Solve angle-relationship problems involving triangles, intersecting by cna67568

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									University of Waterloo                                                                                                                Centre for Education in
Waterloo, ON Canada                                                                                                               Mathematics and Computing
N2L 3G1                                                                                                                               Faculty of Mathematics

                                                                     GEOMETRY
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               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                                                  a
               measure. (They form an X-pattern.)                                                                        b         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
                                                                                                 l
                                    b a
                                     c d                                                                         m
                                                                                                             n       p
                                                                                                r q              o
                                        f e
                                         g h                                                    s t


 2.       Find the measure of each unknown angle.

          a.                                                              b.

                             a                                                                   b a
                          b     c                                                          46          d
                           130                                                                  c
                                                                                                      75



 3.       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.                            1
For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
 4.       What is the measure of the angle that is supplementary to each angle?

          a. 134                                 b. 87                                  c. 102

 5.       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.                                                                       E
               For example, AB and EF are perpendicular lines.
                                                                                                                    A                         B
               Parallel lines are lines in the same plane that do not intersect.
               For example, AB and CD are parallel lines.                                                           C                         D

               A transversal is a line or line segment that intersects two or more lines.                                          F
               For example, EF is a transversal.

               When a transversal intersects two lines, b and f are defined as                                                        b a
               corresponding angles. d and h are another pair of corresponding                                                       c d
               angles. (They form a F-pattern.) Also, d and f are called alternate                                               f e
               angles. c and e are another pair of alternate angles.                                                            g h
               (They form a Z-pattern.)

               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                          a.   a and q
                     c d g e                         b.   g and m
                            h
                           i                         c.   d and t
                          j l
                           k
                                                     d.   k and n
                                                     e.   c and s
                   n m          r q                  f.   i and n
                  o p            s t

 7.       Find the measures of the unknown angles.

          a.                                         b.                                         c.                                b a
                     b a                                                                                                     c
                    122 c
                                                                         96 a                                               35

                                                                         b
                                e d                                          c
                                                                                                                    e d
                                                                                                                f    g


Expectation: i) solve angle-relationship problems involving triangles, intersecting lines, and parallel lines and transversals.                   2
For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
 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.                               b       c


 9.       Find the measure of the unknown interior angles.
                                                                                                                                  Did You Know?
          a.                                     b.                                 c.
                           81                                                                                           The triangle is the only
                                                                                                                         basic stable structure
                                                                                                                         out of all polygons. This
                                                            112                                                         means that out of all
                43
                                                                                                        50              polygons, the triangle
                                                                                                                         can resist the most
                                                                                              A                          pressure to remain the
               Isosceles triangles are triangles with two equal                                                          same shape.
               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.                                                                 B               C

               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                     c.
                 a                                                                                                    32
                                       73
                         70     b                                           b                                      d
                                                                 44                                              88 c
                                                                           106
               51                           c                                                                a       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.                      3
For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
 11.       Find the measures of the unknown angles.

           a.                                                             b.                                        c.                                d
                                   a
                                                                                     a
                                                                                            127                                    25
                                                                                                                                a                 c
                             60                   b                                                                                          b
                                                   133                   101
                                                                                 b                                          49
                     c                                         d
                                                                                                            c


           d.                                                             e.                                        f.
                          40                                                        a           b 127                     a
                         a                         b
                                                                                                                                          e
                                                                                            c                               b c d
                                           c                                                                                110 66
                                       d                                                   85
                                                                                                                                    f
                               e                                                     d                  e       f
                                                                                                                                g       65



                                                                   CHALLENGE YOURSELF!

 12.       What is the sum, in degrees, of the measures of interior angles of figure HOUSE?                                                   H


                                                                                                                                    O                     E



                                                                                                                                    U                     S
                  P
 13.                                           U                   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
                               O                                   the sum of the measures of the angles P, Q, R, S, T and U?
       Q                                                   T



                 R                                     S



                                                                           EXTENSIONS

 14.        B                      C                   D           In the diagram, if BEC = y, then what is the measure of BAC?
                                  x x
                
                                                                                                    y
                                                                   a. 90o  y            b. 90o                    c. 2y
                                                   y                                                2
                                                       E
                                                                   d. 180o  2y          e. y
                         A

Expectation: i) solve angle-relationship problems involving triangles, intersecting lines, and parallel lines and transversals.                           4
For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.

								
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