Lesson 4.3 – Triangle inequalities & Exterior Angles Homework: 4.3/ 1-10, 12-16

Exterior Angle Theorem An exterior angle of a triangle… … is equal in measure to the sum of the measures of its two remote interior angles. remote interior angles Exterior angle

Exterior Angle Theorem (your new best friend) remote interior angles 3 2 1 4 m<1 + m<2 = m<4

Exterior Angle Theorem m<BCD = m<A + m<B m<4= m<1+ m<2

Examples m<G + 60˚ = 111˚ m<G = 51˚ Exterior angle Remote interior angles m<G + 60˚ = 111˚ m<G = 51˚

x = 68° y = 112° Examples Find x & y x 82° 30° y y = 30 + 82 y = 112˚ Remote interior angles x 82° 30° y y = 30 + 82 y = 112˚ Using Linear pair: 180 = 112 + x 68˚ = x x = 68° y = 112°

Examples Find 2x – 5 = x + 70 x – 5 = 70 x = 75 m< JKM = 2(75) - 5

Solve for y in the diagram. Examples Solve for y in the diagram. Solve on your own before viewing the Solution

solution 4y + 35 = 56 + y 3y + 35 = 56 3y = 21 y= 7

Find the measure of in the diagram shown. Examples Find the measure of in the diagram shown. Solve on your own before viewing the Solution

solution 40 + 3x = 5x - 10 Exterior angle: 5x – 10 = 5(25) - 10 = 125 – 10 = 115 25 = x m < 1= 180 - 115 m < 1= 65

Checkpoint: Complete the exercises.

solution Right Scalene triangle x + 70 = 3x + 10 70 = 2x + 10 60 = 2x 3 (30) + 10 = 100˚

Triangle Inequalities

Construct triangle DEF. Make A Triangle Construct triangle DEF. D E D F F E

Construct triangle DEF. Make A Triangle Construct triangle DEF. D E D F F E

Construct triangle DEF. Make A Triangle Construct triangle DEF. D E

Construct triangle DEF. Make A Triangle Construct triangle DEF. D E

Construct triangle DEF. Make A Triangle Construct triangle DEF. D E

Construct triangle DEF. Make A Triangle Construct triangle DEF. 5 3 13 D E Q: What’s the problem with this? A: The shorter segments can’t reach each other to complete the triangle. They don’t add up.

Triangle Inequality Conjecture The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Add the two smallest sides; they MUST be larger than the third side for the triangle to be formed.

Triangle Inequality Conjecture Given any triangle, if a, b, and c are the lengths of the sides, the following is always true: a + b > c a + c > b b + c > a The triangle inequality theorem is very useful when one needs to determine if any 3 given sides will form of a triangle or not. In other words, if the 3 conditions above are not met, you can immediately conclude that it is not a triangle.

Example Three segments have lengths: a= 3 cm, b= 6 cm, and c = 4 cm. Can a triangle be formed with these measures? 3 + 6 = 9 and 9 > 4 3 + 4 = 7 and 7 > 6 6 + 4 = 10 and 10 > 3 So a triangle can be formed!

Example Three segments have lengths: a= 7 cm, b= 16 cm, and c = 8 cm. Can a triangle be formed with these measures? 7 + 16 = 23 and 23 > 8 7 + 8 = 15 , but 15 < 16. This condition is not met because the sum of these two sides is smaller than the third side 16 + 8 = 24 and 24 > 7 Since one of the conditions is not met, a triangle cannot be formed.

Simply: If the two smallest side measures add up to be greater than the largest side, then the sides make a triangle! If the two smallest side measures do not add up to be greater than the largest side, then the sides do not make a triangle!

Can the following lengths form a triangle? Make A Triangle Can the following lengths form a triangle? 1. 6 mm 5 mm 10 mm 2. 2 ft 9 ft 13 ft 3. 5 cm cm 4 cm 𝟐 4. 7 ft 15 ft ft 𝟏𝟑 5. 10 mm 3 mm 6 mm 6. 4 ft 7 ft ft 𝟕 7. 10 mm 13 mm mm 𝟓 8. 8 m 7 m 1 m 9. 9 mm 2 mm 10 mm 10. 12 mm 22 mm mm 𝟏𝟑 11. 5 mm 8 mm mm 𝟏𝟐 12. 1 mm 5 mm 3 mm

Side-Angle Conjecture In a triangle, the longest side is opposite the largest angle; and the shortest side is opposite the smallest angle. Side AB is the shortest, because it's across from the smallest angle (40 degrees).  Also, the side BC is the longest because it is across from the largest angle (80 degrees).

Side-Angle b B a A C b a A B c What’s the biggest side? 60° 100° What’s the biggest side? What’s the biggest angle? b B What’s the smallest side? What’s the smallest angle? a A

Side-Angle Rank the sides from greatest to least. b b c a a c 92° 42° 46° a b c b c a Rank the angles from greatest to least. A C B 7 5 4 C A B

Practice Find x. 25 + x + 15 = 3x - 10 x + 40 = 3x - 10 40 = 2x - 10

Find x and y. 92 = 50 + x 40 = x 92 + y = 180 y= 88 Exterior angle Linear pair of angles

Find the measures of <‘s 1, 2, 3, & 4 LP: 92 + <1 = 180 <1 = 88 LP: 123 + <2 = 180 <2 = 57 EA: <4 = <1 + < 2 <4 = 88 + 57 <4 = 145 LP: 145 + <3 = 180 <3 = 35

Find the measure of each numbered angle in the figure. Exterior Angle Theorem Simplify. linear pairs are supplementary. Substitution Subtract 70 from each side.

Exterior Angle Theorem Substitution Subtract 64 from each side. If 2 s form a linear pair, they are supplementary. Substitution Simplify. Subtract 78 from each side.

Subtract 143 from each side. Angle Sum Theorem Substitution Simplify. Subtract 143 from each side. Answer:

Your Turn: Find the measure of each numbered angle in the figure. Answer: