triangle inequalities theorems .ppt

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 < 8c
b

c
a
andbc ac
then b, a and 0 c If )

1 c
b

c
a
andbc ac
then b, a and 0 c If )

2 36 24
2 18 2 12
18 12


 96
2
18
2
12


18 12

4. Exterior Angle Theorem
You will learn to identify exteriorangles 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
1
2
34
P
Q R
In the triangle below, ∠1, ∠2, and ∠3 are
_______ angles of ΔPQR.interior
Angle 4 is called an _________ angle
of ΔPQR.
exterior

6. Exterior Angle Theorem
1
2
34
P
Q R
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.

7. Exterior Angle Theorem
1
2
34
P
Q R
In ΔPQR, ∠1, and ∠2 are the remote
interior angles with respect to ∠4.
____________________ of a triangle
are the two angles that do notform a
linear pair with the exterior angle.
Remote interior angles

8. Exterior Angle Theorem
1
2
3 45
In the figure below, ∠2 and ∠3 are remote interior angles with
respect to what angle?∠5

9. 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
432
1
ZY
m∠4 = m∠1 + m∠2

12. Exterior Angle Theorem

13. 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
432
1
ZY
m∠4 > m∠1
m∠4 > m∠2

14. 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

15. Exterior Angle Theorem 3 and 1 

16. Exterior Angle Theorem
The feather–shaped leaf is called a pinnatifid.
In the figure, does x = y? Explain.
x = y
?
__ + 81 = 32 + 7828
28°
109 = 110
No! x does not equal y

17. 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⎳ Pm⎳ L<

18. 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⎳Km⎳J<
J
45°
W
K
60°
75°

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

20. Inequalities Within a Triangle A BC
The longest side is
So, the largest angle isL
The largest angle isMN
So, the longest side is

21. 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

22. 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