Geometry Chapter 5 2 and 5 4 Inequalities

Geometry ~ Chapter 5. 2 and 5. 4 Inequalities and Triangles

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

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. n Example: n ¨ <1 is an exterior angle ¨ m < 1 > m <PQR ¨m Holt Geometry < 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. Holt Geometry

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. Holt Geometry

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. Holt Geometry

Example 2 A ~ If m A = 9 x – 7, m B = 7 x – 9 and m C = 28 – 2 x, list the sides of ABC in order from shortest to longest. Draw and label triangle ABC!!! A 101° (9 x – 7) + (7 x – 9) + (28 – 2 x) = 180 (9 x – 7)° B 14 x + 12 = 180 (7 x – 9)° (28 – 2 x)° 75° C 4° 14 x = 168 x = 12 The sides from shortest to longest are AB, AC, BC Holt Geometry

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

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. Holt Geometry

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. Holt Geometry

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. Holt Geometry

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. Holt Geometry

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

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 > – 6 x + 17 > 23 x>6 23 + 17 > x 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. Holt Geometry

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

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. 6 cm < x < 28 cm 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. 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. Holt Geometry