

Q

R

From the Triangle Sum Theorem

Example

55 + (12x+8) + (8x-3) = 180 Substituting from the diagram20x + 60 = 180 Combining like terms

20x = 120 Isolating x using inverse operationsx = 6

Slide 37 / 209

23 Solve for x in the diagram.

Q

R

S2x° 5x°

8x°

What is m Q m R m S

Extension

Click to reveal

Ans

wer

Slide 38 / 209


Q

R

S2x° 5x°

8x°

What is m Q m R m S

Extension

Click to reveal


Ans

wer

2x+5x+8x = 180 15x = 180 x = 12

m Q = 24° m R = 96° m S = 60°

Extension Answer


Solve for x3x-17 +x+40 +2x-5 = 180°

24 What is the measure of angle B?

A

B

C

Hint

Click to reveal

Ans

wer

Slide 39 / 209

Solve for x3x-17 +x+40 +2x-5 = 180°

24 What is the measure of angle B?

A

B

C

Hint

Click to reveal


Ans

wer

3x-17+2x-5+x+40 = 180 6x + 18 = 180 6x = 162 x = 27 m

Corollary to Triangle Sum TheoremThe acute angles of a right triangle are complementary.

A B

C

Since T1. the Triangle Sum Theorem says the interior angles of a triangle must sum to 180°. So, 180° - 90° (the right angle) = 90° left between and .

Recall: two angles that add up to 90° are called complementary

Slide 40 / 209

Example

x°

5x°

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 complementary. So,

x + 5x = 906x = 90x = 15

(using the Triangle Sum Theorem is a little more work)

One acute angle is 15° and the other is 75°

Slide 41 / 209

25 In a right triangle, the two acute angles sum to 90°

True

False

Ans

wer

Slide 42 / 209

25 In a right triangle, the two acute angles sum to 90°

True

False


Ans

wer

True



57°L

M

N

Ans

wer

Slide 43 / 209


57°L

M

N


Ans

wer

x+57 = 90x = 33

Note: we solved this problem earlier using the Triangle Sum Theorem. Use the Corollary to the Triangle Sum this time.


What are the measures of the three angles?

27 Solve for x

ChallengeClick to reveal

A

B C

Ans

wer

Slide 44 / 209

What are the measures of the three angles?

27 Solve for x

ChallengeClick to reveal

A

B C


Ans

wer

3x-1+31 = 903x + 30 = 90 3x = 60 x = 20

Challenge Answerm

28 Solve for x

What are the measures of the three angles?Challenge

Click to reveal

D E

F [This object is a pull tab]

Ans

wer

2x-2+x+5 = 903x + 3 = 90 3x = 87 x = 29

Challenge Answerm

1

23

30 m 1 + m 2 = _______o

Ans

wer

Slide 47 / 209

1

23

30 m 1 + m 2 = _______o


Ans

wer

90


1

23

31 m 1 + m 3 = _________o

Ans

wer

Slide 48 / 209

1

23

31 m 1 + m 3 = _________o


Ans

wer

90

Recall that

Exterior Angle Theorems

Return to Table of Contents

Slide 50 / 209

Interior angle

Interior angle

Interior angle

Exterior angle

Exterior angle

Exterior angle

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.

Slide 51 / 209

Interior angle

Interior angle

Interior angle

Exterior angle

P

QR

The adjacent angles form a straight line so thesum of the two angle measures will be 180o

Slide 52 / 209

page2svg

Interior angle

Interior angle

Interior angle

Exterior angle

P

QR


Slide 53 / 209

Interior angle

Interior angle

Interior angle

Exterior angle

P

Q

R


Slide 54 / 209

Interior angle

Interior angle

Interior angle

Exterior angle

P

QR

The sum of an interior angle and an adjacent exterior angle is 180 degrees

The sum of the interior angles of a triangle add up to 180 degrees

1

P

R Q

m P + m Q + m R = 180o

m Q + m 1 = 180o1

P

R Q

Slide 55 / 209

The Exterior Angle Theorem says : m 1 = m P + m R

The measure of the exterior angle is equal to the sum of the two angles that are not adjacent to the exterior angle.

1

P

R QProof of the Exterior Angle TheoremWe know the following is true :

1. m P + m Q + m R = 180o

2. m Q + m 1 = 180o

This implies that m 1 = m P + m R and the Exterior Angle Theorem is proved true

Slide 56 / 209

Example: Using the Exterior Angle Theorem

140oXo

Xo

P

QR

What is the value of X ?

The measure of the exterior angle is equal to the sum of the two angles that are not adjacent to the exterior angle.

140o = xo + xo

140 = 2x

70 = x

Slide 57 / 209

ExampleSolve for x using the Exterior Angle Theorem

21°

34°x°

The Exterior Angle Theorem says that the exterior angle, marked x°, is equal to the two nonadjacent interior angles.

x = 21 + 34

y°

So, the exterior angle x = 55°

We also know what y is 125o ?What does x° + y° have to equal? 180o

click

click

Slide 58 / 209

Example: What are w and x ? 75 + 50 + x = 180 125 + x = 180 -125 -125 x = 55o

w = 75 + 50 w = 125o

What does w + x equal? 125 + 55 = 180

75o

50owo Xo

Slide 59 / 209

124

3

m 4 = 131 m 3 = 53, fill in all the angles.

53o

49o

131o

78o 180o

127o 49o

131o

180o

127o53o

78o

Slide 60 / 209

33 Solve for the exterior angle, x.

x°60°

55°Y°

Ans

wer

Slide 61 / 209

33 Solve for the exterior angle, x.

x°60°

55°Y°


Ans

wer

1150


34 m 1 = 25 and m 4 = 83 Find m 3 = ?

A 25

B 50

C 58

D 83124

3

Ans

wer

Slide 62 / 209

34 m 1 = 25 and m 4 = 83 Find m 3 = ?

A 25

B 50

C 58

D 83124

3


Ans

wer

C


Y°

35 Find the value of x using the Exterior Angles Theorem?

A 34

B 17

C 60

D 86

Ans

wer

Slide 63 / 209

Y°

35 Find the value of x using the Exterior Angles Theorem?

A 34

B 17

C 60

D 86 [This object is a pull tab]

Ans

wer

B

94 = 60 + 2x34 = 2x17 = x

0


Y°

36 Find the value of y in the figure below.

A 34

B 17

C 60

D 86 Answ

er

Slide 64 / 209

Y°

36 Find the value of y in the figure below.

A 34

B 17

C 60

D 86


Ans

wer

D


37 Using the Exterior Angles Theorem, find the value

of x.

A100

B51

C46

D23

Y°

Ans

wer

Slide 65 / 209

37 Using the Exterior Angles Theorem, find the value

of x.

A100

B51

C46

D23

Y°


Ans

wer

D

100 = 2x +3 +51100 = 2x +54 46 = 2x 23 = x


38 What is the value of Y?

A80

B40

C51

D100

Y°

Ans

wer

Slide 66 / 209

38 What is the value of Y?

A80

B40

C51

D100

Y°


Ans

wer

A


39 Find the value of x.

A40

B37.5

C20

D10

(3x - 5)°

(x + 2)° 33° Ans

wer

Slide 67 / 209


A40

B37.5

C20

D10

(3x - 5)°

(x + 2)° 33°


Ans

wer

C

3x - 5 = (x + 2) + 333x - 5 = x + 35 2x = 40 x = 20


25 o

115o

P

S

R Tw

40 PS bisects RST , what is the value of w?

A100

B110

C115

D125

Ans

wer

Slide 68 / 209

25 o

115o

P

S

R Tw

40 PS bisects RST , what is the value of w?

A100

B110

C115

D125


Ans

wer

C

25 o

115oPR

T

25o

65o

S


ExampleFind the missing angles in the diagram.

Teac

her N

ote

Slide 69 / 209

ExampleFind the missing angles in the diagram.


Teac

her N

ote

Find the measures of all angles togetherm

41 Find the measure of angle 1.

40o

1

24 53

60o


Ans

wer

40 + m

43

40o

1

24 53

60o

Ans

wer

Find the measure of angle 3.

Slide 72 / 209

43

40o

1

24 53

60o



Ans

wer

40 + m

44

40o

1

24 53

60o



Ans

wer m

Isosceles TriangleTheorem


Slide 75 / 209

Parts of an Isosceles TriangleAn isosceles triangle has at least two congruent sides (an equilateral triangle is an isosceles triangle w/three congruent sides)

If an isosceles triangle has exactly two congruent sides, the: - two congruent sides are called legs, - the noncongruent side is called the base, - the two angles adjacent to the base are the base angles,

leg leg

baseangles

vertex angle

base

The vertex angle is the angle opposite the base ORit is the angle included by the legs

Slide 76 / 209

T3. Base Angles Theorem (BAT)If two sides of a triangle are congruent, the angles opposite them are congruent.

If , then

A

B C

Corollary to BAT (T3)

If a triangle is equilateral, then it is equiangular.

A

B C

Slide 77 / 209

page2svg

x°

y°

44°

Examples: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°

52° x = y; Base Angles are Congruent

x + y + 52 = 180; Triangle Sum Th.x + x + 52 = 180; Substitution2x + 52 = 1802x = 128x = 64

Slide 78 / 209

x°

y°

35°

46 Solve for the measurements of the angles x and yA

nsw

er

Slide 79 / 209

x°

y°

35°

46 Solve for the measurements of the angles x and y


Ans

wer x = 35

y = 110


x°

y°

72°

47 Solve for x and y.

Ans

wer

Slide 80 / 209

x°

y°

72°

47 Solve for x and y.


Ans

wer x = 36

y = 72


70°

48 What are the measurements of the base angles?

Ans

wer

Slide 81 / 209

70°

48 What are the measurements of the base angles?


Ans

wer

70 + 2x = 1802x = 110x = 55

Each base angle is 55o

70°

x° x°


49 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle?

A 71° B 38° C 83° D 104°

Ans

wer

Slide 82 / 209

49 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle?

A 71° B 38° C 83° D 104°


Ans

wer

A


T4. Converse of the Base Angles TheoremIf two angles of a triangle are congruent, then the sides opposite them are congruent.

If , then

A

B C

Corollary to Converse of the BAT (T4)

If a triangle is equiangular, then it is equilateral.

A

B C

Slide 83 / 209

D

EF 4

50 What is the measurement of FD?A

nsw

er

Slide 84 / 209

D

EF 4

50 What is the measurement of FD?


Ans

wer

FD = 4

Triangle is equiangular, so it's equilateral too



A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

A

BC

7

40o

Ans

wer

Slide 85 / 209


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

A

BC

7

40o


Ans

wer

B & E



A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G rightA

B

C

4

4

4

Ans

wer

Slide 86 / 209


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G rightA

B

C

4

4

4[This object is a pull tab]

Ans

wer

A, B, D & E


A

B C5

3 3113o


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

Ans

wer

Slide 87 / 209

A

B C5

3 3113o


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right


Ans

wer

B & F



A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

12

12

Ans

wer

Slide 88 / 209


A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

12

12[This object is a pull tab]

Ans

wer

A, B, D & E


ExampleFind the value of x and y

x°

y°

1. First, consider the top triangle. The 3 marks indicate this is an equilateral triangle

2. From the Corollary to the BAT(T3), we know that an equilateral triangle is also equiangular

3. Since the Triangle Sum Theorem (T1) says the interior angles must sum to 180°, y° = 60.

x°

y°

Slide 89 / 209

x°

60°

60°

60°

120°

4. Two adjacent angles whose non-shared sides form a straight line are a linear pair.

5. The supplement to 60° is 120° (60° + 120° = 180°)

6. Using the Base Angles Theorem (T3) and the Triangle Sum theorem (T1), we can determine x°

x°

60°

60°

60°

120°x°

x + x + 120 = 180 2x + 120 = 180 2x = 60 x = 30

Slide 90 / 209

55 What is the value of y?

A 120°

B 70°

C 55°

D 125°

70°

y°

Ans

wer

Slide 91 / 209

55 What is the value of y?

A 120°

B 70°

C 55°

D 125°

70°

y°


Ans

wer

D


50° x°

56 What is the value of x?

A 50°

B 25°

C 30°

D 130°

Ans

wer

Slide 92 / 209

50° x°

56 What is the value of x?

A 50°

B 25°

C 30°

D 130°


Ans

wer

B



A 3 2/3B 14

C 15

D 16

3x - 17

28 Ans

wer

Slide 93 / 209


A 3 2/3B 14

C 15

D 16

3x - 17

28


Ans

wer

C


Congruence &

Triangles


Slide 94 / 209

CongruenceTwo figures are congruent if they have the exact size and shape (They are similar if they have the same shape, but a different size)

Congruent figures have a correspondence between their angles and sides where pairs of corresponding angles are congruent and pairs of corresponding sides are congruent.

A

B

C N

O

P

A

B

C N

O

P

Slide 95 / 209

page2svg

ExampleThe two triangles are congruent , write: 1) a congruence statement 2) identify all congruent corresponding parts

A

B

C

D

E

F

Answer

Slide 96 / 209

A

B

C

D

E

F

Slide 97 / 209

Part Corresponding Side Corresponding Angle

A

B

D

C

E

Slide 98 / 209

page73svg

Problem

Corresponding Sides Corresponding Angles

(If you need, draw a diagram)

Teac

her N

otes

Slide 99 / 209

Problem

Corresponding Sides Corresponding Angles

(If you need, draw a diagram)


Teac

her N

otes Corresponding Sides

AB EFBC FGAC EG

=~=~=~

Corresponding Angles

58 What is the corresponding part to J

A R

B K C Q D P

J

K L R Q

P

JKL PQR=~


Ans

wer

D


59 What is the corresponding part to Q

A R

B K C Q D P

J

K L R Q

P

JKL PQR=~

Ans

wer

Slide 101 / 209

59 What is the corresponding part to Q

A R

B K C Q D P

J

K L R Q

P

JKL PQR=~


Ans

wer

B


60 What is the corresponding part to QP

A JL

B LK C KJ D PQ

J

K L R Q

P

JKL PQR=~

Ans

wer

Slide 102 / 209

60 What is the corresponding part to QP

A JL

B LK C KJ D PQ

J

K L R Q

P

JKL PQR=~


Ans

wer

C


61 Write a congruence statement for the two triangles

A

B

C

D

Z

X

CV

B

Ans

wer

BVC XCZ=~

XCB BCX=~

VBC ZXC=~

CBV CZX=~

Slide 103 / 209

61 Write a congruence statement for the two triangles

A

B

C

D

Z

X

CV

B

BVC XCZ=~

XCB BCX=~

VBC ZXC=~

CBV CZX=~


Ans

wer

C


Y

ZW

X

What else can be marked congruent?

62 Complete the congruence statement

A

B

C

D

XYZ =~

XWZ

ZWX

WXZ

ZXW

Ans

wer

Slide 104 / 209

Y

ZW

X

What else can be marked congruent?

62 Complete the congruence statement

A

B

C

D

XYZ =~

XWZ

ZWX

WXZ

ZXW


Ans

wer

B

XZ ZX

T5. Third Angles TheoremIf two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent.

Q

R

S

T

U

V

Can you give a reason for why this might be true?If the sum of the interior angles is 180 o and both sets of angles are the same, then the third angles will have the same measure. Example: m S = m V = 40o & m R = m U = 80o degrees, then m Q = m T = 60o.Click to reveal

Slide 105 / 209

ExampleFind the value of x.

A

B

C45° 75°

(2x+40)°

W

X

Y

1) From the Third AngleTheorem (T5), we know m B = m Y

2) The m B is easy to find withthe Triangle Sum Theorem (T1),

3) Substitute to find x

Slide 106 / 209

S

Q

R48°117°

H

I

J

63 What is the measurement of J

Ans

wer

Slide 107 / 209

S

Q

R48°117°

H

I

J

63 What is the measurement of J


Ans

wer

48 + 117 + m


62° 78°Q

R

S

(3x+10)°

C

B

A Ans

wer

Slide 109 / 209


62° 78°Q

R

S

(3x+10)°

C

B

A


Ans

wer

62+78+m

SSS Congruence


Slide 111 / 209

From the Congruence and Triangles section, you learned that two triangles are congruent if the 3 corresponding pairs of sides and the 3 corresponding pairs of angles are congruent.

However, we do not always need all 6 pieces of information to prove 2 triangles congruent.

Slide 112 / 209

Postulate:

Side-Side-Side (SSS) CongruenceIf three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

A B

C

E

D

F

Click here to go to the lab titled, "Triangle Congruence SSS"

Slide 113 / 209

page2svghttps://njctl.org/courses/math/geometry/congruent-triangles/triangle-congruence-sss-lab/

Example

A F

K

BGSolution:The congruence marks on the sides show that:

Slide 114 / 209

Example

F

GH

K

J

G

F

H

J

K

JJ

Slide 115 / 209

A

B

C H

J

K

You need to be very careful that you get the corresponding congruent parts in the correct order CAB is not congruent to HKJ

66 The congruence statement is ABC = HJK

True

False

Hint

~

Ans

wer

Slide 116 / 209

A

B

C H

J

K

You need to be very careful that you get the corresponding congruent parts in the correct order CAB is not congruent to HKJ

66 The congruence statement is ABC = HJK

True

False

Hint

~


Ans

wer

True


R

S

T U

67 SRT = SUT

True

False

~A

nsw

er

Slide 117 / 209

R

S

T U

67 SRT = SUT

True

False

~


Ans

wer True

SSS holds becauseST = ST ; Reflexive Property


A

B C Q R

S

3

4

5 3

4

5

68 ABC = _____?

A QRS B SRQ

C ACB

D RSQ

Ans

wer

~

Slide 118 / 209

A

B C Q R

S

3

4

5 3

4

5

68 ABC = _____?

A QRS B SRQ

C ACB

D RSQ

~


Ans

wer

B


SAS Congruence


Slide 119 / 209

page2svg

Included angle: the angle made by two lines with a common vertex

41 °

A

B

C

Included side: the side between two angles

C

D

E

Slide 120 / 209

Postulate:

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.

F

P

B

A

C

Q

Click here to go to the lab titled, "Triangle Congruence SAS"

Slide 121 / 209

Example

1 2

L

M

P

N

OIs there any information you can fill in?

So, listing the corresponding congruent parts:

Slide 122 / 209

https://njctl.org/courses/math/geometry/congruent-triangles/triangle-congruence-sas-lab/

2005 un

its

Why Not SSA? Move the side with the length of 2 units and create a triangle.

2 units

2005 un

its

Can a different triangle be made than the first one made?

Ans

wer

Slide 123 / 209

2005 un

its

Why Not SSA? Move the side with the length of 2 units and create a triangle.

2 units

2005 un

its

Can a different triangle be made than the first one made?


Ans

wer

2005 un

its

2 unitsTwo different shapes with

same sides and length

2005 un

its2 units


69 What is the included angle of the given sides of the triangle?

A J

B K

C L

Hint: Draw the triangle!

JKL, sides KL and JK

Ans

wer

Slide 124 / 209

69 What is the included angle of the given sides of the triangle?

A J

B K

C L

Hint: Draw the triangle!

JKL, sides KL and JK


Ans

wer

B


P

QR

S

TV4 4

5 5

100° 100°

70 List the congruent parts of the triangles below. Is PQR = STV?

Yes

No

~A

nsw

er

Slide 125 / 209

P

QR

S

TV4 4

5 5

100° 100°

70 List the congruent parts of the triangles below. Is PQR = STV?

Yes

No

~


Ans

wer

Yes, by SAS

PQ = ST

F

GH

X

Y Z46° 46°

1010

77

Why?

71 Is FGH = XYZ by SAS?

Yes

No

Ans

wer

~

Slide 126 / 209

F

GH

X

Y Z46° 46°

1010

77

Why?

71 Is FGH = XYZ by SAS?

Yes

No

~


Ans

wer

No, the angles marked congruent are not the included angles in the triangles.

FG = XYHF = ZX

A B

C D

72 Using SAS, what information do you need to show ABC = DCB

A DBC = ACB B B = C

C ABD = DCA D ABC = DCB

~~

~

~

~


Ans

wer

D


73 What type of congruence exists between the two triangles?

A SSS

B SAS

C Not congruent

Ans

wer

Slide 128 / 209


A SSS

B SAS

C Not congruent


Ans

wer

B



A SSS

B SAS

C Not congruent

Ans

wer

Slide 129 / 209


A SSS

B SAS

C Not congruent


Ans

wer

B



A SSS

B SAS

C Not congruent

Ans

wer

Slide 130 / 209


A SSS

B SAS

C Not congruent


Ans

wer

A



A SSS

B SAS

C Not congruent

Ans

wer

Slide 131 / 209


A SSS

B SAS

C Not congruent


Ans

wer

C



A SSS

B SAS

C Not congruent

Ans

wer

Slide 132 / 209


A SSS

B SAS

C Not congruent


Ans

wer

C



A SSS

B SAS

C Not congruent45° 45°

12 12

Ans

wer

Slide 133 / 209


A SSS

B SAS

C Not congruent45° 45°

12 12


Ans

wer

B



ASA Congruence

Slide 134 / 209

Postulate:

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

Q S

T

U

V

Click here to go to the lab titled, "Triangle Congruence ASA"

Slide 135 / 209

page2svghttps://njctl.org/courses/math/geometry/congruent-triangles/triangle-congruence-asa-lab/

ExampleE

FM

G

H90°90°8

8

Vertical anglesare congruent

Slide 136 / 209

First: what data is given to you?

Second: if it is not already marked, check and mark the diagram with that information,

Third: check your congruence postulates - what piece of information are you missing (side/angle) and where does it need to be for your chosen congruence?

Slide 137 / 209

W

X

Y

79 What is the included side for X and W?

A YX

B YW

C XW

Ans

wer

Slide 138 / 209

W

X

Y

79 What is the included side for X and W?

A YX

B YW

C XW


Ans

wer

C


W

X

Y

80 What is the included side for X and Y

A XW

B YX

C YW

Ans

wer

Slide 139 / 209

W

X

Y

80 What is the included side for X and Y

A XW

B YX

C YW


Ans

wer

B


M

N

O

P

81 What piece of information do we need to have ASA congruence between the two triangles?

A

B

C

D

Ans

wer

Slide 140 / 209

M

N

O

P


A

B

C

D


Ans

wer

C

NP = PN is true by the Reflexive Property. If you mark the diagram to show the corresponding congruent parts, you can see NP = PN is needed

~

~


A

B

C

D


A

B

C

D

Ans

wer

Slide 141 / 209

A

B

C

D


A

B

C

D


Ans

wer

B is true by the Reflexive Property. Since BDC is a right angle, BDA must also be a right angle since they form a linear pair. The included segment we need is


E

F G

M

H

83 Why is ?

A ASA

B vertical angles

C included angles

D congruent

Ans

wer

Slide 142 / 209

E

F G

M

H

83 Why is ?

A ASA

B vertical angles

C included angles

D congruent


Ans

wer

B

Mark the congruent vertical angles when you see two lines intersect



A SSS

B SAS

C ASA

D Not congruent

S

Q R

TU

Ans

wer

Slide 143 / 209


A SSS

B SAS

C ASA

D Not congruent

S

Q R

TU[This object is a pull tab]

Ans

wer A


When you have overlapping figures that share sides and/or angles, marking the diagram with the given information & pulling the triangles apart (when needed) makes it much easier to understand the problem.

Slide 144 / 209

85 What type of congruence exists between the two triangles?A SSS

B SAS

C ASA

D Not congruent

J

L

M

N

K

L

Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?

Hints:

Ans

wer

click to reveal

click to reveal

click to reveal

Slide 145 / 209

85 What type of congruence exists between the two triangles?A SSS

B SAS

C ASA

D Not congruent

J

L

M

N

K

L


Hints:click to reveal

click to reveal

click to reveal


Ans

wer B


A

B

C

Q R


A SSS

B SAS

C ASA

D Not congruent

Mark the diagram with the given information. Be careful you don't always use all information

Hint

Ans

wer

click to reveal

Slide 146 / 209

A

B

C

Q R


A SSS

B SAS

C ASA

D Not congruent

Mark the diagram with the given information. Be careful you don't always use all information

Hint

click to reveal


Ans

wer C


C

B

Q R

A

B

Q R


A SSS

B SAS

C ASA

D Not congruent



click to reveal

click to reveal

Ans

wer

Slide 147 / 209

C

B

Q R

A

B

Q R


A SSS

B SAS

C ASA

D Not congruent



click to reveal

click to reveal[This object is a pull tab]

Ans

wer A


vertical


A SSSB SASC ASAD Not congruent

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

Hint

ST

N

D

A

Ans

wer

Click to Reveal Click

Slide 148 / 209

vertical


A SSSB SASC ASAD Not congruent

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

Hint

ST

N

D

A

Click to Reveal Click[This object is a pull tab]

Ans

wer

B - SAS



A SSS

B SAS C ASA D Not Congruent

Ans

wer

Slide 149 / 209


A SSS

B SAS C ASA D Not Congruent


Ans

wer

We have two Sides and one Angle congruent, but they are not in the correct order.

D - Not Congruent



A SSS B SAS C ASA D Not Congruent

C

P M

S

A

Hint:

Mark the given information into your diagram. Identifying vertical angles plays an important part.

Ans

wer

click to reveal

Slide 150 / 209


A SSS B SAS C ASA D Not Congruent

C

P M

S

A

Hint:

Mark the given information into your diagram. Identifying vertical angles plays an important part. click to reveal


Ans

wer

C - ASA


AAS Congruence


Slide 151 / 209

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.

Q

R

S

T

U

V

Slide 152 / 209

Why is AAS a Theorem ?

Given two triangles:

AB

C

P R

K

75° 75°

65° 65°

12 12

The Triangle Sum Theorem (T1) allows us to find the measurement of the third angle in each triangle. 180°-(65°+75°)= 40°

AB

C

P R

K

75° 75°

65° 65°

12 1240°

Since AAS follows from ASA, AAS is a theorem rather than a postulate

Slide 153 / 209

page2svg

Example

C

A

H

T

1) Mark your diagram:C

A

H

T

C

A

H

TSo, by AAS,

congruence statement?

Slide 154 / 209

91 What type of congruence exists, if any, between the two triangles?

A SSS

B SAS

C ASA

D AAS

E Not CongruentD

E

F

GH

Ans

werH

Slide 155 / 209


A SSS

B SAS

C ASA

D AAS

E Not CongruentD

E

F

GH

H


Ans

wer D - AAS.

What is the congruence statement?



A SSS

B SAS

C ASA

D AAS

E Not Congruent

A

B C Q

RS

Ans

wer

Slide 156 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent

A

B C Q

RS


Ans

wer E Not congruent

We need an included (ASA) or nonincluded side (AAS), which we don't have



A SSS

B SAS

C ASA

D AAS

E Not Congruent

Ans

wer

Slide 157 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent


Ans

wer C ASA

The two triangles share a common side which is congruent via the Reflexive Property.



A SSS

B SAS

C ASA

D AAS

E Not Congruent

Q

W

E

R

T

Ans

wer

Slide 158 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent

Q

W

E

R

T[This object is a pull tab]

Ans

wer D AAS

The vertical angles are the key. The congruent side is nonincluded so it cannot be ASA.



A SSS

B SAS

C ASA

D AAS

E Not CongruentA

S

D

F

G

H

Ans

wer

Slide 159 / 209


A SSS

B SAS

C ASA

D AAS

E Not CongruentA

S

D

F

G

H


Ans

wer

B SAS Imagine you are walking around the figures, you must encounter the congruent parts in the correct order to use SAS congruence.



A SSS

B SAS

C ASA

D AAS

E Not Congruent Ans

wer

Slide 160 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent


Ans

wer D AAS

Q: Are there vertical angles in this diagram?



A SSS

B SAS

C ASA

D AAS

E Not Congruent

AB

C

D Ans

wer

Slide 161 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent

AB

C

D


Ans

wer

D AAS Marking the bisected angle and the common side (reflexive), You can see AAS holds.



A SSS

B SAS

C ASA

D AAS

E Not Congruent

Ans

wer

Slide 162 / 209


A SSS

B SAS

C ASA

D AAS

E Not Congruent


Ans

wer

E Not Congruent There is no AAA Congruence. AAA does make them similar (same shape), but the size may be different.


HL Congruence


Slide 163 / 209

page2svg

Theorem (T8):

Hypotenuse-Leg (HL) CongruenceIf 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.

J

K L

M

N O

If you have a right triangle, make sure you check if HL applies

Slide 164 / 209

Why does HL Congruence work?Recall another theorem for right triangles:

c2 = a2 + b2 a

bc

Pythagorean Theorem:

If we know the lengths of two sides of a right triangle, we can solve for the length of the third side. HL Congruence theorem applies when the corresponding hypotenuse and one of the legs is congruent. When this is the case, the two right triangles are congruent.

A

B C E

FG

J

K L

M

N O

Slide 165 / 209

Example Are the two triangles congruent?

A B

C

R S

TThese are right triangles so let's try for HL congruence

A B

C

R S

T

Slide 166 / 209

Q

R S

X

Y Z

Mark the given on your diagram. Note that it is a right triangle.


A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Given: QS = XZ RS = YZ

~~

Ans

wer

HintClick to reveal

Slide 167 / 209

Q

R S

X

Y Z

Mark the given on your diagram. Note that it is a right triangle.


A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Given: QS = XZ RS = YZ

~~

HintClick to reveal


Ans

wer

EQ

R S

X

Y Z

QSR = XZYby HL congruence

~


If they are congruent what is the congruence statement?


A SSSB SASC ASAD AASE HLF Not congruent

L

M

N O

P

Q

Ans

wer

Slide 168 / 209




L

M

N O

P

Q


Ans

wer C

LMN = OPQ~




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

A

B

CD

E

F

Ans

wer

Slide 169 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

A

B

CD

E

F


Ans

wer F




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

T

U

V W

X

Y

Ans

wer

Slide 170 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

T

U

V W

X

Y


Ans

wer

C

TUV = WXY~




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruentQ

W

EY

Ans

wer

Slide 171 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruentQ

W

EY


Ans

wer

D

QWY = EWY~




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

N

M

O J

K

L

Ans

wer

Slide 172 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

N

M

O J

K

L


Ans

wer E

OMN = JKL~




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

H

Ans

wer

Slide 173 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

H


Ans

wer

E

EFH = GFH~




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

H

Ans

wer

Slide 174 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

H


Ans

wer B

EFH = GFH~




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

K F

B M

Ans

wer

Slide 175 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

K F

B M


Ans

wer A

KFB = MBF~


< POY and < UYO


P O

UY

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


A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Ans

wer

Click to Reveal

Slide 176 / 209

< POY and < UYO


P O

UY

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


A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Click to Reveal


Ans

wer D

POY = UYO~




A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

O K

MJ

Ans

wer

Slide 177 / 209



A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

O K

MJ


Ans

wer F





A S

XZ Ans

wer

Slide 178 / 209




A S

XZ


Ans

wer E

ASZ = XZS~


Triangle Congruence Proofs


Slide 179 / 209

Congruent Reasons Summary(Drag ones that don't work out of the chart. Then put HL where it would belong.)

SSS

ASSSSASAS

AASSAAASA

AAA

HL0

3

1

2

Slide 180 / 209

Example

A F

K

BGSolution (two-column):

1) Given

2) SSS Postulate

AF = BG, FK = GK KA = KB~

~ ~1)

2) AFK = BGK~

Statements Reasons

Given: AF = BG, FK = GK & KA = KB~~ ~

Slide 181 / 209

page2svg

Example F

GH

J

K

HF = HJ~

Given

FG = JK~

Given

H is the midpoint of GK.

Given

GH = KH~

Def. of midpoint

FGH = JKH~

SSS

Solution (flow proof):

Teac

her N

otes

Slide 182 / 209

Example F

GH

J

K

HF = HJ~

Given

FG = JK~

Given

H is the midpoint of GK.

Given

GH = KH~

Def. of midpoint

FGH = JKH~

SSS

Solution (flow proof):


Teac

her N

otes

Mark the congruent segments from "Def. of midpoint step" in your diagram

G

F

H

J

K


In two-column proofs, the statements in the left column are justified by the reasons on the right-side column. As we read down the table, we can see the thought process laid out.

A D

ECB

1 2

Example

Statements Reasons

Note: for SAS, corresponding congruent sides and angles are needed, which we have.

Slide 183 / 209

Example

A

B

C

DStatements Reasons

1. Given, AC bisects BCD

click ___________

click ___________

click ___________

click ___________

A

Slide 184 / 209

Problem

Q R

S

T

click click

click ___________

click ___________click ___________

click ___________

Slide 185 / 209

Problem

D

FG

E

Statements Reasons

click ___________

click ___________

click ___________ click ___________

click ___________

Slide 186 / 209

Problem

GHJ

F

H is the midpoint of GJ

click

click ___________

click ___________

click ___________

click ___________

click ___________ click ___________

Slide 187 / 209

Statements Reasons

Problem

A

B

C

DT

click ___________

click ___________click ___________

___________

click ___________

click ___________

click ___________

Slide 188 / 209

Statements Reasons

ProblemD C

A B

lines__ __

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

Slide 189 / 209

ProblemP

Q R S

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

__

__

click

click ___________

click ___________

click ___________click ___________

click ___________

click ___________

Slide 190 / 209

Statements Reasons

1)

2)

3)

4)

5)

1)

2)

3)

4)

5)

Given: AC = BD, E is the midpoint of AB and CD

~

~Prove: AEC = BED

A

B

DC

E

Problem

Def. of midpoint

E is the midpoint of AB and CD

SSS

AC = BD~

Def. of midpoint

AE = BE~

Given~AEC = BED

CE = DE~Given

Teac

her N

otes

Slide 191 / 209

Statements Reasons

1)

2)

3)

4)

5)

1)

2)

3)

4)

5)

Given: AC = BD, E is the midpoint of AB and CD

~

~Prove: AEC = BED

A

B

DC

E

Problem

Def. of midpoint

E is the midpoint of AB and CD

SSS

AC = BD~

Def. of midpoint

AE = BE~

Given~AEC = BED

CE = DE~Given


Teac

her N

otes

Given1)

2)

3)

4)

5)

Statements Reasons

1)

2)

3)

4)

5) SSS

Def. of midpoint

Def. of midpoint

AC = BD~

CE = DE~

AE = BE~

~AEC = BED

E is the midpoint of AB and CD Given

This example can be solved by matching the statements & reasons with their appropriate location.

Ans. is given below.



CPCTCCorresponding Parts of Congruent Triangles are Congruent

Slide 192 / 209

CPCTC says that if two or more triangles are congruent by:

SSS, SAS, ASA, AAS, or HL, then all of their corresponding parts are also congruent.

Corresponding Parts of Congruent Triangles are Congruent

CPCTC

Sometimes, our goal is not to prove two triangles congruent, but to show that a pair of corresponding sides or angles are congruent, or that some other property is true.

Slide 193 / 209

Process for proving that two segments or angles are congruent

1. Find two triangles in which the two sides or two angles are corresponding parts

2. Prove that the two triangles are congruent (SSS, SAS, ASA, AAS, HL)

3. State that the two parts are congruent, using as the reason: "corresponding parts of congruent triangles are congruent"

Slide 194 / 209

page2svg

MN

O

E L

111 Which two triangles might you try to prove congruent in order to prove

A

B

C

D

Ans

wer

Slide 195 / 209

MN

O

E L


A

B

C

D


Ans

wer B and D


MN

O

E L


A

B

C

D

Ans

wer

Slide 196 / 209

MN

O

E L


A

B

C

D


Ans

wer A and C


MN

O

E L


A

B

C

D1 2

Ans

wer

Slide 197 / 209

MN

O

E L


A

B

C

D1 2


Ans

wer B and D


MN

O

E L


A

B

C

D

Ans

wer

Slide 198 / 209

MN

O

E L


A

B

C

D


Ans

wer B and D


3. Given

4.

5.

6.

Statements Reasons

3. C is the midpoint of AD

4.

5.

6.

Problem

A

B

C D

E

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

Slide 199 / 209

Problem A

B

C

D

DB bisects ABC ABD = CBD~

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

_____

Slide 200 / 209

Problem AB

C

D E

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 _________________ ________

_______ ________________________

Slide 201 / 209

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

Statements Reasons

Problem W X

P

Z Y

click ___________click ___________

click ___________

click ___________

click ___________

click ___________

click ___________

click ___________click ___________click ___________

Slide 202 / 209

Triangle Coordinate Proofs


Slide 203 / 209

Coordinate Triangle Proofs

A coordinate proof places a triangle, or any other geometric figure, into a coordinate plane.

A coordinate proof combines: - the geometric postulates, theorems, and properties, and - the Distance Formula and Midpoint Formula.

The only thing that changes from the proofs we have done earlier is you will need to use the Distance and/or Midpoint Formula to calculate side and segment lengths.

Slide 204 / 209

Midpoint Formula TheoremThe midpoint of a segment joining points with coordinates and is the point with coordinates

(x1, y1)(x2, y2)

The Distance FormulaThe distance 'd' between any two points with coordinates and is given by the formula:(x1, y1) (x2, y2)

d =

To refresh your memory:

Slide 205 / 209

page2svg

ExampleA (0,4)

B (3,0)C (-3,0) Q (0,0)

Statements Reasons

2.

5.4.3.

7.

6.

1. AC = 5 and AB = 5= segments have = measure

1. Given~2.

5.4.3.

7.

6.

Slide 206 / 209

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

1. Plot the points2. Use the distance formula to find side lengths3. Does it satisfy the condition for an isosceles

A(4,-1)

B(5,6)

C(1,3)

d =

Distance Formula

Side lengths:

next

Slide 207 / 209

Continued...

triangle

Slide 208 / 209

page186svg

Problem

A(1,1)

B(4,4)

C(6,2)

d =

Distance Formula

After we plot the points, we can see that they form a triangle.

Slide 209 / 209

Documents

Congruent Trianglescontent.njctl.org/.../congruent-triangles-2014-06-03-3-slides-per-page-w-answers.pdf · Scalene triangle - No congruent sides Isosceles Triangle - 2 congruent sides