Geometry ~ Chapter 5.2 and 5.4

Inequalities and Triangles

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Inequalities What are they? Angle measures can be compared using

inequalities: m < a = m < b m < a < m < b m < a > m < b

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Exterior Angle Inequality Theorem:

If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of the remote interior angles.

Example:<1 is an exterior angle m < 1 > m <PQR m < 1 > m<RPQ

1

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The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.

If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

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Ex. 1 - Write the angles in order from smallest to largest.

The angles from smallest to largest are F, H and G.

The shortest side is GH, so the smallest angle is opposite GH…. FThe longest side is FH, so the largest angle is G

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Ex. 2 - Write the sides in order from shortest to longest.mR = 180° – (60° + 72°) = 48° The smallest angle is R, so the shortest side is PQ.The largest angle is Q, so the longest side is PR.The sides from shortest to longest are PQ, QR, and PR.

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Example 2A ~ If m A = 9x – 7, m B = 7x – 9 and m C = 28 – 2x, list the sides of ABC in order from shortest to longest.

Draw and label triangle ABC!!! A

B C

(9x – 7)°

(7x – 9)° (28 – 2x)°

(9x – 7) + (7x – 9) + (28 – 2x) = 180 14x + 12 = 180 14x = 168 x = 12

101°

75° 4°

The sides from shortest to longest are AB, AC, BC

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A triangle is formed by three segments, but not every set of three segments can form a triangle.

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A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

If you take 3 straws of lengths 8 inches, 5 inches and 1 inch and try to make a triangle with them, you will find that it is not possible. This illustrates the Triangle Inequality Theorem.

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Ex. 3 – Applying the Triangle Inequality Thm. 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.

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Ex. 4 - 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.

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Ex. 5 - Tell whether a triangle can have sides with the given lengths. Explain.8, 13, 21

No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

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Finding the RANGE of 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.

Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches.

x + 8 > 13x > 5

x + 13 > 8x > –5

8 + 13 > x21 > x

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Ex. 6 - The lengths of two sides of a triangle are 23 inches and 17 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.

Combine the inequalities. So 6 < x < 40. The length of the third side is greater than 6 inches and less than 40 inches.

x + 23 > 17x > –6

x + 17 > 23x > 6

23 + 17 > x40 > x

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Lesson Wrap Up1. Write the angles in order from smallest to

largest.

2. Write the sides in order from shortest to longest.

C, B, A

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Lesson Wrap Up3. The lengths of two sides of a triangle are 17 cm

and 11 cm. Find the range of possible lengths for the third side.

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.

6 cm < x < 28 cm

5. Ray wants to place a chair so it is 10 ft from his television set. Can the other two distancesshown be 8 ft and 6 ft? Explain.Yes; the sum of any two lengths is greater than the third length.