Download Properties of Parallel Lines

Properties of Parallel Lines
Name(s):
In this investigation, you’ll discover relationships among the angles
formed when parallel lines are intersected by a third line, called a
transversal.
Sketch and Investigate
1. Construct dAB and point C, not on dAB.
Select the line and
the point; then,
in the Construct
menu, choose
Parallel Line.
2. Construct a line parallel to dAB through point C.
3. Construct dCA. Drag points C and A to make sure the three lines are
attached at those points.
C
C
A
Using the Text tool,
click once on a point
to show its label.
Double-click the
label to change it.
To measure an
angle, select three
points, with the
vertex your middle
selection. Then, in
the Measure menu,
choose Angle.
A
B
Step 1
B
Steps 2 and 3
4. Construct points D, E, F, G, and
H as shown at right.
5. Measure the eight angles in your
figure. Be systematic about your
measuring to be sure you don’t
measure the same angle twice.
F
C
D
E
6. Drag point A or B and see
G
A
B
which angles stay congruent.
Also drag the transversal dCA.
H
(Be careful not to change the
point order on your lines. That
Step 4
would change some angles into
other angles.) Observe how many
of the eight angles you measured appear to be always congruent.
Q1
When two parallel lines are crossed by a transversal, the pairs of
angles formed have specific names and properties. The chart on the
next page shows one example of each type of angle pair. Fill in
the chart with a second angle pair of each type, then state what
relationship, if any, you observe between the angles in a pair type.
Exploring Geometry with The Geometer’s Sketchpad
© 2002 Key Curriculum Press
Chapter 1: Lines and Angles • 17
Properties of Parallel Lines (continued)
Angle Type
Pair 1
Corresponding
∠FCE and ∠CAB
Alternate interior
∠ECA and ∠CAG
Alternate exterior
∠FCE and ∠HAG
Same-side interior
∠ECA and ∠BAC
Same-side exterior
∠FCD and ∠HAG
Q2
Pair 2
One of the angle types has more than one pair. Name that angle type
in the chart below, and name the third and fourth pairs of angles of
that type.
Angle Type
Pair 3
Pair 4
7. Next, you’ll investigate the converses of
your conjectures. In a new sketch, draw
two lines that are not quite parallel.
Construct a transversal.
8. Add points as needed, then measure all
eight angles formed by the three lines.
9. Move the lines until you have two sets
of four congruent angles.
Q3
Relationship
Relationship
E
F
D
C
G
A
B
H
If two lines are crossed by a transversal
so that corresponding angles, alternate
interior angles, and alternate exterior
angles are congruent, what can you say about the lines?
Explore More
1. You can use the converse of the parallellines conjecture to construct parallel lines.
Construct a pair of intersecting lines dAB and
dAC as shown. Select, in order, points C, A,
and B. Then, in the Transform menu, choose
Mark Angle. Double-click point C to mark
it as a center for rotation. You figure out the
rest. Explain why this method works.
18 • Chapter 1: Lines and Angles
C
A
B
Exploring Geometry with The Geometer’s Sketchpad
© 2002 Key Curriculum Press
Properties of Parallel Lines (page 17)
Prerequisites: Students should be familiar with
angle notation and angle measurement. This activity
introduces the terms transversal, corresponding
angles, alternate interior angles, alternate exterior
angles, same-side interior angles, and same-side exterior
angles.
Sketchpad Proficiency: Beginner
Activity Time: 25–35 minutes
Example Sketch: Parallel Lines.gsp
Sketch and Investigate
Q1 Students should complete the chart as shown
below.
Angle Type
Pair 1
Pair 2
Rel’ship
Corresponding
∠FCE ,
∠CAB
See Q2
below.
Equal
measure (or
congruent)
Alternate
interior
∠ECA,
∠CAG
∠DCA,
∠CAB
Equal
measure
Alternate
exterior
∠FCE,
∠HAG
∠HAB,
∠DCF
Equal
measure
Same-side
interior
∠ECA,
∠BAC
∠DCA,
∠GAC
Supplementary
Same-side
exterior
∠FCD,
∠HAG
∠FCE,
∠HAB
Supplementary
Q2 There are more than two pairs of corresponding
angles. The three more pairs of corresponding
angles not listed in the second row above are
∠ECA and ∠BAH, ∠HAG and ∠ACD, and
∠GAC and ∠DCF. Students should have
recorded one of these pairs in the chart above.
The relationship between angles in each pair is
that they have equal measure.
Q3 The lines must be parallel.
Explore More
1.
The last step is to rotate dAC by the marked
angle. Because the alternate interior angles were
constructed to be congruent, the new line will be
parallel to dAB.
Exploring Geometry with The Geometer’s Sketchpad
© 2002 Key Curriculum Press
Activity Notes—Chapter 1: Lines and Angles • 213