Download Angles and Triangles Student Notes

CHAPTER 2: PROPERTIES OF ANGLES AND TRIANGLES
1. Exploring Parallel Lines – pg. 70-72
Assignment: pg. 72 #1-6
2. Angles formed by Parallel Lines – pg. 73-82
Assignment: pg. 78-82 #1-4, 7,10, 14, 15, 19, 20
3. Mid-Unit Review – pg. 84-85
Assignment: pg. 85 #1-8
4. Angle Properties in Triangles – pg. 86-93
Assignment: pg. 90-93 #1-5, 7, 9, 11, 13, 14
5. Angle Properties in Polygons – pg. 94-103
Assignment: pg. 99-103 #1-4, 6-8, 10, 11, 13, 16, 17, 19
6. Exploring Congruent Triangles – pg. 104-106
Assignment: pg. 106 #1-4
7. Proving Congruent Triangles - pg. 107-115
Assignment: pg. 112-115 #1, 2, 4-8, 11, 12, 14
8. Chapter Quiz
9. Chapter Review – pg. 119-120
Assignment: pg. 119-120 # 1-10a, 12-16, 18
10. Chapter Exam
LESSON 1: EXPLORING PARALLEL LINES
Learning Outcome: Learn to identify relationships among the measures of angles
formed by intersecting lines.
Work with a partner for the following investigation:
1. Create two intersecting lines on a piece of paper.
With a protractor, measure all angles created by the intersecting lines.
What conclusion can we draw from the angles created by the intersecting lines?
2. Draw two parallel lines.
a b
c d
e
g
f
h
Draw a transversal (a line that intersects two or more other lines at distinct points) that
intersects the pair of parallel lines.
With a protractor, measure all angles created by the transversal.
What conclusion can be made about the angles that are formed by each parallel line
and the transversal?
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Corresponding angles: One interior angle and one exterior angle that are nonadjacent and on the same side of a transversal.
Interior Angles: Any angles formed by a transversal and two parallel lines that lie
inside the parallel lines.
a
b
c
a, b, c, d are
interior angles.
d
Exterior Angles: Any angles formed by a transversal and two parallel lines that lie
outside the parallel lines.
e
f
e, f, g, h are
exterior angles.
g h
3. Draw two non-parallel lines:
Draw a transversal.
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Measure the angles. What conclusions can we make?
Assignment: pg. 72 #1-6
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LESSON 2: ANGLES FORMED BY PARALLEL LINES
Learning Outcome: Learn to prove properties of angles formed by parallel lines and a
transversal, and use these properties to solve problems.
With a partner, complete the following investigation:
Make a conjecture that involves the interior angles formed by parallel lines and a
transversal.
Draw two parallel lines and a transversal:
Measure all the angles, which type of angles did we prove in the previous lesson?
After all measurements are completed, did you notice any other pairs of equal angles?
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Alternate interior angles: Two non-adjacent interior angles on opposite sides of a
transversal.
Alternate Exterior angles: Two exterior angles formed between two lines and a
transversal, on opposite sides of the transversal.
Ex. Determine the measures of a, b, c and d.
b
c
a
d
110˚
∠a =
∠a = ∠b =
Recognize that 180˚ - a = 70˚ (straight line) and that value is the same as c
(corresponding angles)
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∠c = ∠d = (alternate interior angles)
Need to know:
If a transversal intersects two lines such that




The corresponding angles are equal or
The alternate interior angles are equal or
The alternate exterior angles are equal or
The interior angles on the same side of the transversal are supplementary,
Then the lines are parallel.
Supplementary angles: Two angles that add to 180˚.
Complementary angles: Two angles that add to 90̊
Assignment: pg. 78-82 #1-4, 7,10, 14, 15, 19, 20
Mid-Unit Review : pg. 85 #1-8
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LESSON 3: ANGLE PROPERTIES IN TRIANGLES
Learning Outcome: Learn to prove properties of angles in triangles, and use these
properties to solve problems.
In the diagram, ∠MTH is an exterior angle of ΔMAT. Determine the measures of the
unknown angles in ΔMAT.
M
40˚
155˚
A
T
H
With a partner, devise a strategy to solve for all the unknown angles.
What previously learned concepts did we need to know in order to solve this question?
Non-adjacent interior angles: The two angles of a triangle that do not have the same
vertex as an exterior angle.
A
B
C
D
∠ A and ∠ B are non-adjacent interior angles to exterior ∠ACD
∠DCA = ∠CBA + ∠BAC
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Ex. Prove: ∠e = ∠a + ∠b
a
d c
b
e
∠d = ∠ a + ∠ b
∠e = ∠d
∴ ∠𝑒 = ∠𝑎 + ∠𝑏
Examples
1. Find the value of x in the following triangle.
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2. Find the values of x and y in the following triangle.
Ex. Determine the measures of ∠NMO, ∠MNO, and ∠QMO.
L
M
Q
67˚
39˚
N
20˚
O
P
R
∠MNP = ∠LMN (alternate interior angles)
∠MNO =
∠NMO =
(sum of the angles of a triangle)
∠QMO:
NM is a straight line and equals 180˚
(∠NMO) =
Assignment: pg. 90-93 #1-5, 7, 9, 11, 13, 14
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LESSON 4: ANGLE PROPERTIES IN POLYGONS
Learning Outcome: Learn to determine properties of angles in polygons, and use
these properties to solve problems.
A Pentagon has three right angles and four sides of equal length, as shown. What is
the sum of the measures of the angles in the pentagon? Use a protractor and a ruler to
create the object then measure the interior angles. What sum did you get?
How is the number of sides in a polygon related to the sum of its interior angles?
With a partner, draw the polygons listed in the table below. Create triangles to help you
determine the sum of the measures of their interior angles. Record your results below:
Polygon
Number of Sides
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
3
4
5
6
7
8
Number of
Triangles
1
Sum of Angle
Measures
180˚
Given the information you discovered in the table above, can you make a conjecture
about the relationship between the sum of the measures of the interior angles of a
polygon and the number of sides of the polygon?
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Ex. Use your conjecture to predict the sum of the measures of the interior angles of a
dodecagon (12 sides). Can you verify your prediction using triangles?
Convex polygon: A polygon in which each interior angle measures less than 180˚.
The sum of the measures of the interior angles of a convex polygon with n sides can be
expressed as: 180˚(n – 2).
The measure of each interior angle of a regular polygon is:
180°(𝑛−2)
𝑛
.
The sum of the measures of the exterior angles of any convex polygon is 360˚
Ex. Determine the measure of each interior angle of a regular 15-sided polygon (a
pentadecagon).
Given:
What do you notice about the sum of the measures of each exterior angle and its
adjacent interior angle (No protractors)?
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Given the angles below:
z
w a
d
y
c
b
x
Without using a protractor, what do you notice about the relationship between the
interior and exterior angles of the irregular quadrilateral? Explain why.
When a side of a polygon is extended, two angles are created. The angle that is
considered to be the exterior angle is adjacent to the interior angle at the vertex.
Exterior angle
Adjacent Interior angle
Assignment: pg. 99-103 #1-4, 6-8, 10, 11, 13, 16, 17, 19
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LESSON 5: EXPLORING CONGRUENT TRIANGLES
Learning Outcomes: Learn to determine the minimum amount of information needed
to prove that two triangles are congruent.
If we are given the following triangle with measurements:
X
65˚
2.92m
1.95m
75˚
Y
40˚
2.74m
Z
If we wanted to duplicate this triangle, would we have to provide all the measurements?
Which three pieces of information could be provided for duplication to ensure that the
triangle is identical?
Which combinations of given side and angle measurements do not ensure that only one
size and shaped can be produced?
Which combinations of given side and angle measurements ensure that all the triangles
are congruent?
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There are minimum sets of angle and side measurements that, if known, allow you to
conclude that two triangles are congruent.
If three pairs of corresponding sides are equal, then the triangles are congruent. This is
known as side-side-side congruence, or SSS.
A
B
X
C
Y
Z
List the corresponding equal sides:
If two pairs of corresponding sides and the contained angles are equal, then the
triangles are congruent. This is known as the side-angle-side congruence or SAS
A
B
X
C
Y
List the corresponding equal sides and angles:
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Z
If two pairs of corresponding angles and the contained sides are equal, then the
triangles are congruent. This is known as the angle-side-angle congruence or ASA
X
A
B
C
Y
Z
List the corresponding equal sides and angles:
The symbol ∴ represents the word “therefore.” In geometry, this symbol is generally
used when stating a conclusion drawn from preceding facts or deductions.
Assignment: pg. 106 #1-4
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LESSON 6: PROVING CONGRUENT TRIANGLES
Learning Outcomes: Learn to use deductive reasoning to prove that triangles are
congruent.
How can you prove that two or more triangles are congruent? Discuss the possibilities
with a partner.
With the same partner, name as many types of triangles as you can and describe what
makes each triangle different.
Ex. Given: TP ⊥ AC and AP = CP
T
Prove: ΔTAC is isosceles
A
P
∠TPA and ∠TPC are right angles, therefore they are equal
AP = CP are given.
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C
When describing two triangles by their vertices, make sure that the corresponding
vertices are in the same order in both descriptions. For example, when stating that
these two triangles are congruent, you could write:
ΔABC ≅ ∆𝑋𝑍𝑌 or ΔACB ≅ ∆𝑋𝑌𝑍
A
B
X
C
Z
Y
To complete a formal proof that two triangles are congruent, you must show that
corresponding sides and corresponding angles in the two triangles are equal.
Before you can conclude that two triangles are congruent, you must show that the
relevant corresponding sides and angles are equal. You must also state how you know
that these sides and angles are equal.
The pairs of angles and sides that you choose to prove congruent will depend on the
information given and other relationships you can deduce.
Often, proving that corresponding sides or corresponding angles are equal in a pair of
triangles first requires you to prove that the triangles containing these sides or angles
are congruent.
Sometimes, to prove triangles congruent, you must add lines or line segments with
known properties or relationships.
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Ex. Given: AE and BD bisect each other at C.
AB = ED
D
A
Prove: ∠A = ∠E
C
B
Assignment: pg. 112-115 #1, 2, 4-8, 11, 12, 14
Chapter Quiz
Chapter Review: pg. 119-120 #1-10a, 12-16, 18
Chapter Exam
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E