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New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative®

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www.njctl.org

CongruentTriangles

Geometry

2014-06-03

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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

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

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

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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

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2 Classify the triangle with the given information: Side lengths: 3 cm, 2 cm, 3 cm

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

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

A EquilateralB IsoscelesC Scalene

D AcuteE EquiangularF RightG Obtuse

Ans

wer

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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

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5 Classify the triangle with the given information: Angle Measures: 25°, 120°, 35°

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

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6 Classify the triangle with the given information: Angle Measures: 60°, 60°, 60°

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

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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

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8 Classify the triangle with the given information: Side lengths: 3 cm, 3 cm, 3 cmAngle measures: 60°, 60°, 60°

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

Ans

wer

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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

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10 Classify the triangle by sides and angles

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse

L

MN

Ans

wer

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11 Classify the triangle by sides and angles

A Equilateral

B Isosceles

C Scalene

D Acute

E Equiangular

F Right

G Obtuse H

J

K45°

85°

50° Ans

wer

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12 An isosceles triangle is _______________ an equilateral triangle.

A Sometimes

B Always

C Never

Ans

wer

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13 An obtuse triangle is _______________ an isosceles triangle.

A Sometimes

B Always

C Never

Ans

wer

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14 A triangle can have more than one obtuse angle.

True

False

Ans

wer

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15 A triangle can have more than one right angle.

True

False

Ans

wer

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16 Each angle in an equiangular triangle measures 60°

True

False

Ans

wer

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17 An equilateral triangle is also an isosceles triangle

True

False

Ans

wer

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InteriorAngle

Theorems

Return to Tableof Contents

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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"

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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

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A B

C

52°

53°

18 What is the measurement of the missing angle?

m∠B =

Ans

wer

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19 What is the measurement of the missing angle?

57°L

M

N

m∠N =

Ans

wer

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47°

34°

x°

20 What is the measure of the missing angle?

x =

Ans

wer

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(draw a diagram)

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

Ans

wer

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(draw a diagram)

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

Ans

wer

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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

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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

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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

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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

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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°

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

True

False

Ans

wer

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26 What is the measurement of the missing angle?

57°L

M

N

Ans

wer

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What are the measures of the three angles?

27 Solve for x

ChallengeClick to reveal

A

B C

Ans

wer

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28 Solve for x

What are the measures of the three angles?Challenge

Click to reveal

D E

F

Ans

wer

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x°

2x°

G

H

J

29 In the right triangle given, what is the measurement of each acute angle?

Ans

wer

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1

23

30 m 1 + m 2 = _______o A

nsw

er

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1

23

31 m 1 + m 3 = _________o

Ans

wer

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20°

X°

32 Find the value of x in the diagram

Mark your vertical angles!Hintclick to reveal

Ans

wer

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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.

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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

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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

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Interior angle

Interior angle

Interior angle

Exterior angle

P

Q

R

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

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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

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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

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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

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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

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33 Solve for the exterior angle, x.

x°60°

55°Y°

Ans

wer

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

A 25

B 50

C 58

D 83124

3

Ans

wer

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Y°

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

A 34

B 17

C 60

D 86

Ans

wer

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Y°

36 Find the value of y in the figure below.

A 34

B 17

C 60

D 86 Ans

wer

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37 Using the Exterior Angles Theorem, find the value

of x.

A100

B51

C46

D23

Y°

Ans

wer

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38 What is the value of Y?

A80

B40

C51

D100

Y°

Ans

wer

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39 Find the value of x.

A40

B37.5

C20

D10

(3x - 5)°

(x + 2)° 33° Ans

wer

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25 o

115o

P

S

R Tw

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

A100

B110

C115

D125

Ans

wer

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ExampleFind the missing angles in the diagram.

Teac

her N

ote

Slide 69 / 209

41 Find the measure of angle 1.

40o

1

24 53

60o

Ans

wer

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42 Find the measure of angle 2.

40o

1

24 53

60o

Ans

wer

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43

40o

1

24 53

60o

Ans

wer

Find the measure of angle 3.

Slide 72 / 209

44

40o

1

24 53

60o

Find the measure of angle 4.

Ans

wer

Slide 73 / 209

45

40o

1

24 53

60o

Find the measure of angle 5.A

nsw

er

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Isosceles TriangleTheorem

Return to Tableof Contents

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

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 y

Ans

wer

Slide 79 / 209

x°

y°

72°

47 Solve for x and y.A

nsw

er

Slide 80 / 209

70°

48 What are the measurements of the base angles?

Ans

wer

Slide 81 / 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

Slide 82 / 209

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?

Ans

wer

Slide 84 / 209

51 Classify the triangle by sides and angles

A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

A

BC

7

40o

Ans

wer

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52 Classify the triangle by sides and angles

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

B C5

3 3113o

53 Classify the triangle by sides and angles

A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

Ans

wer

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54 Classify the triangle by sides and angles

A equilateral

B isosceles

C scalene

D equiangular

E acute

F obtuse

G right

12

12

Ans

wer

Slide 88 / 209

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

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55 What is the value of y?

A 120°

B 70°

C 55°

D 125°

70°

y°

Ans

wer

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50° x°

56 What is the value of x?

A 50°

B 25°

C 30°

D 130°

Ans

wer

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57 Solve for x in the diagram.

A 3 2/3B 14

C 15

D 16

3x - 17

28 Ans

wer

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Congruence &

Triangles

Return to Tableof Contents

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

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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

Problem

Corresponding Sides Corresponding Angles

(If you need, draw a diagram)

Teac

her N

otes

Slide 99 / 209

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

Slide 100 / 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

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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

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

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

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 A

nsw

er

Slide 107 / 209

I

J

K

80°

32°

P

Q

R(2x+14)°

64 Solve for x

Ans

wer

Slide 108 / 209

65 Find the value of x.

62° 78°Q

R

S

(3x+10)°

C

B

A Ans

wer

Slide 109 / 209

T6. Properties of Congruent Triangles

Reflexive Property of Congruent TrianglesEvery triangle is congruent to itself A

B

C A

B

C

Symmetric Properties of Congruent Triangles

A

B

C D

E

F A

B

CD

E

F

Transitive Property of Congruent Triangles

A

B

C D

E

F D

E

F J

K

L A

B

C J

K

L

Slide 110 / 209

SSS Congruence

Return to Tableof Contents

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

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

~A

nsw

er

Slide 116 / 209

R

S

T U

67 SRT = SUT

True

False

~

Ans

wer

Slide 117 / 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

~

Slide 118 / 209

SAS Congruence

Return to Tableof Contents

Slide 119 / 209

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

200

5 units

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

2 units

200

5 units

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

Ans

wer

Slide 123 / 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

Slide 124 / 209

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

F

GH

X

Y Z46° 46°

1010

77

Why?

71 Is FGH = XYZ by SAS?

Yes

No

Ans

wer

~

Slide 126 / 209

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

Slide 127 / 209

73 What type of congruence exists between the two triangles?

A SSS

B SAS

C Not congruent

Ans

wer

Slide 128 / 209

74 What type of congruence exists between the two triangles?

A SSS

B SAS

C Not congruent

Ans

wer

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75 What type of congruence exists between the two triangles?

A SSS

B SAS

C Not congruent

Ans

wer

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76 What type of congruence exists between the two triangles?

A SSS

B SAS

C Not congruent

Ans

wer

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77 What type of congruence exists between the two triangles?

A SSS

B SAS

C Not congruent

Ans

wer

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78 What type of congruence exists between the two triangles?

A SSS

B SAS

C Not congruent45° 45°

12 12

Ans

wer

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Return to Tableof Contents

ASA Congruence

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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"

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ExampleE

FM

G

H90°90°8

8

Vertical anglesare congruent

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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?

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W

X

Y

79 What is the included side for X and W?

A YX

B YW

C XW

Ans

wer

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W

X

Y

80 What is the included side for X and Y

A XW

B YX

C YW

Ans

wer

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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

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A

B

C

D

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

A

B

C

D

Ans

wer

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E

F G

M

H

83 Why is ?

A ASA

B vertical angles

C included angles

D congruent

Ans

wer

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84 What type of congruence exists between the two triangles?

A SSS

B SAS

C ASA

D Not congruent

S

Q R

TU

Ans

wer

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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.

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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

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A

B

C

Q R

86 What type of congruence exists between the two triangles?

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

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C

B

Q R

A

B

Q R

87 What type of congruence exists between the two triangles?

A SSS

B SAS

C ASA

D Not congruent

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

Hints:click to reveal

click to reveal

click to reveal

Ans

wer

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vertical

88 What type of congruence exists between the two triangles?

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

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89 What type of congruence exists between the two triangles?

A SSS

B SAS C ASA D Not Congruent

Ans

wer

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90 What type of congruence exists between the two triangles?

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

AAS Congruence

Return to Tableof Contents

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

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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

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Example

C

A

H

T

1) Mark your diagram:C

A

H

T

C

A

H

TSo, by AAS,

congruence statement?

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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

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

A SSS

B SAS

C ASA

D AAS

E Not Congruent

A

B C Q

RS

Ans

wer

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

A SSS

B SAS

C ASA

D AAS

E Not Congruent

Ans

wer

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

A SSS

B SAS

C ASA

D AAS

E Not Congruent

Q

W

E

R

T

Ans

wer

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

A SSS

B SAS

C ASA

D AAS

E Not CongruentA

S

D

F

G

H

Ans

wer

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

A SSS

B SAS

C ASA

D AAS

E Not Congruent Ans

wer

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

A SSS

B SAS

C ASA

D AAS

E Not Congruent

AB

C

D Ans

wer

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

A SSS

B SAS

C ASA

D AAS

E Not Congruent

Ans

wer

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HL Congruence

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Slide 163 / 209

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

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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

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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

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Q

R S

X

Y Z

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

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Given: QS = XZ RS = YZ

~~

Ans

wer

HintClick to reveal

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If they are congruent what is the congruence statement?

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

A SSSB SASC ASAD AASE HLF Not congruent

L

M

N O

P

Q

Ans

wer

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If they are congruent what is the congruence statement?

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

A

B

CD

E

F

Ans

wer

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If they are congruent what is the congruence statement?

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

T

U

V W

X

Y

Ans

wer

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If they are congruent what is the congruence statement?

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Q

W

EY

Ans

wer

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If they are congruent what is the congruence statement?

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

N

M

O J

K

L

Ans

wer

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If they are congruent what is the congruence statement?

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

H

Ans

wer

Slide 173 / 209

If they are congruent what is the congruence statement?

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

E

F

G

H

Ans

wer

Slide 174 / 209

If they are congruent what is the congruence statement?

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

K F

B M

Ans

wer

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< POY and < UYO

If they are congruent what is the congruence statement?

P O

UY

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

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

Ans

wer

Click to Reveal

Slide 176 / 209

If they are congruent what is the congruence statement?

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

A SSS

B SAS

C ASA

D AAS

E HL

F Not congruent

O K

MJ

Ans

wer

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If they are congruent what is the congruence statement?

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

A SSSB SASC ASAD AASE HLF Not congruent

A S

XZ Ans

wer

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Triangle Congruence Proofs

Return to Tableof Contents

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

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

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

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Problem

Q R

S

T

click click

click ___________

click ___________click ___________

click ___________

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Problem

D

FG

E

Statements Reasons

click ___________

click ___________

click ___________ click ___________

click ___________

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Problem

GHJ

F

H is the midpoint of GJ

click

click ___________

click ___________

click ___________

click ___________

click ___________ click ___________

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Statements Reasons

Problem

A

B

C

DT

click ___________

click ___________click ___________

___________

click ___________

click ___________

click ___________

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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

Return to Tableof Contents

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"

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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

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

A

B

C

D

Ans

wer

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MN

O

E L

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

A

B

C

D1 2

Ans

wer

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MN

O

E L

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

A

B

C

D

Ans

wer

Slide 198 / 209

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 ___________

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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

Return to Tableof Contents

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

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

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