1 The sum of 2 sides of the triangle greater than the other side? Ordering the angles of a triangle?...

1

The sum of 2 sides of the trianglegreater than the other side?

Ordering the angles of a triangle?

Ordering the sides of a triangle?

SAS Inequality

SSS Inequality

PROBLEM 1 PROBLEM 2

PROBLEM 5 PROBLEM 6

PROBLEM 3

PROBLEM 7STANDARD 6

PROBLEM 4

END SHOW

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2

STANDARD 6:

Students know and are able to use the Triangle Inequality Theorem.

Los estudiantes conocen y son capaces de usar el Teorema de Desigualdad del Triángulo.

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3

The sum of the lengths of any two sides of a triangle is greater than the third side.

5 12

15

5+12 >15 or 17>15

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4

The sum of the lengths of any two sides of a triangle is greater than the third side.

5 12

15

5+15 >12 or 20>12

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5

The sum of the lengths of any two sides of a triangle is greater than the third side.

5 12

15

12+15 >5 or 27>5

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6

The sum of the lengths of any two sides of a triangle is greater than the third side.

5 12

15

5+12 >15 or 17>15

5+15 >12 or 20>12

12+15 >5 or 27>5

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7

The measures of two sides of a triangle are 15 and 8. Between what two numbers is the third side.

X

15+8 > X 15+X > 8 8+X > 15

STANDARD 6

23 > X

X < 2315+X > 8-15 -15

X > -7

8+X > 15-8 -8

X > 7

0 5 10 15 20 25x

-5-10-15-20

x

x

x

-7

7

23

X | 7<X<23

815

The third side will be any value between 7 and 23.

7 23

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8

If a triangle has sides of measure x, x+4, 3x-5, find all possible values of x

(X+4)+(3X-5) > X (X+4 )+X > (3X-5)

X

X+4 3X-5

STANDARD 6

4X -1 >X-4X -4X

-1 >-3X-3 -3

.3 <X

X>.3

2X +4 > 3X-5-2X -2X

4 > X-5

+5 +5

9 > X

X < 9

Sign (>) changes when dividing by (-3)

0 5 10 15 20 25x

-5-10-15-20

x

x

x

9

3

.3

(3X-5) +X > (X+4 )4X – 5 > X +4 -X -X3X – 5 > 4

+5 +5

3X > 93 3

X > 3

3 9X | 3<X< 9

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9

If one side of a triangle is the longest then

A

B

C

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10

If one side of a triangle is the longest then

The opposite angle to this side is the largest

A

B

C

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11

And the angle opposite to the shortest side

A

B

C

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12

And the angle opposite to the shortest side is the smallest

A

B

C

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13

And the angle opposite to the shortest side is the smallest

A

B

C

The opposite angle to this side is the largest

If one side of a triangle is the longest then

m B > m C > m ASTANDARD 6

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14

In STU, ST=37-X, TU=2X-16, SU=X+13. The perimeter of the triangle is 90. List the angles in order from smallest to largest.

S

T

U

37-X 2X-16

X+13

=90

STANDARD 6

37-X +2X-16

+X+13

37 – 16 + 13 –X +2X +X = 90

34 +2X = 90-34 -34

2X = 562 2

X=28

ST=37-X

Substituting X:

=37 – ( )28

= 99

TU=2X-16

= 2( )-1628

=56 -16

= 40

SU=X + 13

= ( ) + 1328

= 41

41

40

41 is the longest side and it is opposite to T

So T is the largest

9 is the shortest side and it is opposite to U so then U is the smallest.

Then:

m U < m S < m T

The perimeter is 90, so:

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15

If one angle of a triangle is the largest then

A

B

C

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16

The opposite side to this angle is the longest

A

B

C

If one angle of a triangle is the largest then

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17

And the side opposite to the smallest angle

A

B

C

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18

And the side opposite to the smallest angle is the shortest

A

B

C

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19

A

B

C

The opposite side to this angle is the longest

If one angle of a triangle is the largest then

And the side opposite to the smallest angle is the shortest

AC > AB> BCSTANDARD 6

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20

25°92°

33°

D

C

A

B

What is the shortest side in the figure below?

STANDARD 6

180°- 92°-33°= 55°

Finding missing angles:

55°

180°- 90°-25°= 65°

65°

So, which angle’s measure is the smallest?

25°

So, the opposite side to this angle is DCand it is the shortest side in the figure.

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63°

27°

E

21

In JKL, m J=12x+11, m K=9x+3, m L=7x+26. List the sides in order from longest to shortest.

m J + m K + m L = 180°

STANDARD 6

(12x+11)+(9x+3)+(7x+26)=180°

12x+119x+3

28X + 40 = 180°-40 -40

28X = 140°28 28

X = 5

Finding the angles:

Adding the interior angles in the triangle:

m J =12x + 11=12( ) + 115= 60 + 11

= 71°

m K=9x+3

=9( ) + 35

= 45 + 3

= 48°

m L =7x+26

= 7( )+265

= 35 + 26

= 61°

7x+26

61°

71°48°

The largest angle is J and opposite segment LK is the longest side.

Then:

LK > KJ> JL

K

L

J The smallest angle is K and opposite segment JL is the shortest side.

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22

If two sides of a triangle are congruent to two sides in another triangle

K

L

MA

B

C

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23

If two sides of a triangle are congruent to two sides in another triangle

And the included angle between the sides in one triangle is larger than

K

L

MA

B

C

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24

If two sides of a triangle are congruent to two sides in another triangle

The included angle between the sides of the other triangle

And the included angle between the sides in one triangle is larger than

K

L

MA

B

C

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25

If two sides of a triangle are congruent to two sides in another triangle

And the included angle between the sides in one triangle is larger than

The included angle between the sides of the other triangle

Then the opposite side to the largest angle is also larger:

K

L

MA

B

C

AC > KM by SAS InequalitySTANDARD 6

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26

If two sides of a triangle are congruent to two sides in another triangle

K

L

MA

B

C

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27

If two sides of a triangle are congruent to two sides in another triangle

And the third side is larger in one than in the other

K

L

MA

B

C

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28

If two sides of a triangle are congruent to two sides in another triangle

Then the included angle opposite to the larger

K

L

MA

B

C

And the third side is larger in triangle than in the other

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

29

If two sides of a triangle are congruent to two sides in another triangle

Then the included angle opposite to the larger is greater than the angle opposite to the shorter.

K

L

MA

B

C

And the third side is larger in one triangle than in the other

STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

30

If two sides of a triangle are congruent to two sides in another triangle

Then the included angle opposite to the larger is greater than the angle opposite to the shorter:

K

L

MA

B

C

And the third side is larger in one triangle than in the other

m B > m L by SSS Inequality

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31

Write an inequality or pair of inequalities to describe the possible values of x.

14

7

7

115°

8 89

(3x+5)°

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14

32

Write an inequality or pair of inequalities to describe the possible values of x.

STANDARD 6

14

7

7

115°

8 89

(3x+5)°

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14

33

Write an inequality or pair of inequalities to describe the possible values of x.

14

7

7

115°

8 89

(3x+5)°

115 > 3x+5 by SSS inequality

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14