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Transcript
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