Angle Relationships
10.1
Outcomes

E7 – make and apply generalizations
about angle relationships.
Always true?
Sometimes true?
Never true?
1.
2.
3.
4.
5.
6.
The diagonals of a parallelogram are
congruent.
Parallelograms are rectangles.
Vertically opposite angles are across from
each other.
The angles in a triangle can measure 100o.
If two rectangles have the same area, the
rectangles are congruent.
Angles in a square are 90o.
Always true?
Sometimes true?
Never true?
1.
2.
3.
4.
5.
6.
The diagonals of a parallelogram are always
congruent.
All parallelograms are rectangles.
Vertically opposite angles are across from
each other.
None of the angles in a triangle can
measure 100o.
If two rectangles have the same area, the
rectangles are congruent.
All angles in a square are 90o.
Stand up and show me using
your arms….

The following angles







Acute
Obtuse
90o
180o
0o
Reflex
Now take two Geostrips and repeat
Supplementary or
Complementary?


Supplementary angles – two angles
whose sum is 180o.
Complementary angles – two angles
whose sum is 90o.
Join two pattern blocks to
show

A pair of supplementary angles.




Sketch the pattern blocks to illustrate your
answers for this and each of the following.
A pair of congruent pattern blocks.
A pair of non-congruent pattern blocks.
A different pair of non-congruent
pattern blocks.
Use a Power Polygon square to
verify that the angles are
complementary.



Draw diagrams to illustrate your
answers.
Find a combination of two Power
Polygons that show complementary
angles.
Find two other possible solutions, using
two different Power Polygons.
Vertically Opposite Angles

Vertically opposite angles are nonadjacent angles formed when two lines
cross.
A
E
B
D
C
How are vertically opposite
angles related?





Join a pair of Geostrips so that they form an X, as
shown
Take a pattern block of your choice. Adjust the
Geostrip model so that one of the angles is congruent
to one of the angles of the pattern block.
Without changing the model, check the size of the
vertically opposite angles. How are the angles
related?
Investigate the other pair of opposite angles using
pattern blocks.
Repeat using different pattern blocks or Power
Polygons.
Method 1: Sum of the interior
angles of a triangle!
Cut off the corner of your page and label
each of the corners as A, B and C. Tear
off the corners and put them together.
Sum of interior angles in
Polygons
A + B + C = 180o
Use a straight edge to draw any
quadrilateral.
Draw one diagonal in it.
Explain how you can use this diagram to
find the sum of the measures of the
interior angles of the quadrilateral.
Method 2:
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Select three congruent triangles (other than
equilateral triangles) from the Power Polygons set.
Trace one of the triangles onto paper and label them
a, b, c
Place the three triangles in a way that shows that the
sum of their interior angles is 180o. Record your work
by tracing around the three triangles
Explain how your model demonstrates this property.
Repeat using a different triangle
Does this convince you that the sum of the interior
angles of a triangle is 180o for all triangles?
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
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Use a straight edge to draw any pentagon
(does not have to be regular).
From one vertex only, draw all possible
diagonals.
How many diagonals meet at this vertex?
Explain how you can use this model to find
the sum of the interior angles of a pentagon.
Find this sum.
Does this sum change if you change the size
or shape of the pentagon?
Complete this table
Polygon
Diagram
Number of
sides
Number of
Diagonals
from one
vertex
Number of
Triangles
Formed
Sum of
Interior
Angles
Triangle
3
0
1
180o
Quadrilateral
4
1
2
Pentagon
5
Polygon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Hendecagon
n-gon
Diagram
Number of
sides
Number of
Diagonals
from one
vertex
Number of
Triangles
Formed
Sum of
Interior
Angles
Transversal

A line that crosses two other lines or
line segments
Angles Formed by Parallel
Lines and a Transversal

Build the following using Geostrips.
Sketch your construction into your
notebook and label it as shown.
G
A
B
C
E
D
F
H
transversal
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Use pattern blocks to find congruent angles in
your construction. Record the congruent
angles.
Use pattern blocks to find the supplementary
angles.
Name pairs of angles that always appear to
be congruent.
Name pairs of angles that always appear to
be supplementary
Use your last sketch to answer
the following.
1.
2.
3.
4.
5.
6.
Which angle pair are vertically opposite? What
would CBH measure if ABG measured 40o? Why
Place your construction on top of your drawing.
Slide the construction down so the CBG coincides
with FEG of your drawing.
What can you say about these two angles?
These angles are named corresponding angles.
Why might they be called this?
Work with a partner to find other pairs of
corresponding angles.
Corresponding Angles


Angles either both above or both below
two lines on the same side of the
transversal
If the lines are parallel, corresponding
angles are congruent
x
y

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What is the relationship between angles
ABH and DEG?
Why might these angles be called
interior angles?
Work with a partner and using both of
your constructions find other pairs of
interior angles.
Interior angles


Angles between two lines on the same
side of a transversal.
If the lines are parallel, interior angles
are supplementary.
a >
b
a + b = 180o
>



What is the relationship between angles
ABH and FEG?
Why might these angles be called
alternate interior angles?
Work with a partner and using both of
your constructions find other pairs of
alternate interior angles.
Alternate interior angles


Angles between two lines on either side
of a transversal
If the lines are parallel, alternate
interior angles are congruent.
a
b
>
>
What happens to angle relationships
when a transversal crosses two lines that
are not parallel?

Remove the pair of opposite sides that
are not crossed by the transversal.
Move one of the strips so that the lines
are no longer parallel.

Examine what happens to the following
angle relationships when a transversal
crosses two non-parallel lines:


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Alternate interior angles
Corresponding angles
Interior angles
Vertically opposite angles
Example 1: Parallel Lines and
a Transversal

Find the measures of EHB,
and EHG
D
A
C
F
G
131o
FHG,
H
B
E
Line segment AB is a transversal.
Example 2: Apply Angle Relationships
to Find Unknown Angles
D
E
A
x
B
29o
C
A
B
64o
F
C
x
x
D
E
A
B
C z
D
80o
G
y
z
F
E
Communicate Key Ideas

Page 431 #1 and 2
Find the measure of ABC,
CBD, and ECD. Explain your
reasoning.
C
A
35
B
80
E
D
Answer
C
ABC = 145o
CBD = 35o
ECD = 65o
A
35
B
80
E
D
Question #2
a) Describe a method for determining the sum
of the interior angles of a polygon, without
having to measure them.
b) Does this method work for polygons that are
not regular as well as regular polygons?
Explain why or why not.
c) Suppose you know the number of sides of a
polygon. Can you determine the measures of
its interior angles, assuming that the polygon
is regular and not regular? Explain your
answer.
Answers
a)
b)
c)
S = 180o (n-2) , where S represents the
sum of the interior angles and n represents
the number of sides of the polygon.
Yes, the sum of interior angles is the same
for regular and not regular polygons
Yes, all the interior angles will have the
same measure, so simply divide the sum of
the interior angles by the numbers of sides.
No, the interior angles will have different
measures.
Question #3
a
c
b
d
e f
g
h
a) List the corresponding angles, alternate interior angles, and interior
angles on the same side of the transversal.
b) Why do you think these names make sense?
Answers
a)
Corresponding angles: a and e,
c and f,
d and h
alternate interior angles: c and f,
e and d
interior angles: c and e,
d and f
Check Your Understanding
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Page 432 – 433
#1, 2, 4, and 5
Page 432 #1a)
(labelled D in textbook)
a)
C
B
d
62.0
G
e
A
e
F
E
D
Question 1b)
b)
L
M
g
P
f Q
110
N
h
h
i
R
S
Answer
Question 1c)
J
C
D
w
c)
x B
E
y
A
F
I
G
z
H
L
K
Answer
Q.#2 Classify as always true,
sometimes true, never true
If you pick ‘always true’ or ‘never true’ explain how you know.
If you pick ‘sometimes true’ describe the conditions necessary to make
it true.
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a) the sum of the interior angles in a triangle is 180o
b) alternate interior angles on a transversal are congruent
c) a triangle has two right angles
d) interior angles on the same side formed by an transversals
and two parallel lines are equal in measure
e) a regular polygon has at least one pair of parallel sides.
f ) a quadrilateral has four acute angles.
g) three of the angles of a quadrilateral can be acute
h) vertically opposite angles are complementary
Answers
Question #4

I am a quadrilateral. I have two pairs of
congruent angles and two pairs of
supplementary angles.
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Who am I, and how do you know?
Is there more than one possible answer?
Explain, using words and diagrams.
Two of my sides are equal in length, the
other two are different. Which of these
quadrilaterals am I? How do you know?
Answer
#5

I am a right triangle. One of my two
complementary angles is twice the
measure of the other.


What are the measure of my three angles?
Explain how you know.
Draw me, using only pattern blocks to
measure the angles. Explain your method.