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DG4GSP_897_02.qxd
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Page 27
Lesson 2.6 • Special Angles on Parallel Lines
In this activity you’ll discover relationships among the angles formed when
you intersect parallel lines with a third line called a transversal.
Investigation 1: Which Angles Are Congruent?
Sketch
Step 1
and point C, not on AB
.
In a new sketch, construct AB
Step 2
through point C.
Construct a line parallel to AB
Step 3
. Drag points C and A to make sure
Construct AC
you attached the three lines correctly.
C
A
B
Step 1
C
A
B
Steps 2 and 3
Step 4
Construct points D, E, F, G, and H as shown.
Step 5
Measure the eight angles in your figure. (Remember
that to measure angles you need to select three points
on the angle, making sure the middle point is always
the vertex.)
F
C
D
A
G
Investigate
1. When two parallel lines are cut by a transversal, the pairs
of angles formed have specific names and properties. Drag
point A or B and determine which angles stay congruent.
. Describe how many of the
Also drag the transversal AC
eight angles you measured appear to be always congruent.
E
B
H
Step 4
2. Angles FCE and CAB are a pair of corresponding angles.
a. List all the pairs of corresponding angles in your construction.
b. Write a conjecture describing what you observe about
corresponding angles (CA Conjecture).
3. Angles ECA and CAG are a pair of alternate interior angles.
a. List all the pairs of alternate interior angles in your construction.
b. Write a conjecture describing what you observe about alternate
interior angles (AIA Conjecture).
(continued)
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press
CHAPTER 2
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DG4GSP_897_02.qxd
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1:07 PM
Page 28
Lesson 2.6 • Special Angles on Parallel Lines (continued)
4. Angles FCE and HAG are a pair of alternate exterior angles.
a. List all the pairs of alternate exterior angles in your construction.
b. Write a conjecture describing what you observe about alternate
exterior angles (AEA Conjecture).
5. Combine the three conjectures you made in Questions 2–4 into a single
conjecture about parallel lines that are cut by a transversal (Parallel
Lines Conjecture).
6. Suppose, in a similar sketch, all you knew was that the angle pairs
described above had the properties you observed. Could you be sure
that the original pair of lines were parallel? Try to answer this question
first without using the computer.
Investigation 2: Is the Converse True?
Sketch
Step 1
Step 2
Step 3
In a new sketch, construct two lines that are not
quite parallel. Intersect both lines with a transversal.
F
Measure all eight angles formed by the three lines. Add
points if you need them.
Move the lines until the pairs of angles match the
conjectures you made in the previous investigation.
C
D
G
E
A
B
H
Investigate
1. Lines with equal slopes are parallel. To check if your lines are
parallel, measure their slopes. Write a new conjecture summarizing
your conclusions (Converse of the Parallel Lines Conjecture).
EXPLORE MORE
1. Angles ECA and BAC in Step 4 in Investigation 1 are sometimes
called consecutive interior angles. In a new sketch, find all pairs
of consecutive interior angles and make a conjecture describing
their relationship.
2. Angles FCD and HAG in that same figure are sometimes called
consecutive exterior angles. Find pairs of consecutive exterior angles
in the figure and make a conjecture describing their relationship.
3. You can use the Converse of the Parallel Lines Conjecture
to construct parallel lines. Construct a pair of intersecting lines AB
as shown. Select, in order, points C, A, and B, and choose
and AC
Transform ⏐ Mark Angle. Double-click point C to mark it as a center
for rotation. You figure out the rest. Explain why this works.
C
B
A
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CHAPTER 2
Discovering Geometry with The Geometer’s Sketchpad
©2008 Key Curriculum Press