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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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
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.
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
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.
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