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Triangle InequalitiesTriangle InequalitiesTriangle InequalitiesTriangle 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
Segments, Angles, and InequalitiesSegments, Angles, and Inequalities
You will learn to apply inequalities to segment and angle measures.
1) Inequality
Inequalities
Segments, Angles, and InequalitiesSegments, Angles, and Inequalities
The Comparison Property of Numbers is used to compare two line segments ofunequal measures.
The property states that given two unequal numbers a and b, either:
a < b or a > b
The same property is also used to compare angles of unequal measures.
T U2 cm
V W4 cm
The length of is less than the length of , or TU < VW TU VW
Segments, Angles, and InequalitiesSegments, Angles, and Inequalities
J
133°K
60°
The measure of J is greater than the measure of K.
The statements TU > VW and J > K are called __________ becausethey contain the symbol < or >.
inequalities
Postulate 7 – 1
ComparisonProperty
For any two real numbers, a and b, exactly one of the following statements is true.
a < b a = b a > b
Segments, Angles, and InequalitiesSegments, Angles, and Inequalities
6420-2
S D N
Replace with <, >, or = to make a true statement.
SN DN
6 – (- 1) 6 – 2
7 4>
>
Lesson 2-1Finding Distanceon a number line.
Segments, Angles, and InequalitiesSegments, Angles, and Inequalities
Theorem7 – 1
If point C is between points A and B, and A, C, and B are collinear, then ________ and ________.
A C B
AB > AC AB > CB
A similar theorem for comparing angle measures is stated below.This theorem is based on the Angle Addition Postulate.
Segments, Angles, and InequalitiesSegments, Angles, and Inequalities
Theorem7 – 2
then,EF and EDbetween is EP If
and DEPm DEFm PEFm DEFm
D
P
F
E
A similar theorem for comparing angle measures is stated below.This theorem is based on the Angle Addition Postulate.
Segments, Angles, and InequalitiesSegments, Angles, and Inequalities
108°
149°45°
40°
18°
A
B
C
D
Replace with <, >, or = to make a true statement.
mBDA mCDA
45° 40° + 45°
<
<
Use theorem 7 – 2 to solve the following problem.
CDAm BDA m
then,DA and DCbetween is DB Since
Check:
CDA BDA mm
45° 85°
Segments, Angles, and InequalitiesSegments, Angles, and Inequalities
Property
TransitiveProperty
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.
Segments, Angles, and InequalitiesSegments, Angles, and Inequalities
Property
Addition andSubtractionProperties
Multiplicationand DivisionProperties
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 < 31 + 5 < 3 + 5
6 < 8
c
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
962
18
2
12
18 12
Exterior Angle TheoremExterior Angle Theorem
You will learn to identify exterior angles and remote interiorangles of a triangle and use the Exterior Angle Theorem.
1) Interior angle
2) Exterior angle
3) Remote interior angle
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 ofΔPQR.
interior
Angle 4 is called an _______ angle of ΔPQR.exterior
An exterior angle of a triangle is an angle that forms a _________ with one ofthe 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 forma linear pair with the exterior angle.Remote interior angles
In ΔPQR, 1, and 2 are the remote interior angles with respect to 4.
Exterior Angle Theorem Exterior Angle Theorem
1
2
3 4 5
In the figure below, 2 and 3 are remote interior angles with respect towhat angle? 5
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
432
1
ZY
m4 = m1 + m2
Exterior Angle Theorem Exterior Angle Theorem
Exterior Angle Theorem Exterior Angle Theorem
Theorem 7 – 4
Exterior Angle
InequalityTheorem
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
m4 > m1m4 > m2
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 – 5If a triangle has one right angle, then the other two angles
must be _____.acute
Exterior Angle Theorem Exterior Angle Theorem
3 and 1
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 + 7828
28°
109 = 110
No! x does not equal y
Inequalities Within a Triangle Inequalities Within a Triangle
You will learn to identify the relationships between the _____and _____ of a triangle.
sidesangles
Nothing New!
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
811
L
P
M
in the same order
LP < PM < ML
mM < mPmL <
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
mW < mKmJ <
J
45°W K
60°
75°
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
35
4 Y
W
X
WY > XY
Inequalities Within a Triangle Inequalities Within a Triangle
A
The longest side is BC
So, the largest angle is
LThe largest angle is
MNSo, the longest side is
Triangle Inequality Theorem Triangle Inequality Theorem
You will learn to identify and use the Triangle Inequality Theorem.
Nothing New!
Triangle Inequality Theorem Triangle Inequality Theorem
Theorem 7 – 9
TriangleInequalityTheorem
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
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