Triangle Inequalities (Textbook Lesson 5.5)

1. MM1G3b -Understand and use the triangle inequality, the side-angle inequality, and the exterior angle inequality. Triangle Inequalities(Textbook Lesson 5.5)

2. Important Triangle Facts • A triangle has 6 parts: • 3 sides • 3 angles • The sum of the angles in a triangle is always 180°. • A triangle is named by its vertices. Name the triangle and its parts. A ∆ ABC R ∆ RST AB BC AC RS ST RT <ACB <RST T B C S <BAC <TRS <CBA <STR

3. Classifying Triangles • Classify triangles by the side lengths. • Equilateral • Isosceles • Scalene • Classify triangles by angle measures. • Right • Acute • Obtuse • Equiangular • Do Parts I and II of the Triangle Notes Handout. – all sides are equal – at least two sides are equal – no sides are equal – has one 90⁰ angle – all angles are less than 90⁰ – has one angle greater than 90⁰ – all angles are equal

4. Triangle Side Angle Inequalities • Smaller angles are opposite shorter sides. • Larger angles are opposite longer sides. Example. 1. In ∆ABC , list the sides in order from smallest to largest. 2. In ∆JKL , list the angles in order from smallest to largest. 11 in 98⁰ 15 in 47⁰ 35⁰ 24 in <KJL, <JKL <JLK, AB, BC, AC

5. Handout straight edges and compasses • Construct triangles: • 3”, 4”, 5” • 2”, 3”, 6”

6. Triangle Inequality Theorem • The sum of any two lengths of any two sides of a triangle is greater than the length of the third side. • Could say that the sum of the two shorter sides must be greater than the longest side. • If the 3rd side is equal to or less than the sum of the 2 other sides, then it can not form a triangle. Examples – • Can these three sides form a triangle? • 5, 8, 16 • 6, 11, 14 • 8, 13, 5 5 + 8 < 16 NO 6 + 11 > 14 YES 5 + 8 = 13 NO

7. Triangle Inequality Theorem • If two sides of a triangle are given, describe the possible lengths of the third side. • 2 yd, 6 yd What possible values would work? Compare the sum of 2 shorter sides to longest side. So the third side has to be bigger than 4 and less than 8 or 4 < x < 8. • Yes • Yes • No • No • No • If 3rd side is 1: 1 + 2 > 6 • If 3rd side is 2: 2 + 2 > 6 • If 3rd side is 3: 3 + 2 > 6 • If 3rd side is 4: 4 + 2 > 6 • If 3rd side is 5: 5 + 2 > 6 • If 3rd side is 6: 6 + 2 > 6 • If 3rd side is 7: 6 + 2 > 7 • If 3rd side is 8: 6 + 2 > 8 • If 3rd side is 9: 6 + 2 > 9 • If 3rd side is 10: 6 + 2 > 10 • No • No • No • No • Yes

8. Triangle Inequality Theorem • If two sides of a triangle are given, describe the possible lengths of the third side. • 2 yd, 6 yd So the third side has to be bigger than 4 and less than 8 or 4 < x < 8. In other words, 2 + x > 6 or 2 + 6 > x – 2 – 2 x > 4 or Therefore, the range is going to be x has to greater than the difference or less than the sum of the two given sides. 8 > x x < 8

9. Triangle Inequality Theorem • If two sides of a triangle are given, describe the possible lengths of the third side. • 4 in, 12 in 4 + x > 12 and 4 + 12 > x – 4 – 4 x > 8 and C. 3 ft, 18 ft 3 + x > 18 and 3 + 18 > x – 3 – 3 16 > x 21 > x x < 15 x < 16 x < 21 15 < x < 21 8 < x < 16

10. Exterior Angle Inequality Theorem • The measure of an exterior angle of a triangle is greater than the measure of either of the nonadjacent (remote) interior angles. • The measure of an exterior angle is the sum of the remote interior angles. Example What relationships do we know about the angles listed? 65° 6 2 3 (3x – 5)° 53 3 4 1 What do you know about x? 5 3x – 5 > 53 3x – 5 > 65 <5 > <2 <5 > <3 <5 = <2 + <3 3x – 5 = 53 + 65 <4 > <1 <4 > <2 <4 = <1 + <2 3x – 5 = 118 + 5 = +5 <6 > <3 <6 > <1 <6 = <1 + <3 3x = 123 x = 41

11. Classwork/Homework • Textbook p287 (4-9,13-24 all)