Inequalities and Triangles

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Presentation transcript:

Inequalities and Triangles Geometry ~ Chapter 5.2 and 5.4 Inequalities and Triangles

Inequalities What are they? Angle measures can be compared using inequalities: m < a = m < b m < a < m < b m < a > m < b

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

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.

The angles from smallest to largest are F, H and G. Ex. 1 - Write the angles in order from smallest to largest. The shortest side is GH, so the smallest angle is opposite GH…. F The longest side is FH, so the largest angle is G The angles from smallest to largest are F, H and G.

Ex. 2 - 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 PQ. The largest angle is Q, so the longest side is PR. The sides from shortest to longest are PQ, QR, and PR.

Draw and label triangle ABC!!! 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 101° (9x – 7) + (7x – 9) + (28 – 2x) = 180 14x + 12 = 180 14x = 168 x = 12 (9x – 7)° (7x – 9)° (28 – 2x)° 4° 75° The sides from shortest to longest are AB, AC, BC

A triangle is formed by three segments, but not every set of three segments can 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. A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

Ex. 3 – Applying the Triangle Inequality Thm 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.

Yes—the sum of each pair of lengths is greater than the third length. 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.

Ex. 5 - Tell whether a triangle can have sides with the given lengths 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.

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. x + 8 > 13 x + 13 > 8 8 + 13 > x x > 5 x > –5 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.

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. x + 23 > 17 x + 17 > 23 23 + 17 > x x > –6 x > 6 40 > x Combine the inequalities. So 6 < x < 40. The length of the third side is greater than 6 inches and less than 40 inches.

Lesson Wrap Up 1. Write the angles in order from smallest to largest. 2. Write the sides in order from shortest to longest. C, B, A

Lesson Wrap Up 3. 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. 6 cm < x < 28 cm 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.