Understanding Exterior Angles in Polygons
This content explains the concept of exterior angles in polygons and the Exterior Angle Theorem. It covers how exterior angles are formed when the sides of a polygon are extended, their relationship with interior angles, and how to calculate their measures using the Exterior Angle Theorem. Various examples are provided to demonstrate the application of the theorem in solving for unknown angle measures.
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Angles When the sides of a polygon are extended, other angles are formed. The original angles are the interior angles. The angles that form linear pairs with the interior angles are the exterior angles.
EXTERIOR ANGLE THEOREM An exterior angle of a triangle is equal in measure to the sum measures of its two remote interior angles. sum of the remote remote interior interior angles angles Exterior angle Exterior angle
EXTERIOR ANGLE THEOREM (YOUR NEW BEST FRIEND) The measure of an exterior angle in a triangle is the sum sum of the measures of the 2 remote interior remote interior angles exterior exterior angle angle remote remote interior interior angles angles 2 2 1 1 3 3 4 4 m<1 + m<2 = m<4 m<1 + m<2 = m<4
EXTERIOR ANGLE THEOREM m 1 = m A + m B
EXAMPLES m<FHG = 180 111 = 69 linear pair m<G = 60 + 69
EXAMPLES Find x & y y = 30 + 82 y = 112 82 30 x y 180 = 30 + 82 + x 180 = 112 + x 68 = x x = 68 y = 112
EXAMPLES Find m JKM 2x 5 = x + 70 x 5 = 70 x = 75 m< JKM = 2(75) - 5 m< JKM = 150 - 5 m< JKM = 145
EXAMPLES Solve for y in the diagram. Solve on your own Solve on your own before viewing the before viewing the Solution
SOLUTION 4y + 35 = 56 + y 3y + 35 = 56 3y = 21 y= 27 y= 27
EXAMPLES Find the measure of in the diagram shown. 1 Solve on your own Solve on your own before viewing the before viewing the Solution
SOLUTION 40 + 3x = 5x - 10 40 = 2x - 10 50 = 2x 25 = x 25 = x m < 1= 5x - 10 m < 1= 5(25) - 10 m < 1= 125 - 10 m m < 1= 115 < 1= 115
SOLUTION x + 70 = 3x + 10 70 = 2x + 10 60 = 2x 30 = x Right Scalene triangle 3 (30) + 10 = 100
Make A Triangle Construct triangle DEF. D E D F F E
Make A Triangle Construct triangle DEF. D E D F F E
Make A Triangle Construct triangle DEF. D E
Make A Triangle Construct triangle DEF. D E
Make A Triangle Construct triangle DEF. D E
Make A Triangle Construct triangle DEF. 5 3 D E 13 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 larger than the third side for the triangle to be formed.
Make A Triangle Can the following lengths form a triangle? 2.2 ft 2 ft 9 ft 9 ft 13 ft 13 ft 1.4 mm 4 mm 5 mm 5 mm 10 mm 10 mm 3.5 cm 5 cm cm 4 cm 4 cm 4.7 7 ft ft 15 15 ft ft ft ft ?? cm ? 6.7 7 ft ft 7 7 ft ft ft ft 5.10 mm 10 mm 3 mm 3 mm 6 mm 6 mm 7.10 mm 10 mm 13 mm 13 mm mm mm ? 8.8 m 8 m 7 m 7 m 1 m 1 m ? 10. 12. 9. 11. 1 mm 1 mm 5 mm 5 mm 3 mm 3 mm 9 mm 9 mm 2 mm 2 mm 1 mm 1 mm 12 mm 12 mm 22 mm 22 mm mm mm ?? 7 mm 7 mm 8 mm 8 mm mm mm ??
Side-Angle Conjecture In a triangle, if one side is longer than another side, then angle opposite the longer side is larger than the other.
Side-Angle C b a A B c What s the biggest side? What s the biggest angle? b B a A What s the smallest side? What s the smallest angle?
Side-Angle Rank the sides from greatest to least. 46 b b c a a 42 92 c Rank the angles from greatest to least. A C A B 7 4 B C 5