5-5 Inequalities in Triangles 1/13/17 Objective: To use inequalities involving angles of triangles and sides of triangles. COMPARISON PROPERTY OF INEQUALITY If a = b + c and c > 0, then a > b (If c were negative, b would be > a)

b + c > b + 0 2) Addition Property of Inequality (+b each side) Do not write in notes PROOF Given: a = b + c, c > 0 Prove: a > b STATEMENTS REASONS c > 0 1) Given b + c > b + 0 2) Addition Property of Inequality (+b each side) b + c > b 3) Simplify a = b + c 4) Given a > b 5) Substitution (a for b + c in statement 3)

COROLLARY TO THE TRIANGLE EXTERIOR ANGLE THEOREM Do not write the proof in notes COROLLARY TO THE TRIANGLE EXTERIOR ANGLE THEOREM The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles m 1 > m 2 and 3 m 1 > m 3 2 1 Informal Proof: The Exterior Angle Theorem says that 1 = 2 + 3. Since both 2 and 3 are > 0, the Comparison Property of Inequality shows that 1 is greater than both 2 and 3.

Ex: m 2 = m 1 by Isosceles Triangle Theorem. Explain why m 2 > m 3. By the Corollary to the Exterior Angle Theorem, m 1 > m 3. So, m 2 > m 3 by Substitution. O 3 P 1 4 2 T Y Explain why m OTY > m 3 We know 2 > 3, and OTY = 2 + 4, so OTY > 3

Two VERY important theorems. Do not write the proof, however. If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. Y If XZ > XY, then m Y > m Z X Z THEOREM 5-11 If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. B If m A > m B, then BC > AC C A GIVEN: m A > m B PROVE: BC > AC by CONTRADICTION!!! STEP 1: Assume BC > AC. That is, assume BC < AC or BC = AC. STEP 2: If BC < AC, then m A < m B (Theorem 5-10). This contradicts the given fact that m A > m B. Therefore BC < AC must be false. If BC = AC, then m A = m B. (Isosceles Triangle Theorem). This also contradicts m A > m B. Therefore BC = AC must be false. STEP 3: The assumption BC > AC is false so BC > AC.

Ex: In TUV, which side is the shortest? T m T = 60 by Triangle Sum Theorem. U is the smallest angle. 58o 62o Side TV is shortest by Theorem 5-11 U V List the sides of XYZ in order from shortest to longest. Y X 40o 60o Z . 80° YZ < YX < XZ

Ex: Can a triangle have sides with the given lengths? Explain. Another VERY important theorem. THEOREM 5-12 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 Y YZ + ZX > YX ZX + XY > ZY X Z Ex: Can a triangle have sides with the given lengths? Explain. a) 3 ft, 7 ft, 8 ft b) 3 cm, 6 cm, 10 cm 3 + 7 > 8 10 + 3 > 6 8 + 7 > 3 10 + 6 > 3 3 + 8 > 7 YES 3 + 6 > 10 NO 2 m, 7 m, 9 m 4 yd, 6 yd, 9 yd 2 + 7 = 9… NO: it must be greater 4 + 6 > 9… YES Just check the two smallest sides. If their sum is > than the third side, it’s a Δ.

Longer than 2 cm and shorter than 18 cm Ex: A triangle has sides of lengths 8 cm and 10 cm. Describe the lengths possible for the third side. x = the third side x can be 8 + 10 “at most” x is “at least” 10 – 8 Longer than 2 cm and shorter than 18 cm 2 < x < 18 {do NOT use <} 1) A triangle has sides 3 in, 12 in. Describe the lengths possible for the third side. 9 < x < 15 (12 – 3)   (12 + 3)

Use ABC to describe the possible lengths of AC 1. The exterior angle is larger than either of the remote interior angles. Angle 3 is opposite AB. Angle 1 is opposite BC. AB > BC so Angle 3 > Angle 1. A 1 25 3 4 27o B 12 C D Explain why m 4 > m 1 Explain why m 3 > m 1 Use ABC to describe the possible lengths of AC _____________________________________________ Can a triangle have lengths of 2 mm, 3 mm, and 6 mm? Explain. In XYZ, XY = 5, YZ = 8, and XZ = 7. Which angle is the largest? In PQT, m P = 50 and m T = 70. Which side is shortest? 13 < AC < 37 NO: 2 + 3 < 6 Angle X, since it’s opposite the largest side, YZ. Angle Q = 60, so the side opposite Angle P is the shortest. That side is QT.

USING A VARIABLE pg 280 A B C Points A, B, and C are collinear. BC is 6 less than twice AB, and AC = 30. What is the length of BC? Let x = AB AC = 30 “BC is 6 less than twice AB” so BC = 2x - 6 x + 2x – 6 = 30 3x – 6 = 30 3x = 36 x = 12, so AB = 12 BC = 2 • 12 – 6 BC = 18

Assignment: Page 276 #4 – 21, 32, 34 – 36