Page 1
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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Page 2
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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Page 3
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
Page 4
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
Page 5
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
Page 6
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
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 7
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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Page 8
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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Page 9
9
If one side of a triangle is the longest then
A
B
C
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 10
10
If one side of a triangle is the longest then
The opposite angle to this side is the largest
A
B
C
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 11
11
And the angle opposite to the shortest side
A
B
C
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 12
12
And the angle opposite to the shortest side is the smallest
A
B
C
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 13
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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Page 14
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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Page 15
15
If one angle of a triangle is the largest then
A
B
C
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 16
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
Page 17
17
And the side opposite to the smallest angle
A
B
C
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 18
18
And the side opposite to the smallest angle is the shortest
A
B
C
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 19
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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Page 20
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
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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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Page 22
22
If two sides of a triangle are congruent to two sides in another triangle
K
L
MA
B
C
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 23
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
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 24
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
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 25
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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Page 26
26
If two sides of a triangle are congruent to two sides in another triangle
K
L
MA
B
C
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 27
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
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 28
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
Page 29
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
Page 30
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
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Page 31
31
Write an inequality or pair of inequalities to describe the possible values of x.
14
7
7
115°
8 89
(3x+5)°
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
14
Page 32
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)°
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14
Page 33
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
STANDARD 6PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
14