Unit 5 – Congruent Triangles

Day 1 – Triangles Basics

Objectives: SWBAT classify triangles by their sides and angles.

A. Classifying Triangles: Ways of describing Triangles

Equilateral Triangle: All three sides of a triangle are congruent.

Isosceles Triangle: Exactly two sides of a triangle are congruent.

Scalene Triangle: No sides of a triangle are congruent.

Acute Triangle: A triangle where all three angles of a triangle are acute.

Equiangular Triangle: A triangle where all angles are congruent (60 degrees)

Right Triangle: A triangle where exactly one angle is a right angle.

Obtuse Triangle: A triangle where exactly one angle is an obtuse angle.

B. Parts of a Triangle Vertex : A point where two sides of an angle meet.

Side: a line segment forming a triangle

Angle: The space formed by two sides

Examples:

Classify each triangle by its angles and its sides (Hint: Has more than one name). A. B. C.

Scalene Acute Obtuse Right Triangle Equilateral / Equiangular

Need one word about the sides, and one word about the angles. The word

“triangle” doesn’t count as a descriptive word. Draw the following Triangles if possible.

A. Right Isosceles B. Obtuse Scalene C. Acute Equilateral

D. Equilateral Scalene E. Acute Obtuse Triangle

Not Possible Need a side word

Find x and the unknown measure of the given triangles.

1. 2.

3 4 2 7

2 2

4 7

4 4

11

LN NM

x x

x x

x

x

5 6 5

6 6

5

1 1

5

RQ RS

w w

w w

w

w

58

65

57

130

B

A

C

D

EF

60

60 60

17

3 4x

2 7x

Q

R

S

5w

3 10w

6 5w

Day 2 – Triangles and their Angles

Objectives: SWBAT examine and find the measure of internal and exterior angles of a Triangle

A. Triangle Sum Theorem

The sum the angles in a triangle is 180 degrees. Examples: Solve for x A. B. C.

Find the measure of each angle.

A.

)60(

)30(

xCm

xBm

xAm

B.

)15(

)23(

)116(

xCm

xBm

xAm

C.

)33(

)22(

90

xCm

xBm

Am

9 48 51 180

9 99 180

9 9

9 81

9 81

9 9

9

x

x

x

x

x

90 2 180

90 3 180

90 90

3 90

3 90

3 3

30

x x

x

x

x

x

80 3 24 180

4 56 180

56 56

4 124

4 124

4 4

31

x x

x

x

x

x

30 60 180

3 90 180

3 90

30

30

30 30 60

30 60 90

x x x

x

x

x

m A

m B

m C

6 11 3 2 5 1 180

14 12 180

14 168

12

6 12 11 83

3 12 2 38

5 12 1 59

x x x

x

x

x

m A

m B

m C

90 2 2 3 3 180

5 95 180

5 85

17

90

2 17 2 36

3 17 3 54

x x

x

x

x

m A

m B

m C

x2

x

80 x

(3 24)x

9x

51

48

External Angle Theorem

The sum of the remote internal angles equal the external angle

1 2 3m m m

Solve for the following variables

Find the measure of each numbered angle.

The following problems could be done a number of different ways. Pick the one that works for you.

Here is one way:

1. 2.

3.

36 34

70

x

x

97 22

119

x

x

61 180

119

x

x

OR

5 4 5 6 120

10 10 120

10 10

10 110

10 110

10 10

11

x x

x

x

x

x

29 1 105

29 29

1 76

m

m

29 1 2 180

29 76 2 180

105 2 180

2 75

m m

m

m

m

90 1 31 180

121 1 180

2 59

m

m

m

90 3 28 180

118 3 18

2 62

m

m

m

90 28 2 180

118 2 180

2 62

m

m

m

63 96 3 180

159 3 180

3 21

m

m

m

96 1 180

1 84

m

m

53 1 2 180

53 84 2 180

137 2 180

2 43

m m

m

m

m

36°

34°

x97°

22°

61°

x

1205x+6

5x+4

29°

105°

1

231°

1

28°

28°2

3

63°

1

2

53°

96°

3

Day 3 – Isosceles and Equilateral Triangles

Objectives: SWBAT Use properties of isosceles and equilateral triangles.

A. Properties of Isosceles Triangles

Base Angles Theorem~

If a triangle is isosceles, then the base angles are congruent.

Base Angles Theorem Converse~

If the base angles of a triangle are congruent, then the triangle is isosceles. B. Properties of Equilateral Triangles

1. All equilateral triangles are also equiangular, and vice versa.

2. If a triangle is equiangular, then each angle is 60 degrees.

Examples: 1. 2. 3.

22

Iso Triangle

x

4 72

18

Iso Triangle

x

x

12

12

Equilateral Triangle

x

y

Solve for x and y. 4. 5. 6.

7. Find all the variables

6 6 60

6 6

6 54

6 54

6 6

9

x

x

x

x

2 60

2 60

2 2

30

y

y

y

3 7

3 7 50 180

3 3 130

6 130

21.667

x y

x x

x

x

x y

3 21.667

3

7

5 7

5

7

6

8

y

y

y

y

x

60

30

x

y

6 6( )x

4 25x

y2x3 7( )y

x y50

5

46

88

75

30

4

9

6

59

v

r

e

u

m

t

o

180 62 118

118

2

5

1

9

Iso Triangle

base angles are

m e m t

m e m t

75

180 75 75 30

0 30

2

Iso Triangle

base angles are

m r

m

3

180

75 59 180

46

46

180 46 46 88

88

m m m r m e

m m

m m

Iso Triangle

base angles are

m m m v

m u

Day 4 – Congruence and Triangles

Objectives: SWBAT Identify congruent figures and corresponding parts. SWBAT prove that two triangles are congruent.

A. Congruent Triangles~ Triangles with the exact same angle and side measurements.

Corresponding Angles:

Angles with the same angle measure, and are in the same position relative to each

triangle

Corresponding Sides:

Sides with the same length, and are in the same position relative to each triangle

B. Using Properties of Congruent Figures

Given ABC DEF, find the values of all angles and sides.

A.

Match the pieces of each triangle based on their relative position in the Given.

mA= 87 AB= 10m mD= 87 DE= 10m

mB= 42 BC= 11m mE= 42 EF= 11m

mC= 54 AC= 14m mF= 54 DF= 14m

A

B

C P

Q

R

A

B

C

E

F

D87 42

m11

m14

QABC P R

m10

1

2

3

st

nd

rd C

A P

R

B Q

BC QR

AB PQ

CA RP

A D

B

F

E

C

BC EF

AB ED

AC DF

EABC D F

Third Angle Theorem:

If two pairs of angles of a triangle are congruent, then the third angles are also congruent.

Find the values of x.

A. B. B D, C F

Builders use the King Post Truss, below to the left, for the top of a simple structure. In

this truss, ∆ABC ≅ ADB. Label the diagram and list the corresponding congruent parts.

...

A D

B E

therefore

C F

180

22 87 180

109 180

71

71

m A m B m C

m C

m C

m C

so

m F

...

B D

C F

therefore

A E

180

70 11 22 180

92 11 180

11 88

8

m F m E m D

x

x

x

x

AC AD

CB BD

AB AB

CAB DAB

C D

ABC ABD

22

87

A

B

C

D E

F )302( x 22 70 11x

A

B

CD

EF

Proving Triangles are Congruent

Objectives: SWBAT prove triangles congruent using SSS, SAS, AAS, ASA and HL

List the 5 shortcuts for proving triangles are congruent and draw a picture for each.

1. Side – Side - Side

If 3 sides of a triangle are congruent

to 3 sides of another triangle, then the triangles are congruent.

2. Side – Angle (included) - Side

If 2 sides of a triangle are congruent and the included angle is congruent 2 sides of another triangle and the

included angle, then the triangles are congruent.

3. Hypotenuse Leg (Right Triangles ONLY)

If a leg and the hypotenuse of a right Triangle is congruent to the leg and Hypotenuse of another right triangle, then

Those triangles are congruent.

4. Angle – Angle – Side (not included side)

If 2 angles of a triangle are congruent

and the not included side is congruent 2 angles of another triangle and the not

included angle, then the triangles are congruent.

5. Angle – Side (included) - Angle

If 2 angles of a triangle are congruent

and the included side is congruent 2 angles of another triangle and the

included angle, then the triangles are congruent.

Determine if the following Triangles are congruent and if so why? If not why? *****SIMPLY PUTTING SSS AND SAS IS NOT SUFFICIENT*****

Ex I Ex II

Ex III Ex IV

Ex V Ex VI

Given two pairs of congruent sides; the two triangles share side DB; therefore the triangles are congruent by SSS.

Both triangles are right triangles. Given one pair of right angles is congruent, they share EG, and all right angles are congruent. The triangles are congruent by SAS

Given two pairs of congruent sides, and the included angle is also congruent. Hence, the triangles are congruent because of SAS.

Given three pairs of congruent sides, then the triangles are congruent by SSS.

Given one pair of congruent angles, and one pair of congruent sides, they triangles share one side. So the triangles are congruent by SAS.

Given three sides are congruent, then the triangles are congruent by SSS. OR Given two sides are congruent and <𝑿𝒀𝑾 ≅ < 𝒁𝒀𝑼 by Vertical Angles are congruent, so they are congruent by SAS.

A

DC

B

FE

G

H

1 2

1 2

C

B

A

T

D

E

U

V

W

D

A

B

C

W

X

Y

U

Z

Determine if the following Triangles are congruent and if so why? If not why? *****SIMPLY PUTTING HL IS NOT SUFFICIENT*****

Ex I Ex II

Ex III Ex IV

Ex V Ex VI

F is the Midpoint of

Both triangles are right triangles. Given one pair of congruent legs, and they share the hypotenuse. Therefore, the triangles are congruent by HL.

Both triangles are right triangles. Given one pair of congruent hypotenuses, and they share a leg. Therefore, the triangles are congruent by HL.

There is not enough information to prove that these triangles are congruent (can’t assume right angles for HL).

Both triangles are right triangles. Given one pair of congruent hypotenuses, and they share a leg. Therefore, the triangles ∆𝑻𝑺𝑼 ≅ ∆𝑼𝑸𝑻 are congruent by HL.

Because there are multiple triangles, you must specify which triangles are congruent.

Both triangles are right triangles. Given congruent hypotenuses; midpoint means

that AF FE . Therefore triangles are congruent by HL.

Both triangles are right triangles. Given one pair of congruent hypotenuses, and they share a leg. Therefore, the triangles are congruent by HL.

A

DC

B

FE

G

H

N

M

O

L S Q

T U

R

TQ SU

B

EA

D

F

AEW

X

Z

Y

Determine if the following Triangles are congruent and if so why? If not why? *****SIMPLY PUTTING ASA AND AAS IS NOT SUFFICIENT*****

Ex I Ex II

Ex III Ex IV

Ex V Ex VI

Using Congruent Triangles

Given two pairs of congruent angles, and the two triangles share the non-included side; then the triangles are congruent by AAS.

Given a congruent side; an angle since all right angles are congruent, and < 𝑴𝑶𝑵 ≅<𝑷𝑶𝑸 are Vertical Angles. Then the triangles are congruent by AAS.

Not enough information to prove the triangles congruent. One triangle is trying to use ASA, and the other AAS. They have to use the same one

Given ∆𝑮𝑲𝑳 ≅ ∆𝑯𝑳𝑲 are right triangles and all right angles are congruent. Given one pair of congruent sides, and they share KL. Therefore the triangles are congruent by SAS.

Given that all right angles are congruent and < 𝑮𝑯𝑳 ≅< 𝑱𝑯𝑲 are Vertical Angles, and the included side is congruent. The triangles are congruent by ASA.

Given that all right angles are congruent, a given non included side, and a given angle; then the triangles are congruent by AAS.

W

XY

Z M N

O

P Q

B E

A D

F

K L

G H

J

B

E

A D

C

Objectives: SWBAT Use congruent triangles to find other information about those

triangles

A. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Once you proven that triangle are congruent, then the right of the corresponding parts of the triangles are also congruent.

Tell how the following Triangles are congruent. Then, find the missing variable.

1. 2.

3. 4.

5. 6.

10

SAS

x

2 18

9

AAS

w

w

3 2 12

12

HL

y y

w

60

SSS

z

50 9 5

45 9

5

SAS

b

b

b

2

2

5 41

36

6

ASA

x

x

x

10 x

18

2w

2y+12

3y

60

z

6 6

6 6

50

9 5b

41 2 5x