Discussion on Triangle Inequalities

1. Triangle Inequalities
Triangle Inequalities
 §
§ 7.1 Segments, Angles, and Inequalities
7.1 Segments, Angles, and Inequalities
 §
§ 7.4 Triangle Inequality Theorem
7.4 Triangle Inequality Theorem
 §
§ 7.3 Inequalities Within a Triangle
7.3 Inequalities Within a Triangle
 §
§ 7.2 Exterior Angle Theorem
7.2 Exterior Angle Theorem

2. Segments, Angles, and Inequalities
Segments, Angles, and Inequalities
Property
Transitive
Property
For any numbers a, b, and c,
1) if a < b and b < c, then a < c.
2) if a > b and b > c, then a > c.
if 5 < 8 and 8 < 9, then 5 < 9.
if 7 > 6 and 6 > 3, then 7 > 3.

3. Segments, Angles, and Inequalities
Segments, Angles, and Inequalities
Property
Addition and
Subtraction
Properties
Multiplication
and Division
Properties
For any numbers a, b, and c,
For any numbers a, b, and c,
1) if a < b, then a + c < b + c
and a – c < b – c.
2) if a > b, then a + c > b + c
and a – c > b – c.
1 < 3
1 + 5 < 3 + 5
6 < 8
c
b
c
a
and
bc
ac
then
b,
a
and
0
c
If
)




1
c
b
c
a
and
bc
ac
then
b,
a
and
0
c
If
)




2 36
24
2
18
2
12
18
12





9
6
2
18
2
12


18
12

4. Exterior Angle Theorem
Exterior Angle Theorem
You will learn to identify exterior angles and remote interior
angles of a triangle and use the Exterior Angle Theorem.
1) Interior angle
2) Exterior angle
3) Remote interior angle

5. Exterior Angle Theorem
Exterior Angle Theorem
1
2 3 4
P
Q R
In the triangle below, recall that 1, 2, and 3 are _______ angles of
ΔPQR.
interior
Angle 4 is called an _______ angle of ΔPQR.
exterior
An exterior angle of a triangle is an angle that forms a _________ with one of
the angles of the triangle.
linear pair
In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3.
____________________ of a triangle are the two angles that do not form
a linear pair with the exterior angle.
Remote interior angles
In ΔPQR, 1, and 2 are the remote interior angles
with respect to 4.

6. Exterior Angle Theorem
Exterior Angle Theorem
1
2
3 4 5
In the figure below, 2 and 3 are remote interior angles with respect to
what angle? 5

7. Exterior Angle Theorem
Exterior Angle Theorem
Theorem 7 – 3
Exterior
Angle
Theorem
The measure of an exterior angle of a triangle is equal to sum
of the measures of its ___________________.
remote interior angles
X
4
3
2
1
Z
Y
m4 = m1 + m2

8. Exterior Angle Theorem
Exterior Angle Theorem

9. Exterior Angle Theorem
Exterior Angle Theorem
Theorem 7 – 4
Exterior
Angle
Inequality
Theorem
The measure of an exterior angle of a triangle is greater than
the measures of either of its two ____________________.
remote interior angles
X
4
3
2
1
Z
Y
m4 > m1
m4 > m2

10. Exterior Angle Theorem
Exterior Angle Theorem
1 and 3
74°
1 3
2
Name two angles in the triangle below that have measures less than 74°.
Theorem 7 – 5
If a triangle has one right angle, then the other two angles
must be _____.
acute

11. Exterior Angle Theorem
Exterior Angle Theorem
3
and
1 


12. Exterior Angle Theorem
Exterior Angle Theorem
The feather–shaped leaf is called a pinnatifid.
In the figure, does x = y? Explain.
x = y
?
__ + 81 = 32 + 78
28
28°
109 = 110
No! x does not equal y

13. Inequalities Within a Triangle
Inequalities Within a Triangle
You will learn to identify the relationships between the _____
and _____ of a triangle.
sides
angles
Nothing New!

14. Inequalities Within a Triangle
Inequalities Within a Triangle
Theorem 7 – 6
If the measures of three sides of a triangle are unequal,
then the measures of the angles opposite those sides
are unequal ________________.
13
8
11
L
P
M
in the same order
LP < PM < ML
mM < mP
mL <

15. Inequalities Within a Triangle
Inequalities Within a Triangle
Theorem 7 – 7
If the measures of three angles of a triangle are unequal,
then the measures of the sides opposite those angles
are unequal ________________.
in the same order
JK < KW < WJ
mW < mK
mJ <
J
45°
W K
60°
75°

16. Inequalities Within a Triangle
Inequalities Within a Triangle
Theorem 7 – 8
In a right triangle, the hypotenuse is the side with the
________________.
greatest measure
WY > XW
3
5
4 Y
W
X
WY > XY

17. Inequalities Within a Triangle
Inequalities Within a Triangle
A

The longest side is BC
So, the largest angle is
L

The largest angle is
MN
So, the longest side is

18. Triangle Inequality Theorem
Triangle Inequality Theorem
You will learn to identify and use the
Triangle Inequality Theorem.
Nothing New!

19. Triangle Inequality Theorem
Triangle Inequality Theorem
Theorem 7 – 9
Triangle
Inequality
Theorem
The sum of the measures of any two sides of a triangle is
_______ than the measure of the third side.
greater
a
b
c
a + b > c
a + c > b
b + c > a

20. Triangle Inequality Theorem
Triangle Inequality Theorem
Can 16, 10, and 5 be the measures of the sides of a triangle?
No! 16 + 10 > 5
16 + 5 > 10
However, 10 + 5 > 16