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Centre for Education in Mathematics and Computing Faculty of Mathematics University of Waterloo Waterloo, ON Canada N2L 3G1 Name: _______________________ GEOMETRY This resource may be copied in its entirety, but is not to be used for commercial purposes without permission from the Centre for Education in Mathematics and Computing, University of Waterloo. Opposite angles are two angles that have a common vertex but not a common side that are formed by intersecting lines. Opposite angles formed by straight lines have the same measure. For example, ∠b and ∠d are opposite angles and have the same measure. (They form an X-pattern.) b a d c Supplementary angles are two angles whose sum is 180°. For example, ∠a and ∠b are supplementary angles. Complementary angles are two angles whose sum is 90°. For example, ∠c and ∠d are complementary angles. 1. Name all pairs of opposite and supplementary angles for each set of intersecting lines. a. b. j i k b a c d l n r q s t f e g h 2. o p Find the measure of each unknown angle. a. b. b a a b c 130° 3. m 46° c d 75° What is the measure of the angle that is complementary to each angle? a. 14° b. 57° c. 81° Expectation: i) solve angle-relationship problems involving triangles, intersecting lines, and parallel lines and transversals. For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca. 1 4. What is the measure of the angle that is supplementary to each angle? a. 134° 5. b. 87° c. 102° a. If the measure of an angle is 30° more than the measure of its supplement, what is the measure of each angle? b. If the measure of an angle is 30° more than the measure of its complement, what is the measure of each angle? Perpendicular lines intersect at a 90° angle. For example, AB and EF are perpendicular lines. E Parallel lines are lines in the same plane that do not intersect. For example, AB and CD are parallel lines. A B C D F A transversal is a line or line segment that intersects two or more lines. For example, EF is a transversal. b a c d When a transversal intersects two lines, ∠b and ∠f are defined as corresponding angles. ∠d and ∠h are another pair of corresponding angles. (They form a F-pattern.) Also, ∠d and ∠f are called alternate angles. ∠c and ∠e are another pair of alternate angles. (They form a Z-pattern.) f e g h Moreover, when a transversal intersects parallel lines, corresponding angles and alternate angles are equal. 6. Identify each pair of angles as alternate or corresponding angles. b a f c d g e h i j l k n m o p 7. r q s t a. b. c. d. e. f. ∠a and ∠q ∠g and ∠m ∠d and ∠t ∠k and ∠n ∠c and ∠s ∠i and ∠n Find the measures of the unknown angles. a. b. b a 122° c c. b a c 35° 96° a b e d c f e d g Expectation: i) solve angle-relationship problems involving triangles, intersecting lines, and parallel lines and transversals. For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca. 2 8. A transversal intersects two parallel lines. Label a pair of corresponding angles a and b. If the two angles are complementary, what are their measures? An interior angle of a polygon is the angle formed at a vertex of the polygon that lies inside the region enclosed by the polygon. a ∠a+∠b+∠c=180° The sum of the measures of the interior angles of any triangle is 180°. 9. b c Find the measure of the unknown interior angles. a. b. 81° Did you know? c. The triangle is the only basic stable structure out of all polygons. This means that out of all polygons, the triangle can resist the most pressure to remain the same shape. 112° 43° 50° Isosceles triangles are triangles with two equal sides. Alternately, isosceles triangles can be defined as triangles with two equal angles. For example, AB and AC are equal, so ∆ABC is an isosceles triangle. Also, angles ABC and ACB are equal. A C B Equilateral triangles are triangles with all three sides equal. (Using the fact isosceles triangles have two equal angles, we find that all the angles of an equilateral triangles are equal to 60º (180º ÷ 3 = 60º). 10. Find the measures of the unknown angles. a. b. a a c. 32° 73° 70° 51° b 44° b 106° c a d 88° c b Slice of History The name isosceles comes from the Greek iso, which means “same,” and skelos, which means “leg.” (A leg of a triangle is one of its sides.) Expectation: i) solve angle-relationship problems involving triangles, intersecting lines, and parallel lines and transversals. For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca. 3 11. Find the measures of the unknown angles. a. b. a c. d a 127° b 133° 60° 101° b c 25° a b c 49° d c d. e. a40° f. a b a b 127° b c d 110° 66° c c d 85° e f d e e f g 65° CHALLENGE YOURSELF! 12. What is the sum, in degrees, of the measures of interior angles of figure HOUSE? P 13. U O Q T H O E U S Lines PS, QT and RU intersect at a common point O, as shown. P is joined to Q, R to S, and T to U, to form triangles. What is the value of the sum of the measures of the angles P, Q, R, S, T and U? S R EXTENSIONS 14. B C x x ° ° D y A E In the diagram, if ∠BEC = y, then what is the measure of ∠BAC? a. 90 ! ! y b. 90 ! ! d. 180 ! ! 2y e. y y 2 c. 2y Expectation: i) solve angle-relationship problems involving triangles, intersecting lines, and parallel lines and transversals. For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca. 4