3.2 inequalities in one triangle.notebook
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3.2 Inequalities in One Triangle.notebook October 22, 2013
3.2 Inequalities in One Triangle
Lesson Objectives: Students will recognize that only certain side lengths will form a
triangle. To use inequalities involving angles and sides of triangles.
3.2 Inequalities in One Triangle.notebook October 22, 2013
Angle Inequalities: In algebra, you learned about the inequality relationship between two real numbers. This relationship is often used in proofs.
Definition of Inequality:
For any real numbers a and b, a > b if and only if there is a positive number c such that
a = b + c.
Ex: If 5 = 2 + 3, then 5 > 2 and 5 > 3.
3.2 Inequalities in One Triangle.notebook October 22, 2013
Properties of Inequality for Real Numbers
Comparison Property of Inequality a < b, a = b, or a > b
Transitive Property of Inequality1. If a < b and b < c, then a < c.
2. If a > b and b > c, then a > c.
Addition Property of Inequality1. If a > b, then a + c > b + c.
2. If a < b, then a + c < b + c.Subtraction Property of Inequality
1. If a > b, then a c > b c.
2. If a < b, then a c < b c.
The following properties are true for any real numbers a, b, and c
3.2 Inequalities in One Triangle.notebook October 22, 2013
Let's take a look at this figure. Consider <1, <2, and <3.
1
2
3
Exterior angle: an angle formed
by one side of a triangle and the
extension of another side.
Exterior Angle Theorem:
m<1 = m<2 + m<3
Since the angle measures are positive numbers, we can also say that
by the definition of inequality. Which results in the next theorem.
3.2 Inequalities in One Triangle.notebook October 22, 2013
Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles.
Example: A
B
C1
3.2 Inequalities in One Triangle.notebook October 22, 2013
Use the Exterior Angle Inequality TheoremJ K
M
L7
4
6
5
1 2
3
8a. measures less than
b. measures greater than
3.2 Inequalities in One Triangle.notebook October 22, 2013
Your Turn!1. measures less than
2. measures greater than
2
1 8 6
7
3 4
5
Q
P R
3.2 Inequalities in One Triangle.notebook October 22, 2013
AngleSide Inequalities: We know that if two sides of a triangle are congruent, it is an isosceles triangle. Then the angles opposite those sides are also congruent. What kind of relationship exists if the sides are not congruent? Let's examine the longest and shortest sides and smallest and largest angles of a scalene obtuse triangle.
A
C
B
longest side
largest angle
A
C
Bshortest side
smallest angle
Notice that the longest side and largest angle are opposite each other. Likewise the shortest side and smallest angle are opposite each other.
3.2 Inequalities in One Triangle.notebook October 22, 2013
The sideangle relationships in an obtuse scalene triangle are true for all triangles, and are stated using inequalities in the theorems below.
AngleSide Relationships in TrianglesIf one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.
Example: XY > YZ, so m<Z > m<X.
X
Y
Z
9 7
If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
Example: m<J > m<K, so KL > JL.
J
K
L
3.2 Inequalities in One Triangle.notebook October 22, 2013
Identify Arithmetic Sequences:
List the angles of in order from smallest to largest.The sides from shortest to longest
are The angles opposite
these sides are and , respectively. So the angles from smallest to largest are
and . R
P
Q
8.8
7.5
9.1
3.2 Inequalities in One Triangle.notebook October 22, 2013
Your turn:
List the angles and sides of in order from smallest to largest. A
C
B3.7
4.8
7
3.2 Inequalities in One Triangle.notebook October 22, 2013
Order Triangle Side LengthsList the sides of in order from shortest to longest.
First find the missing angle measure using the Triangle Angle Sum Theorem. F
GH
3.2 Inequalities in One Triangle.notebook October 22, 2013
Your Turn!
List the angles and sides of in order from smallest to largest.
W
Y
X