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Objectives. State the inequalities that relate angles and lengths of sides in a triangle State the possible lengths of three sides of a triangle. Angle-Side Relationships in a Triangle. If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

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Objectives

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

  • State the possible lengths of three sides of a triangle


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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 > XY, then m∠Y > m∠Z.

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

If m∠A > m∠B, thenBC > AC


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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.


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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 > XZ

YZ + ZX > YX

ZX + XY > ZY


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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 > 9 Yes

12 + 9 > 15 Yes

15 + 9 > 12 Yes

So the three lengths satisfy the Triangle Inequality Theorem and are possible in a triangle.


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Triangle Inequality Theorem Example 2

Given that two sides of a triangle measure 9 and 15, what are the possible values of the third side’s lengths?

Lower limit = 15 – 9 = 6 (x + 9 > 15)

Upper limit = 15 + 9 = 24 (15 + 9 > x)

6 < x < 24


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