:SB :SH1127A D y B :SH0431C :SH0537E :SH0932k :SP189089210089210032210032 :SP258040210089254089254089 :SF B. Complementary and Supplementary Angles and Angle Sums. Make sure you know these theorems, definitions and assumptions. NOTE: < = Angle ^ = Degrees 1. Two angles whose sum is a right angle are complementary. <k + <y = 90^ 2. Two angles whose sum is 180^ are supplementary. <ADE + <y = 180^ :RA :SD :SB :SH0828A a b B :SH0536C :SH1230D :SH1030d c :SP196064252064252064252064 :SP247040224064210088210088 :SF 3. The sum of the angles about a point is 360^. <a + <b + <c + <d = 360^ 4. The sum of the angles about a point on one side of a straight line is 180 degrees. <a + <b = 180^ <c + <d = 180^ :RA :SD :SB :SH0927A C :SH0333B :SP189064224028259064189064 :SF 5. The sum of the angles of a triangle is 180^. <A + <B + <C = 180^ :RA :SD :SB :SH0827A :SH0735B :SH0939C :SH1432D :SP189064244056266072217104 :SP217104189064189064189064 :SF 6. The sum of the angles of a quadrilateral is 360^. <A + <B + <C + <D = 360^ :RA :SD :SB :SP196024210016245032252056 :SP252056217064196048196024 :SH0328A :SH0231B :SH0436C :SH0737D :SH0932E :SH0628F :SF 7. The sum of the angles of a polygon of n sides is 180 (n - 2) degrees. Note: In the diagram (hexagon ABCDEF), the sum of the angles is: 180(6 - 2) degrees = 720^ :RA :SD :SB :SH0829A E B :SH0431C :SH0636D :SP203055252055252055252055 :SP200032224055245050245050 :SF :Q 1. Given: CE PERPINDICULAR ED; <AEC = 70^. Find: <BED (a) 20^ (b) 70^ (c) 90^ (d) 110^ (e) 180^ :RCA 1. (a) 20^ Ans. The sum of the angles about a point on one side of a straight line is 180^ therefore: <BED + <DEC + <AEC = 180^ Since CE PERPINDICULAR ED, <DEC = 90^ <BED + 90^ + 70^ = 180^ <BED = 20^ Ans. :RA :SD :Q :SB :SH0828A B :SH0338C :SH1038D :SH0931E :SP190060257060257060257060 :SP254025210060250070250070 :SF 2. Given: <CED is a right angle. <CEB is 50^ Find: <AED (a) 40^ (b) 90^ (c) 130^ (d) 140^ (e) It cannot be determined from the information given. :RCD 2. (d) 140^ Ans. <CEB + <BED = 90^ 50^ + <BED = 90^ <BED = 40^ Since the sum of the angles about a point on one side of a straight line is 180^, <BED + <AED = 180^ 40^ + <AED = 180^ <AED = 140^ Ans. :RA :Q 3. <CED is a right angle, and <AEC = <AED. Find the number of degrees in <CEB. (a) 45 (b) 90 (c) 135 (d) 180 (e) 20 :RCA 3. (a) 45 Ans. The sum of the angles around a point is 360^. Therefore: <CED + <AEC + <AED = 360^ 90^ + <AEC + <AED = 360^ <AEC + <AED = 270^ 270^ <AEC = <AED = ---- 2 = 135^ :RA <AEC + <CEB = 180^ 135^ + <CEB = 180^ <CEB = 45^ Ans. :RA :SD :Q 4. In {ABC, <A:<B:<C = 1:2:3. Find the number of degrees in <B. (a) 15 (b) 30 (c) 45 (d) 60 (e) 90 :RCD 4. (d) 60 Ans. <A + <B + <C = 180^ y + 2y + 3y = 180 6y = 180 y = 30 <B = 2y = 60^ Ans. :RA :Q 5. In {ABC, <A = 3<B, and <B = 2<C. Find the number of degrees in <C. (a) 10 (b) 20 (c) 40 (d) 60 (e) 120 :RCB 5. (b) 20 Ans. <A + <B + <C = 180^ 6y + 2y + y = 180 9y = 180 y = 20 <C = y = 20^ Ans. :RA :Q :SB :SP196055266055217032210055 :SP266055279055279055279055 :SH0828D A B E :SH0432C :SF 6. If <DAC = 120^ and <EBC = 150^, then {ABC is: (a) equiangular (b) equilateral (c) isosceles (d) right (e) It cannot be determined from the information given :RCD 6. (d) right Ans. Since the sum of angles about a point on one side of a straight line is 180^ <DAC + <CAB = 180^ 120^ + <CAB = 180^ <CAB = 60^ and <EBC + <ABC = 180^ 150^ + <ABC = 180^ <ABC = 30^ :RA The sum of the angles of a triangle is 180^. Therefore: <CAB + <ABC + <ACB = 180^ 60^ + 30^ + <ACB = 180^ <ACB = 90^, making {ABC a right triangle. Ans. :RA :SD :Q :SB :SH0330A :SH1127D C B :SH1238E :SP203079203024273096273096 :SP182079256079256079256079 :SF 7. If <DCA = 2<A, and <CBE = 3<A, then the number of degrees in <A is: (a) 15 (b) 30 (c) 45 (d) 74 (e) 75 :RCC 7. (c) 45^ Ans. The sum of the angles of a triangle is 180^ <A + <B + <C = 180^ Since the sum of the angles about a point on one side of a straight line is 180^: <C = (180^ - 2A) <B = (180^ - 3A) <A + (180^ - 3A) + (180^ - 2A) = 180^ <A = 45^ Ans. :RA :SD :Q :SB :SP189064244056266072217104 :SP217104189064189064189064 :SH0827A :SH0735B :SH0939C :SH1432D :SF 8. In quadrilateral ABCD, <A = y^, <B = 2y^, <C = y - 30^, and <D = y + 30^ Find the number of degrees in <A. (a) 36 (b) 43 (c) 72 (d) 102 (e) 144 :RCC 8. (c) 72 Ans. <A + <B + <C + <D = 360^ y + 2y + y - 30 + y + 30 = 360 5y = 360 y = 72^ <A = y = 72^ Ans. :RA :SD :Q 9. Each angle of an equiangular pentagon (a polygon of 5 sides) is: (a) 72^ (b) 90^ (c) 108^ (d) 180^ (e) 540^ :RCC 9. (c) 108^ Ans. Note: The sum of the angles of a polygon = (N - 2)180 (5 - 2)180 = (3)180 = 540^ in all 540/5 = 108^ each Ans. :RA :Q :SB :SH0528A B :SH0928C D :SH0739E :SP196036245036266052245068 :SP245068196068196068196068 :SF 10. Given: AB || CD; <B = 140^; <D = 150^. Find: <E. (a) 10^ (b) 30^ (c) 40^ (d) 50^ (e) 70^ :RCE 10. (e) 70^ Ans. NOTE: Draw a line perpindicular to lines AB and CD. The sum of the angles in polygon ABEDC = (5 - 2) x 180^ = 540^ Both <A and <C = 90^ 90^ + 140^ + <E + 150^ + 90^ = 540^ <E = 70^ Ans. :ET :ET
