Indirect Proof and Inequalities
5-5 in One Triangle
The positions of the longest and shortest sides of
a triangle are related to the positions of the
largest and smallest angles.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 1: Ordering Triangle Side Lengths and Angle
Measures
Write the angles in order from
smallest to largest.
The shortest side is
smallest angle is F.
, so the
The angles from smallest to largest are F, H and G.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 2: Ordering Triangle Side Lengths and Angle
Measures
Write the sides in order from
shortest to longest.
mR = 180° – (60° + 72°) = 48°
The smallest angle is R, so the
shortest side is
.
The sides from shortest to longest are
Holt Geometry
48°
Indirect Proof and Inequalities
5-5 in One Triangle
Example 3:
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 4:
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
A triangle is formed by three segments, but not
every set of three segments can form a triangle.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
A certain relationship must exist among the lengths
of three segments in order for them to form a
triangle.
NOTE: Just check that the sum of the two
shorter sides is greater than the longest
side.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 5: Applying the Triangle Inequality
Theorem
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 5: Applying the Triangle Inequality
Theorem
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 6: Finding Possible Side Lengths
The lengths of two sides of a triangle are 8
inches and 13 inches. Find the range of
possible lengths for the third side.
Let x represent the length of the third side. Then
apply the Triangle Inequality Theorem.
x + 8 > 13
x>5
8 + 13 > x
21 > x
Combine the inequalities. So 5 < x < 21. The length
of the third side is greater than 5 inches and less
than 21 inches.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 7
The lengths of two sides of a triangle are 22
inches and 17 inches. Find the range of possible
lengths for the third side.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
You can also use side lengths to classify a
triangle as acute or obtuse.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Holt Geometry