triangle inequalities

1. Triangle
Inequalitie
s
Balagtas national agricultural high school

2. 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
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
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
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
1
2 3 4
P
Q R

7. 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

8. 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

9. Exterior Angle Theorem

10. 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

11. 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

12. Exterior Angle Theorem
3
and
1 


13. 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

14. 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
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
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
A

BC
The longest side is
So, the largest angle is
L

The largest angle is
MN
So, the longest side is

18. 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

19. 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