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2. Use your knowledge of angle relationships to solve for x in the
diagrams below. Justify your solutions by naming the
geometric relationship.
corresponding Consecutive int.
5x + 7 = 9x – 63 32x + 20 = 180
x=5
17.5 = x
No, Need
arrows to
show parallel
lines
2.4
How Can I Use It?
Pg. 16
Angles In a Triangle
2.4 – How Can I Use It?_____________
Angles In a Triangle
So far in this chapter, you have
investigated the angle relationships
created when two lines intersect, forming
vertical angles. You have also investigated
the relationships created when a
transversal intersects two parallel lines.
Today you will study the angle
relationships that result when three nonparallel lines intersect, forming a triangle.
2.23 – ANGLE RELATIONSHIPS
Marcos decided to study the angle
relationships in triangles by making a
tiling. Find the pattern below.
a. Color in one of the angles with a pen
or pencil. Then use the same color to
shade every angle on the pattern that is
equal to the shaded angle.
b. Repeat this process for the other
two angles of the triangle, using a
different color for each angle in the
triangle. When you are done, every
angle in your tiling should be shaded
with one of the three colors.
c. Now examine your colored tiling. What
relationship can you find between the
three different-colored angles? You may
want to focus on the angles that form a
straight angle. What does this tell you
about the angles in a triangle?
"If a polygon is a triangle, then the sum
180°
of the interior angles is ___________"
d. Let us see if this works for any
triangle. Each team member will
cut out a different type of triangle:
isosceles, scalene, right, or
obtuse. Rip off the angles of the
triangle and put them together to
form a straight line. Do these
three angles all add to 180°?
e. How can you convince yourself
that your conjecture is true for all
triangles? Match the reasons to the
proof below to show that the sum of
the interior angles of any triangle
adds to 180°.
Statements
1.
2.
3.
4.
a  d
Reasons
1.Alternate interior =
c  e
2.Alternate interior =
d  b  e  180 3. Make Straight line
a  b  c  180 4. substitution
2.23 – TRIANGLE ANGLE
RELATIONSHIPS
Use your proof about the angles in a
triangle to find x in each diagram
below.
x + 40 + 80 = 180
x + 120 = 180
x = 60°
2x + x + 12 + 96 = 180
3x + 108 = 180
3x = 72
x = 24°
CLASSIFICATION BY SIDES
Type of ∆
Equilateral
Triangle
Definition
All sides
are
congruent
Picture
CLASSIFICATION BY SIDES
Type of ∆
Isosceles
Triangle
Definition
2 sides
congruent
Picture
leg
leg
base
CLASSIFICATION BY SIDES
Type of ∆
Scalene
Triangle
Definition
No sides
are
congruent
Picture
CLASSIFICATION BY ANGLES
Type of ∆
Equiangular
Triangle
Definition
All angles
are
congruent
Picture
CLASSIFICATION BY ANGLES
Type of ∆
Acute
Triangle
Definition
All angles
are acute
Picture
CLASSIFICATION BY ANGLES
Type of ∆
Right
Triangle
Definition
One right
angle
Picture
leg
leg
CLASSIFICATION BY ANGLES
Type of ∆
Obtuse
Triangle
Definition
One obtuse
angle and 2
acute
angles
Picture
Classify the triangle by the best definition
based on sides and angles.
scalene
right
equilateral
equiangular
isosceles
obtuse
2.25 – TRIANGLE SUM THEOREM
What can the Triangle Angle Sum
Theorem help you learn about
special triangles?
a. Find the measure of each angle in
an equilateral triangle. Justify your
conclusion.
180°
3
60°
60°
60°
b. Consider the isosceles right
triangle at right. Find the measures
of all the angles.
180 – 90
2
45°
45°
c. What if you only know one angle of
an isosceles triangle? For example, if
m A  34, what are the measures
of the other two angles?
180 – 34
2
34°
73° 73°
d. Use the fact that when the triangle is
isosceles, the base angles are congruent
to solve for x and y.
2(22)+11
55
55
3x – 11 = 2x + 11
x – 11 = 11
x = 22
2y + 110 = 180
2y = 70
y = 35
9x – 8 = 28
9x = 36
x=4
2.26 – TEAM REASONING
CHALLENGE
How much can you figure out about
the figure using your knowledge of
angle relationships? Work with your
team to find the measures of all the
labeled angles in the diagram.
123°
99° 81°
123° 57° 81° 99°
42°
57°
123° 57°
2.27 – ANGLE AND LINE RELATIONSHIPS
Use your knowledge of angle relationships
to answer the questions below.
a. In the diagram at right, what is the sum
of the angles x and y?
180°
b. While looking at the diagram
below, Rianna exclaimed, "I think
something is wrong with this
diagram." What do you think she is
referring to? Be prepared to share
your ideas.
E
0°
Yes,
112° + 68° = 180°
d. Write a conjecture based on your
conclusion to this problem.
"If the measures of same-side interior
angles are __________________,
then
supplementary
parallel
the lines are _____________."
e. State the angle relationship
shown. Then find the value of x that
makes the lines parallel.
Alternate interior
4x – 10 = 3x + 13
x – 10 = 13
x = 23
corresponding
8x – 14 = 7x + 2
x – 14 = 2
x = 16
Consecutive interior
20x – 2 + 6x = 180
26x – 2 = 180
26x = 182
x=7
Alternate exterior
6x + 17 = 143
6x = 126
x = 21
Vocabulary Project
10 words
Definition
Mark in real life picture
notation
Perpendicular Line
Lines that cross at right angles
AB  CD
C
A
B
D
Coplanar Points
Points on the
same plane
points A, B, C
C
A
B
Vertical Angles
Angles opposite each other that are
congruent
ACB  DCE
A
C
D
E
B
Triangle
Polygon with 3 sides
B
A
C
ABC