Geometry Lesson 6 4 Inequalities for One Triangle

Geometry Lesson 6. 4 Inequalities for One Triangle Tomorrow we will learn inequalities For Two Triangles

The Main Idea You can think of an angle as the jaw bones of an alligator. When you prop the jaw open with a stick, you form a triangle.

The larger the stick, the larger the angle formed by the jawbones. The smaller the stick, the smaller the angle formed by the jawbones.

You could prop a very long stick in the mouth of a hippo, creating a very large angle. long stick large angle

It follows logically that the longest side of a triangle would be opposite the largest angle

And the shortest side would be opposite the smallest angle.

Theorem If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. A If BC > AB, then small m<A > m<C. C big B

Crikey, that seems easy! Longest side opposite biggest angle, shortest side opposite smallest angle, medium side opposite medium angle.

Examples: What ’s are the largest and smallest? A 2. A 1. x -3 x + 10 10 6 B B C 9 Largest : _____; Smallest : ______ C x Largest : _____; Smallest : ______ Examples: What sides are the largest and smallest? 3. A 4. A 90 B 60 5 x 30 C Largest side: _____; Smallest side: _____ B 2 x 10 x C Largest side: _____; Smallest side: _____

Consider two sides of a triangle, each side 6 inches long 6 6 What is the maximum length the third side can be? 3 5 9

6 6 3 5 6 6 11. 999 6 6 12 6 6 9 How long could the third side be? Could it be 12 or more inches long? No – ∆ could NOT be drawn

Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 9 + 11 = 20 11 + 14 = 25 14 + 9 = 23 20 > 14 A 25 > 9 14 23 > 11 C 9 11 B

Example The lengths of two sides of a triangle are 3 and 5. The length of the third side must be greater than ____ and less than _______. Let x be the length of the third side. x+3>5 x> 2 3+5>x 8>x x< 8 x+5>3 x > -2

Examples: Is this a triangle with these side measures possible? 5. 10, 9, 8 6. 16, 11, 5

Sample Problems 1. 2. A x B 3. ABCD is parallelogram A A x-5 x+3 x C x-3 D 54 52 C B B Largest : ____; Smallest : ____ 4. 5. Largest : _____; Smallest : ____ 6. A A C 53 60 A B 68 D 3 x B 54 Largest side: _____ Smallest side: _______ C D 58 Largest side: ____ Smallest side: ____ 59 4 x 11 x C C Longest Side: ____ B

Given the possible sides listed, do we have a triangle: The 3 rd side of this triangle must be between what two numbers: 7. 9. 11. 8. 5, 11, 15 10, 0. 5 Sides: 5, 15 Sides: 4, 5 _____ < 3 rd side <______ 10. Sides: 20, 14 12. _____ < 3 rd side <_____ Sides: 3 x, 7 x _____ < 3 rd side <______

Homework pg. 222 #1 -17

Corollaries The segment from a point to a line is the shortest segment from the point to the line. ● The segment from a point to a plane is the shortest segment from the point to the plane. ●

The following is an Anti- Drug Message…….

Examples A 6 B Which is the largest angle? 10 Which is the smallest angle? C 8 A 20˚ 120˚ B 40˚ C Name the side lengths from shortest to longest. BC AB AC <B <C

Nude crack man in alligator rescue Thursday, November 30, 2006 In further news of the misadventures of people who are naked, a pulled from the jaws of an alligator in Florida on Wednesday. naked man on crack was He was only saved when sheriff's deputies jumped into the lake where he was being eaten, and wrestled with the alligator until it let go of the man, before pulling the man about 40 m back to the shore. They had responded to several reports from members of the public that screams had been heard coming from the lake, at around 4 o'clock in the morning. The man – identified as 45 -year-old Adrian Apgar – lost his left arm in the alligator attack, as well as suffering severe injuries to his right arm and left leg. His right leg is believed to be fine. 'He was totally naked, ' noted Sheriff Grady Judd. He admitted to the deputies after he was pulled from the water that he'd been smoking crack. No explanation has yet been offered for why he was naked, although police have not ruled out the possibility that he might have decided to go for a swim. Judd added: 'He admitted that he'd been smoking crack cocaine. But still, it's a human life. Our deputies don't ask questions, they respond and they save people. ' At the time of writing, the condition of the alligator is unknown.

Getty Images In Memory Steve Irwin 1962 - 2006