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0.3.11 Alternate Interior Angles
Lesson Objectives
Required Materials
Show that alternate interior angles are congruent.
geometry toolkits
Setup: Angle Pairs (5 minutes)
Students in groups of 2. Access to protractors.
Statement
1.
and
Anticipated Responses
are lines that intersect at
. Estimate
1. Answers vary. Sample response: I think that angle
the measures of angles
and
in the
diagram without using a protractor. Share your
estimate with a partner and revise if needed.
measures
and angle
measures
.
2. Answers vary. Sample response: I was close. My
angle sum was correct.
3.
.
4. Angle
measures
measures
.
and angle
5. Sketch should show a right angle, bisected. Each
angle measures
2. Use a protractor to measure angles
and
.
. How close were your estimates? Was the
sum of your two estimates accurate?
3. Find the measure of angle
your reasoning.
. Explain or show
4. Find and label a second
degree angle in the
diagram. Find and label an angle congruent to
angle
.
5. Sketch a diagram that shows two angles that are
both congruent to each other and
complementary. What are their angle measures?
Setup: Cutting Parallel Lines With a Transversal (20 minutes)
Students in groups of 2, 1 minute quiet time, partners to complete task.
Statement
Lines
and
transversal
Anticipated Responses
are parallel. They are cut by
.
1. Explanations vary. Sample response: With a
protractor and supplementary and vertical angles.
1. Work with your partner to find the seven unknown
angle measures in the diagram. Explain your
reasoning.
2. What do you notice about the eight angles at
points and ?
3. Using what you noticed, find the measures of the
2. Answers vary. Sample response:The angles in the
same place relative to the transversal have the
same measure.
four angles at point in the second diagram.
Lines
and
are parallel.
3. Answers vary. Sample response: Angle
is a
34 degree angle and angle
is a 146 degree
angle.
4. At vertex
at vertex
the angles measure
they measure
and
and
.
and
5. Answers vary. Sample response: In both pictures,
the two pair of vertical angles at each vertex are
congruent. Also adjacent angles at each vertex are
supplementary. In the first picture, the angle
measures at the two vertices are the same while in
the second picture they are different.
4. The next diagram resembles the first one but the
lines form slightly different angles. Work with your
partner to find the 6 unknown angles with vertices
at points and .
5. What do you notice about the angles in this
diagram as compared to the previous diagram?
How are the two diagrams different? How are they
the same?
Setup: Alternate Interior Angles are Congruent (10 minutes)
Access to ruler and tracing paper.
Statement
Anticipated Responses
Suppose and are parallel lines. The point is on ,
and the point is on , as shown in the diagram.
1. If I rotate the picture 180 degrees with center ,
line angles
and
trade places. This
means that they are congruent.
2. If and
are not parallel, then I can not show
that angles
and
are congruent with
rigid motions.
is the midpoint of segment
, which is contained
in line , which is a transversal line to and .
1. Find a rigid transformation showing that angles
and
are congruent.
2. In this picture, lines and are no longer parallel.
is still the midpoint of segment
.
Does your argument in the previous problem apply in
this situation? Explain.
Setup: All The Rest (5 minutes)
Cool-down (5 minutes)
Anticipated Responses
The diagram shows two parallel lines cut by a
transversal. One angle measure is shown.
a:
, b:
, c:
, d:
, e:
, e:
, g:
.
Find the values of , , , , ,
and .
Lesson Summary (5 minutes)
What are alternate interior angles? When we have two parallel lines cut by a transversal, how can we find the
measures of other angles if we know the measure of one of them? How did we use rigid transformations to help
find these angle measures?