If you do not have YES next to your student #, either you did not send me
GSP constructions or I am waiting for corrections.
KSU#
432043
397469
374787
445707
441983
483280
394078
312014
406936
443837
339238
401035
420456
GSP for HW 3
YES
YES
YES
YES
YES
YES
KSU#
GSP for HW 2
350057
428483
338465
340871
427222
427055
341525
493249
401794
449724
437486
382516
414398
YES
YES
YES
YES
YES
YES
YES
Warm-up
1. Using Geometer’s Sketchpad, construct a rectangle whose length
and width are in the ratio of 2:1 and display the ratio.
2. Using compass and straight edge, construct a right angle using
only the angle bisector construction (Basic Construction 4).
Two angles form a linear pair if they are adjacent and their exterior sides
are opposite rays.
D
A
B
C
Theorem: The angles of a linear pair are supplementary.
Theorem: If two angles are congruent and supplementary, they are
right angles.
From HW # 3
1. Using Geometer’s Sketchpad
a. Construct triangle ABC.
b. Construct the angle bisector of BAC
c. Construct a line through point C parallel to AB. Label its intersection with
the angle bisector point D.
d. Make a conjecture about the relationship between the length of AC and the
length of CD . It is not necessary to prove your conjecture.
Conjecture: AC  CD
B
D
A
C
B
Conjecture: the length of AM is
three times the length of MP .
P
M
A
D
C
From HW # 3
4. Construct a triangle congruent to triangle ABC.
C
A
B
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?
Conjecture: m2 = m1 + m3
D

C

1
2


 Q
F
E

A
3

B
1++blue
3
2 = red
How can we be sure that our
1. Construct a circle using point B as center,
conclusion
isat point
correct?
BA at point P and BC
intersecting
Q.
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 A B.
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 A B.
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
P
l
B
A
Q
(Prove: AMP and BMP are right angles)
They
radii of congruent
circles
Radiiare
of congruent
circles are
congruent.
Reflexive property
4. PAQ  PBQ
5. APQ  BPQ
SSS
P
CPCTC
Same as 3
7. PAM  PBM
8. AMP  BMP
SAS
CPCTC
l
A
M
B
9. AMP is supplementary to BMP.
The angles of a linear pair are supplementary
10. AMP and BMP are right angles.
If two angles are congruent and supplementary,
they are right angles.
Q
Theorems that should make perfect sense to you
1. If two angles are complements of congruent angles (or of the same angle),
then the two angles are congruent.
2. If two angles are supplements of congruent angles (or of the same angle),
then the two angles are congruent.
3. Vertical angles are congruent.
4. If two lines intersect, then they intersect in exactly one point.
5. Every segment has exactly one midpoint.
6. Every angle has exactly one bisector.
The sum of the measures of the angles of a triangle is 180°.
Proof: Construct PQ parallel to AB.
Q
C
P
B
A
B
A + ACB + QCB
PCA
= 180°
Related Corollaries and Theorems
• Through a point outside a line, exactly one perpendicular can be drawn
…to the line.
• P
l
Related Corollaries and Theorems
• Through a point outside a line, exactly one perpendicular can be drawn
…to the line.
• If two angles of one triangle are congruent to two angles of another
…triangle, then the third angles are congruent.
• Each angle of an equiangular triangle has measure 60o.
• In a triangle, there can be at most one right angle or one obtuse angle.
• The acute angles of a right triangle are complementary.
In the diagram, ABD  DCA and BD  BC . If the
measure of DCB is 50, what is the measure of A?
D
A
B
C
Related Corollaries and Theorems
• Through a point outside a line, exactly one perpendicular can be drawn
…to the line.
• If two angles of one triangle are congruent to two angles of another
…triangle, then the third angles are congruent.
• Each angle of an equiangular triangle has measure 60o.
• In a triangle, there can be at most one right angle or one obtuse angle.
• The acute angles of a right triangle are complementary.
• If one side of a triangle is extended, then the measure of the exterior
…angle(s) formed is equal to the sum of the measures of the two remote
…interior (non-adjacent interior) angles.
If one side of a triangle is extended, then the measure of the exterior
angle(s) formed is equal to the sum of the measures of the two remote
interior (non-adjacent interior) angles.
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?
Conjecture: m2 = m1 + m3
D

C

1
2


 Q
F
E

A
3

B
1++blue
3
2 = red
Homework:
Download, print, and complete Homework # 4