Download Geometry Summer Institute 2014 Parallel Lines and Angles

Geometry Summer Institute 2014
Parallel Lines and Angles
Summary
Goals
Having learned what it
means for two geometric
figures to be congruent,
participants now get to see
some immediate
applications. Since we are
already in possession of all
the general criteria for
triangle congruence, we are
free to develop some of the
concept on geometry at this
point as in the classical
treatment handed down to
us by Euclid. However, in
the spirit of the hands-on
approach to geometry, we
will continue to provide
proofs of parallel lines and
angles, when it is
appropriate, using basic
rigid motions.
Participant Handouts
 Investigate the parallel
lines and transversal
theorems.
 Experience working on
proofs by rigid motions
 Comments on proofs
provided by other
participants
1. Parallel Lines and
Angles
Materials
Technology
Source
Estimated Time
Paper
Pencil
Patty Paper
LCD Projector
Facilitator Laptop
GeoGebra
Teaching Geometry 90 minutes
in Grade 8 and High
School According to
the Common Core
Standards
H. Wu
Mathematics Standards
Common Core State Standards for Mathematics
G-CO. 6, 7, 8: Build on rigid motions as a familiar starting point for development of
geometric proofs.
6. Use geometric descriptions of rigid motions to transform figures and to predict the
effect of a given rigid motion on a given figure; given two figures, use the
definition of congruence in terms of rigid motions to decide if they are congruent.
7. Use the definition of congruence in terms of rigid motions to show that two
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triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
G-CO. 9, 10: Prove Geometric theorems
8. Prove theorems about lines and angles. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles are
congruent and corresponding angles are congruent; points on a perpendicular
bisector of a line segment are exactly those equidistant from the segment’s
endpoints.
Standards for Mathematical Practice
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use tools appropriately
Instructional Plan
Having learned what it means for two geometric figures to be congruent, participants now
get to see some immediate applications. Since we are already in possession of all the
general criteria for triangle congruence, we are free to develop some of the concept on
geometry at this point as in the classical treatment handed down to us by Euclid.
However, in the spirit of the hands-on approach to geometry, we will continue to provide
proofs, when it is appropriate, using basic rigid motions.
Participants will work on 4 proofs: vertical angles, corresponding and alternate interior
angles, converse of the parallel lines and transversal, and sum of measures of interior
angles in a triangle. For each one, participants will get the formal definition, work on the
concepts using patty paper and GeoGebra, and work on the formal proof.
Divide participants in groups of 4 while they work on each of the concepts.
Vertical Angles
(Slide 2)
1. Definition
First, we show that two intersecting lines L and l give rise to some congruent
angles. Suppose they meet at a point O. Let A, B be two points on L that lie on
opposite sides of l and, similarly, let C, D be two points on l that lie on opposite
sides of L.
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Then ∠AOC and ∠BOD are called opposite angles at O (sometimes called vertical
angles).
2. Patty paper investigation:
i.
Fold a line on a patty paper. Unfold. Fold a second line intersecting the
first line. Unfold.
ii. What seems to be congruent of vertical angles? Fold the paper to see if the
sides of one angle of a pair of vertical angles can be made to fit over the
sides of the other angle of the pair. Or place a second patty paper over the
first and copy one angle of a pair of vertical angles. Rotate the copy to see
how well it fits over the second angle of the vertical angle pair.
iii. What do you notice about adjacent angles?
3. GeoGebra investigation
i.
Open a new GeoGebra page and select “Algebra and Geometry”
perspective. Hide the coordinate system in the Graphics window.
ii. Construct two lines AB and CD that intersect.
iii. Construct a point of intersection of the two lines (E).
iv.
Measure the 4 angles formed (counterclockwise)? What do you observe?
v.
Move the position of one of the two lines, or both if you like. Is your
conjecture still holding true?
vi.
What do you notice about adjacent angles? Is your conjecture the same as
the one you formulated during the patty paper investigation?
4. Formal Proof: vertical angles are congruent (using rigid motions):
If R denotes the 180-degree rotation of the plane around O, then R moves the ray
OA to the ray OB and the ray OC to the ray OD. Thus R moves the ∠AOC to
∠BOD. Because basic rigid motions are assumed to preserve degree, the opposite
angles ∠AOC and ∠BOD are equal.
Parallel Lines, Alternate Interior Angles, and Corresponding Angles
(Slide 3, 4)
1. Definitions:
Let lines L1 and L2 be given and let l be a line that intersects L1 and L2 at P1 and
P2, respectively. The line l is called a transversal of the lines L1 and L2. Let C1, D2
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Geometry Summer Institute 2014
be points on L1 and L2, respectively, so that they lie on opposite sides of l.
Then ∠C1P1P2 and ∠P1P2D2 are said to be alternate interior angles of the
transversal l with respect to L1 and L2. If E, D1 are points on l and L1,
respectively, so that ∠EP1D1 and ∠C1P1P2 are opposite angles, then ∠EP1D1 and
∠EP2D2 are said to be corresponding angles of the transversal l with respect to
L1 and L2. We now come to one of the characteristic properties of the plane.
2. Patty paper investigation
i.
First we need to fold two parallel lines. There are several ways to do this.
Start with one line and a point not on the line. How could you fold the
parallel line to the one on the paper that goes through your point? Share
your method with the other participants.
ii. Fold another line that intersects the two parallel lines. Identify all the
alternate interior angles and the corresponding angles.
iii. Copy the lines on a second patty paper. What do you notice about the
alternate interior angles? What do you notice about the corresponding
angles? How many of the corresponding angles have the same measure?
3. GeoGebra investigation
i.
Draw a line and a point not on the line.
ii. Construct the parallel line to the first one and a new transversal line.
iii. Construct new points at the intersections of the lines.
iv.
Measure the 8-formed angles. What do you notice?
v.
Move the original line or the transversal line. Do your conjectures hold
true?
4. Formal Proof:
Alternate interior angles and corresponding angles of a transversal with respect to
a pair of parallel lines are equal.
Solution: Let the parallel lines be L1 and L2 and let a transversal l intersect L1 and
L2 at P1 and P2, respectively. Let the alternate interior angles be ∠C1P1P2 and
∠P1P2D2, as shown. It suffices to prove that these angles are equal because we
have established that vertical angles are congruent then takes care of the statement
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about the corresponding angles ∠EP1D1 and ∠EP2D2.
Let O be the midpoint of the segment P1P2 and let R be the 180-degree rotation
around O. Furthermore, let R map the line L2 to R(L2). Now consider the two
lines R(L2) and L1. We have R(L2) ∥ L2, and also L1 ∥ L2 by hypothesis. Of
course L1 passes through the point P1, but so does R(L2) because a rotation
preserves distance and therefore R maps P2 to P1, so that R(L2) contains P1. By the
Parallel Postulate, R(L2 ) = L1. Because R is a 180-degree rotation, it maps the ray
P2 D2 to the ray P1C1. Therefore R maps ∠P1P2D2 to ∠P2P1C1. Since a rotation
preserves degree, we see that ∠P1P2D2 is equal to∠P2P1C1.
Converse of the parallel lines and transversal conjecture
(Slide 5)
The parallel line conjecture says that if two parallel lines are intersected by a transversal,
the alternate interior angles and corresponding angles are congruent. Is the converse also
true? That is, if a transversal intersects two lines and corresponding angles are congruent,
are the two lines parallel?
1. Patty paper investigation
i.
Draw or fold two intersecting lines
ii. Copy these lines onto another patty paper
iii. Place the original patty paper over the copy so that part of the original
lines lie on the copy. Then trace the lines and angles from the copy onto
the original paper. (When you do this, you are constructing sets of
corresponding angles as well as sets of alternate interior angles)
iv.
Determine whether the two lines are parallel. You can do this by using
another patty paper to show that the distance between the two lines always
remain the same.
2. GeoGebra investigation
i.
Draw two intersecting lines and a point in the intersection. Measure all 4
angles
ii. Draw another line intersecting one of the original two and the point of
intersection. Measure the angles.
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iii.
iv.
v.
Move the last line until the last set of angles measure the same as the first
set of angles.
Draw perpendicular lines (2 or more) to the first drawn line that is one of
the supposedly parallel lines. Draw points at the intersections between the
perpendicular line and the parallel lines. Measure the distance between the
parallel lines? Are they the same? Are the lines parallel?
Another way is to draw a parallel line through the point of intersection?
Does the real parallel line overlap the third line?
3. Formal Proof:
If a pair of alternate interior angles or a pair of corresponding angles of a
transversal with respect to two lines are equal, then the lines are parallel.
Solution: Let ∠C1P1P2 and ∠P1P2D2 be equal alternate interior angles of the
transversal l with respect to the lines L1 and L2, and we have to prove that L1 ∥ L2.
As before, let O be the midpoint of the segment P1P2 and let R be the 180-degree
rotation around O. If the rotated image of L2 is denoted by R(L2) and the rotated
image of D1 is denoted by C′, then R(L2) passes through P1 and ∠C′P1P2 is equal
to ∠P1P2D2 (rotation preserves degree). By hypothesis, ∠C1P1P2 is also equal to
∠P1P2D2. Hence ∠C′P1P2 and ∠C1P1P2 are equal angles with a common side
P1P2. Moreover, since both C1 and C′ are on the opposite side of line LP1P2 = l
relative to D2, we see that C1 and C′ are on the same side of l. Therefore the rays
P1C1 and P1C′ coincide, or what is the same thing, the lines L1 and R(L2) coincide.
But R(L2) ∥ L2, so L1 ∥ L2 after all.
Angle Sum Theorem
(Slide 6)
1. Definition
The sum of the interior angles of any triangle is equal to 180 degrees.
2. Patty paper investigation
i.
Draw a triangle and try to make the three angles different sizes. Cut the
triangle from the paper
ii. Color the three interior angles with different colors and label them A, B,
and C.
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Geometry Summer Institute 2014
iii.
iv.
Tear off the three colored angles to have three separate angles.
Attempt to line up the angles so their vertices coincide. What do you
notice? What is the measure of the new angle?
3. GeoGebra Investigation
i.
Construct any triangle and measure the interior angles
ii. Using the text tool, add up the measures of the three angles.
iii.
Move any of the vertices of the triangle, what happens to the sum? Notice
how the measure of each angle changes in the algebra window but the sum
remains the same.
4. Formal Proof: The angle sum of a triangle is 180 degrees.
Solution: Let △ABC be given. On the ray BC, let a point D be chosen so that B
and D lie on opposite sides of line LAC. Then ∠ACD is called an exterior angle
of △ABC. Let CE be the line parallel to line AB and passing through C.
Since we know that corresponding angles are congruent (∠B is equal to ∠ECD)
as well as alternate interior angles (∠A is equal to ∠ACE), the angle sum of
△ABC is:
|∠A|+|∠B|+|∠C| = |∠ACE|+|∠ECD|+|∠C| = 180◦
5. Formal proof: An exterior angle is equal to the sum of its remote interior angles.
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Solution: same as before
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