Angle: An angle is the union of two line segments (or two rays) with a common endpoint, called a vertex.
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1 MATH 008: Angles Angle: An angle is the union of two line segents (or two rays) with a coon endpoint, called a vertex. A B C α Adjacent angles: Adjacent angles are two angles that share a vertex, have a coon side, but whose interiors do not intersect. A B D C Classification of Angles according to their easureents: Acute angle: angle easuring less than 90 Right angle: angle easuring 90 Obtuse angle: angle easuring ore than 90 but less than 80 Straight angle: angle easuring 80 Reflex angle: angle easuring ore than 80
2 2 MATH 008: ANGLES Vertical Angles: Opposite angles fored by two intersecting lines are called vertical angles. IMPORTANT: Vertical angles always have the sae easureent. 2 4 In the above figure, and are vertical angles; 2 and 4 are vertical angles. Copleentary Angles: Two angles whose su is 90 are called copleentary angles. If A and B are copleentary angles, then A is the copleent of B and B is the copleent of A. 2 In the above figure, and 2 are copleentary angles. Suppleentary Angles: Two angles whose su is 80 are called suppleentary angles. If A and B are suppleentary angles, then A is the suppleent of B and B is the suppleent of A. 2 In the above figure, and 2 are suppleentary angles.
3 MATH 008: ANGLES Exaple : Find the easure of each arked angle. (a) (x - 4) (4x +9) (b) (x + 25) (6x - 7) (c) (x-0) (6x+)
4 4 MATH 008: ANGLES ANGLES ASSOCIATED WITH PARALLEL LINES: Corresponding Angles have the sae location relative to lines l, and transversal t. (IMPORTANT: l if and only if corresponding angles fored by l,, and t are congruent.) In Figure A-, and 5 are corresponding angles. The following pairs are also corresponding angles: 2 and 6; and 7; 4 and 8. t l Figure A- Alternate Interior Angles are nonadjacent angles fored by lines l,, and transversal t, the union of whose interiors contain the region between l and. (IMPORTANT: l if and only if alternate interior angles fored by l, and t are congruent.) In Figure A-, and 6 are alternate interior angles. Likewise, 4 and 5 are also alternate interior angles. Alternate Exterior Angles are angles on the outer sides of two lines cut by a transversal, but on opposite sides of the transversal (IMPORTANT: l if and only if alternate exterior angles fored by l, and t are congruent.) In Figure A-, 2 and 7 are alternate exterior angles. Siilarly, and 8 are alternate exterior angles. Interior Angles on the sae side of the transversal are interior angles whose interiors are the sae. (IMPORTANT: l if and only if the interior angles on the sae side of the transversal are suppleentary.) In Figure A-, and 5, as well as 4 and 6, are interior angles on the sae side of the transversal.
5 MATH 008: ANGLES 5 Exaple 2: In the diagra below, l and r s. Find the easureent of each nubered angle l r s NOTES: The su of the angles inside a triangle is 80. The su of the angles inside a quadrilateral is 60. Exaple : In the diagra below, BG EF. Find the easure of each angle. B E C A H G F
6 6 MATH 008: ANGLES Exercises In # #6 find the easure of each arked angle.. 4. (x - 7) (7x+27) (5x + ) (4x + 6) (x+) (4x - 56) (8x + 7) (x - 4). (x - 5) (5x + ) 6. (x-) (4x+7)
7 MATH 008: ANGLES 7 7. Using the diagra below, nae the relationship between the following pairs of angles, given l and n p n p l a) and 5 b) and 5 c) and 9 d) and 0 e) 2 and 6 f) 6 and 8 g) and 7 h) 4 and 0 i) 7 and 0 8. In the diagra below, n. Find the easure of each nubered angle n 9. In the diagra below, t s and n. Find the easureent of each nubered angle. t s n
8 8 MATH 008: ANGLES 0. In the diagra below, l and r s. Find the easureent of each nubered angle l t r s. In the following figure, n and r s. Given the angle easures indicated on the figure, find the easure of each lettered angle. r 50 n 05 a b c e d s f 2. Find the easure of x x. In the following figure, AB CD. Find the easure of x and y. B x D 70 A y 50 C
9 MATH 008: ANGLES 9 4. In the figure below, n. Find the easure of each labeled angle. n e d c f b a 6 5. Find the easure of a and b. b a Find the of each labeled angle. 50 a b c i 60 d 0 0 e f g h In the figure below, ( BF C) = 55, ( AF D) = 50, and ( BF E) = 20. Deterine the easures of AF B and CF D. B C D A F E
10 0 MATH 008: ANGLES ANSWERS. Both angles are , 2. 54, , Both angles are , (a) alternate exterior angles (b) alternate exterior angles (c) corresponding angles (d) interior angles on the sae side of the transversal (suppleentary angles) (e) alternate interior angles (f) corresponding angles (g) alternate interior angles (h) alternate interior angles (i) interior angles on the sae side of the transversal (suppleentary angles) 8. ( ) = 82, ( 2) = 58, ( ) = 58, ( 4) = 82, ( 5) = 40, ( 6) = 58, ( 7) = 40, ( 8) = ( ) = 65, ( 2) = 5, ( ) = 65, ( 4) = 5, ( 5) = 5, ( 6) = ( ) = 05, ( 2) = 75, ( ) = 05, ( 4) = 05, ( 5) = 9, ( 6) = 6, ( 7) = 05, ( 8) = 9, ( 9) = 6, ( 0) = 05. ( a) = 40, ( b) = 75, ( c) = 55, ( d) = 45, ( e) = 5, ( f) = ( x) = 8. ( x) = 50, ( y) = ( a) = 8, ( b) = 64, ( c) = 08, ( d) = 72, ( e) = 55, ( f) = 5 5. ( a) = 58, ( b) = ( a) = 70, ( b) = 0, ( c) = 00, ( d) = 20, ( e) = 20, ( f) = 20, ( g) = 80, ( h) = 60, ( i) = ( AF B) = 60, ( CF D) = 5
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