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G•
E
•
•
C
•
•
•
A
In the diagram, AB CD. The measures of
GED and GFB are shown.
F
D
1.
Find the value of the variable x.
•
2.
Find m(BFH)
x = 23
124
B
•H
B
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In the diagram, AB CD, and PQ bisects CPB.
P
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A
•
D
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Q
C
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If m(PCQ) = 38°, what is the measure of PQD?
108
From HW # 2
7. Using Geometer’s Sketchpad
a. Construct the diagram shown below, in which two pairs of line segments are parallel.
(The letters a, b, and x do not need to appear on the sketch.) Please include a hard-copy
of the sketch when you turn in this assignment.
b. Use the measurement tool to display the measures of the angles marked a, b and x.
c. Make a conjecture about the relationship between the measures of the three angles.
d. Explain why your conjecture will always be true.
a
x
b
I'm working on problem 7 on HW2 and this is what it looks like so far:
I was wondering how to cut the parts of the segment that are below the horizontal and
the parts above the endpoints.
Math 3395
HW 2
Koppelman
From
HW # 2
Name__________________________
In problems 1 - 4 below, parallel lines are indicated. Find the measure of the angle
marked x in each.
1.
2.
x
49
x
42
x = 131
x = 48
3.
4.
50
x
150
5x
x
x = 20
4x
x = 22.5
From HW # 2
Paper and pencil constructions
5. Construct a parallel through point S to line m.
S
m
From HW # 2
Paper and pencil constructions
6. Construct a triangle two of whose three angles have the same measure as A.
A
From HW # 2
Conjecture: a + b – x = 180
7. Using Geometer’s Sketchpad
a. Construct the diagram shown below, in which two pairs of line segments are parallel.
(The letters a, b, and x do not need to appear on the sketch.) Please include a hard-copy
of the sketch when you turn in this assignment.
b. Use the measurement tool to display the measures of the angles marked a, b and x.
c. Make a conjecture about the relationship between the measures of the three angles.
d. Explain why your conjecture will always be true.
E
D
a
A
F
x
P
C
b
B
From HW # 2
8. Using Geometer’s Sketchpad
a. Construct the diagram below, in which AC is parallel to BD. Your diagram should
look exactly like the one shown.
b. Use the measurement tool to display the measures of BCD, ADC, A, and B.
c. Display the sum of the measures of these four angles.
d. Use your cursor to move point B. Does the sum change?
B
A
C
D
Conjecture: The measures have a sum of 180
From HW # 2
8. Using Geometer’s Sketchpad
a. Construct the diagram below, in which AC is parallel to BD. Your diagram should
look exactly like the one shown.
b. Use the measurement tool to display the measures of BCD, ADC, A, and B.
c. Display the sum of the measures of these four angles.
d. Use your cursor to move point B. Does the sum change?
B
4
A
3
5
C
1
2
6
D
Last class, we used Geometer’s Sketchpad to investigate the following problem.
1. Use Geometer’s Sketchpad to construct the following diagram, in which
line DC is parallel to line AB and point Q is randomly chosen between them.
D

C

1
2

A
3
 Q

B
2. Display the measures of <1, <2, and <3
3. Make a conjecture about how the three measures are related to one another.
Conjecture: m2 = m1 + m3
4. Drag point Q and verify your conjecture or form a new conjecture.
5. Can you prove the conjecture?
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Converse
If two lines are cut by a transversal and corresponding
angles are congruent, then the lines are parallel.
Converses of Parallel Lines Theorems
If two lines are cut by a transversal and corresponding
angles are congruent, then the lines are parallel.
If two lines are cut by a transversal and alternate interior
angles are congruent, then the lines are parallel.
If two lines are cut by a transversal and same side interior
angles are supplementary, then the lines are parallel.
If a transversal is perpendicular to each of two lines, then
the two lines are parallel.
More Basic Constructions
Basic Construction 3: Steps for constructing a parallel to a line l (or
AB) through a point P not on the line.
1. Construct a line through P that intersects line AB at point Q.
2. Follow the steps for “copying an angle” to construct an angle QPR
that is congruent to PQB and having PQ as one of its sides.
Conclusion: PR is parallel to AB
P
R
A
Q
B
l
Basic Construction 4: Steps for constructing the bisector of a given
angle, ABC.
1. Construct a circle using point B as center,
intersecting BA at point P and BC at point Q.
2. Construct congruent circles with centers at P and Q.
Use a radius that will cause the two circles to intersect.
Call the intersection point N.
3. Construct BN .
Conclusion: BN is the bisector of ABC.
A
P
B
N
Q
C
Using Geometer’s Sketchpad, construct
the diagram shown, in which
AB is parallel to CD,
MP bisects BMN, and
NP bisects DNM.
A
M
B
P
C
N
D
Using Geometer’s Sketchpad, construct
the diagram shown, in which
AB is parallel to CD,
MP bisects BMN, and
NP bisects DNM.
A
M
B
P
C
N
Make a conjecture about the measure of P.
Explain why your conjecture will always be true.
D
Construction 2: Copying a given angle ABC:
How can we Basic
be
sure
that our conclusion is correct?
PQ
1. “Construct” a ray
.
(in2.other
whyradius
does
this
process
guarantee
Constructwords,
a circle of convenient
with point
B as center.
Call the
intersection
of the
circle with BAis
point
M and the intersection
the circle
that
SPR
congruent
toof ABC?
with BC point N.
1. Construct a congruent circle with point P as center. Call its intersection with
PQ point R.
4. Construct a circle with center M and radius MN .
5. Construct a circle congruent to the one in step 5 with R as center. Call the
intersection of this circle and circle P, point S.
4. “Construct” PS.
Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.
A
M
S
B
N
P
C
R
Q
Congruent triangles are triangles with all pairs of corresponding sides
and all pairs of corresponding angles congruent.
B
D
F
A
C
E
ABC  EDF
The three triangle congruence postulates:
If three sides of one triangle are congruent to three sides of
another triangle, then the triangles are congruent (SSS).
If two sides of one triangle are congruent to two sides of
another triangle, and the angles between these pairs of sides
are congruent, then the triangles are congruent (SAS).
If two angles of one triangle are congruent to two angles of
another triangle, and the sides between these pairs of angles
are congruent, then the triangles are congruent (ASA).
Corresponding parts of congruent triangles are congruent.
(CPCTC)
Proving that the basic constructions do what
we claim they do.
Basic Construction 2: Copying a given angle ABC:
1. “Construct” a ray PQ .
2. Construct a circle of convenient radius with point B as center. Call the
intersection of the circle with BA point M and the intersection of the circle
with BC point N.
1. Construct a congruent circle with point P as center. Call its intersection with
PQ point R.
4. Construct a circle with center M and radius MN .
5. Construct a circle congruent to the one in step 5 with R as center. Call the
intersection of this circle and circle P, point S.
4. “Construct” PS.
Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.
A
M
S
B
N
P
C
R
Q
Proof of the construction
BN  PR, and BM  PS because they are radii of congruent
circles. Similarly, MN  SR. Therefore, MBN  SPR (SSS)
and SPR is congruent to ABC (CPCTC).
A
M
S
B
N
P
C
Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.
R
Q
How can we be sure that our
1. Construct a circle using point B as center,
conclusion
correct?
intersecting
at point
Q.
BA at point P and BCis
Basic Construction 4: Constructing the bisector of a given angle ABC.
2. Construct congruent circles with centers at P and Q.
Use a radius that will cause the two circles to intersect.
Call the intersection point N.
3. Construct BN .
Conclusion: BN is the bisector of ABC.
A
P
N
B
Q
B
Proof of the construction
BP  BQ because they are radii of congruent circles. Similarly,
PN  QN. Since BN  BN (Reflexive Postulate), PBN  QBN
(SSS) and PBN is congruent to QBN (CPCTC).
A
P
N
B
Q
B
AB
PQ
AB
PQ
P
B
A
Q
Basic Construction 6: Steps for constructing a perpendicular to a line l
through a point P on the line.
1. Construct a circle with center at point P intersecting line l in two
points, A and B.
2. Construct congruent circles with centers at A and B, and radii at
least as long as AB .
3. Call the intersection of the two congruent circles, point Q.
4. Construct PQ.
Q
Conclusion: PQ is perpendicular to line l.
l
A
P
B
Basic Construction 7: Steps for constructing a perpendicular to a line l
through a point P not on the line.
1. Construct a circle with center at point P intersecting line l in two
points, A and B.
2. Construct congruent circles with centers at A and B, and radii at
least as long as AB.
3. Call the intersection of the two congruent circles, point Q.
4. Construct PQ.
P
Conclusion: PQ is perpendicular to line l.
l
B
A
Q
Homework:
Download, print, and complete Homework # 3
(Constructions Continued)