Download Document

Indirect Proof and Inequalities
5-5 in One Triangle
Objectives
Apply inequalities in one triangle.
Holt Geometry
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 2A: Ordering Triangle Side Lengths and
Angle Measures
Write the angles in order from
smallest to largest.
The shortest side is
smallest angle is F.
The longest side is
, so the
, so the largest angle is G.
The angles from smallest to largest are F, H and G.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 2B: 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 largest angle is Q, so the longest side is
The sides from shortest to longest are
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.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 3A: Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the
given lengths. Explain.
7, 10, 19
No—by the Triangle Inequality Theorem, a triangle
cannot have these side lengths.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Example 3B: Applying the Triangle Inequality
Theorem
Tell whether a triangle can have sides with the
given lengths. Explain.
2.3, 3.1, 4.6


Yes—the sum of each pair of lengths is greater
than the third length.
Holt Geometry

Indirect Proof and Inequalities
5-5 in One Triangle
Example 4: Finding 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
x + 13 > 8
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 5: Travel Application
The figure shows the
approximate distances
between cities in California.
What is the range of distances
from San Francisco to Oakland?
Let x be the distance from San Francisco to Oakland.
x + 46 > 51 x + 51 > 46 46 + 51 > x Δ Inequal. Thm.
x>5
x > –5
97 > x
Subtr. Prop. of
Inequal.
5 < x < 97 Combine the inequalities.
The distance from San Francisco to Oakland is
greater than 5 miles and less than 97 miles.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Lesson Quiz: Part I
1. Write the angles in order from smallest to
largest.
C, B, A
2. Write the sides in order from shortest to
longest.
Holt Geometry
Indirect Proof and Inequalities
5-5 in One Triangle
Lesson Quiz: Part II
3. The lengths of two sides of a triangle are 17 cm
and 12 cm. Find the range of possible lengths for
the third side.
5 cm < x < 29 cm
4. Tell whether a triangle can have sides with
lengths 2.7, 3.5, and 9.8. Explain.
No; 2.7 + 3.5 is not greater than 9.8.
5. Ray wants to place a chair so it is
10 ft from his television set. Can
the other two distances
shown be 8 ft and 6 ft? Explain.
Yes; the sum of any two lengths is
greater than the third length.
Holt Geometry