Download 1.1.5 - schsgeometry

1. Find the missing variables. Explain your reasoning.
80°
x = ________________
vertical
Angle Relationship: _________________
80°
y = _________________
corresponding
Angle Relationship: _______________
1. Find the missing variables. Explain your reasoning.
60°
x = ________________
Supplementary
Angle Relationship: _________________
120°
y = _________________
Alternate ext.
Angle Relationship: _______________
1. Find the missing variables. Explain your reasoning.
50°
x = ________________
Same-side int.
Angle Relationship: _________________
130°
y = _________________
vertical
Angle Relationship: _______________
2. Use your knowledge of angle relationships to solve for x in the
diagrams below. Justify your solutions by naming the
geometric relationship.
corresponding Same-side int.
5x + 7 = 9x – 63 32x + 20 = 180
x=5
17.5 = x
3.
Looking at the diagram, John says that m
BCF  m EFH .
a. Do you agree with John? Why or why not?
No, lines not
parallel
b. Jim says, "You can't be sure
those angles are equal. An
important piece of information
is missing from the diagram!"
What is Jim talking about?
Need arrows
to show
parallel lines
Examine the diagram.
a. Finish labeling the figure, given the following:
lines a and b are cut by transversal t
3 and 8 are same-side interior angles
4 and 6 are corresponding angles
7 and 3 are corresponding angles
5 and 3 are alternate interior angles
t
1 2
3 4
8 5
76
a
b
b. Complete the statements
about 8. Justify your
answers.
t
1 2
3 4
8 5
76
4 or
(1) 8 ≅ ______
1 or 6
5, 7, 3, 2
(3) 8 + ______ = 180°
a
b
4 or
(2) 8 ≅______
1 or 6
5, 7, 3, 2
(4) 8 + ______ = 180°
c. If m1 = 105°, find the measure of the other
seven angles.
75° 105°
105° 75°
75° 105°
105° 75°
5. Examine the diagrams below. What is the geometric relationship
between the labeled angles? What is the relationship of their
measures? Then, use the relationship to write an equation and
solve for x.
vertical
supplementary
5x – 57 = 3x + 5 6x + 150 = 180
x=5
x = 31
2x + 54 = 90
x = 18
38°
3x + 27 = 90
x = 21
Scoring Your Homework
• Count how many problems you missed
or didn’t do
•
•
•
•
•
0-1 missed = 10
2-3 missed = 9
4-5 missed = 8
6-7 missed = 7
8-9 missed = 6
• 10-11 missed = 5
• 12-13 missed = 4
• 14-15 missed = 3
• 16-17 missed = 2
• 18-19 missed = 1
• 20-21 missed = 0
2.4
How Can I Use It?
Pg. 14
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.
2.22 – 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. Supplementary
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°
2.24 – 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. Find the measure of each indicated
angle in a regular octagon. Start by
finding the central angle, then remember
that each triangle inside is equal.
x° = 360 = 45°
8
y° = 180 – 45
2
x°
y& z= 67.5°
y°
z°
2.25 – 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°
2.26 – 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.
c. Maria is still not convinced that
the lines in must be parallel. She
decides to assume that they are not
parallel and draws the diagram
below. Where do lines and
intersect in Maria's diagram?
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. Complete the conjectures below
based on this information.
"If the measures of corresponding
congruent then the
angles are __________,
parallel
lines are _____________."
e. Complete the conjectures below
based on this information.
"If the measures of alternate interior
congruent then the
angles are __________,
parallel
lines are _____________."