Objectives

1
Objectives

• State the inequalities that relate angles and
  lengths of sides in a triangle
• State the possible lengths of three sides of a
  triangle

2
Angle-Side Relationships in a Triangle

• If two sides of a triangle are not congruent,
  then the larger angle lies opposite the longer
  side.

If XZ gt XY, then m?Y gt m?Z.
If two angles of a triangle are not congruent,
then the longer side lies opposite the larger
angle.
If m?A gt m?B, then BC gt AC
3
Angle-Side Relationships Example

• List the angles from least to greatest in
  measure.
• Since the sides can be ordered by 1.8cm, 2.7cm,
  and 3.9cm, the angles opposite to those sides
  respectively are ?V, ?M, and ?O.

4
Possible Lengths of Sides of a Triangle

• Triangle Inequality Theorem
• The sum of the lengths of any two sides of a
  triangle is greater than the length of the third
  side.
• XY YZ gt XZ
• YZ ZX gt YX
• ZX XY gt ZY

5
Triangle Inequality Theorem Example 1

• Is it possible to have a triangle with side
  lengths of 15, 12, 9 ?
• The sum of any two sides must be greater than the
  remaining side.
• 15 12 gt 9 Yes
• 12 9 gt 15 Yes
• 15 9 gt 12 Yes
• So the three lengths satisfy the Triangle
  Inequality Theorem and are possible in a triangle.

6
Triangle Inequality Theorem Example 2

• Given that two sides of a triangle measure 9 and
  15, what are the possible values of the third
  sides lengths?
• Lower limit 15 9 6 (x 9 gt 15)
• Upper limit 15 9 24 (15 9 gt x)
• 6 lt x lt 24