warm-up using geometer’s sketchpad, construct a rectangle whose length

KSU# GSP for HW 3

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KSU# GSP for HW 2

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If you do not have YES next to your student #, either you did not send me GSP constructions or I am waiting for corrections.

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

BA C

Theorem: The angles of a linear pair are supplementary.

Theorem: If two angles are congruent and supplementary, they are right angles.

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 . Label its intersection with the angle bisector point D.   d. Make a conjecture about the relationship between the length of and the length of . It is not necessary to prove your conjecture.

AB

ACCD

D

A

C

BConjecture: CDAC

From HW # 3

A

B

CD

M

P

Conjecture: the length of is three times the length of .

AMMP

4. Construct a triangle congruent to triangle ABC.                  A

C

From HW # 3

B

BA

C

3

2

1D

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.

2. Display the measures of <1, <2, and <3

3. Make a conjecture about how the three measures are related to one another.

4. Drag point Q and verify your conjecture or form a new conjecture.

5. Can you prove the conjecture?

Q

Conjecture: m2 = m1 + m3

BA

C

3

2

1D

E

F

2 = red + blue 1 + 3

Conjecture: m2 = m1 + m3

Q

Basic Construction 4: Constructing the bisector of a given angle ABC.

1. Construct a circle using point B as center, intersecting at point P and 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 .

Conclusion: is the bisector of ABC.

BA BC

BN

BN

N

Q

P

B

A

B

How can we be sure that our conclusion is correct?

N

Q

P

B

A

B

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).

Proof of the construction

AB

PQ

ABPQ

A B

P

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 .

3. Call the intersection of the two congruent circles, point Q.

4. Construct .

Conclusion: is perpendicular to line l.

AB

PQ

PQ

A P B

Q

l

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 .

3. Call the intersection of the two congruent circles, point Q.

4. Construct .

Conclusion: is perpendicular to line l.

AB

PQ

PQ

P

A B

Q

l

P

A B

Q

l

P

A B

Q

l

They are radii of congruent circlesRadii of congruent circles are congruent.

Reflexive property

4. PAQ PBQ SSS

5. APQ BPQ CPCTC

M

Same as 3

7. PAM PBM SAS

8. AMP BMP CPCTC

9. AMP is supplementary to BMP. 10. AMP and BMP are right angles.

The angles of a linear pair are supplementary

(Prove: AMP and BMP are right angles)

If two angles are congruent and supplementary, they are right angles.

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.

A

B

C

P

Q

+ +PCA ACB QCB = 180° A B

The sum of the measures of the angles of a triangle is 180°.

PQ ABProof: Construct parallel to .

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 . If the

measure of DCB is 50, what is the measure of A?

BCBD

A

B C

D

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.

BA

C

3

2

1D

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.

2. Display the measures of <1, <2, and <3

3. Make a conjecture about how the three measures are related to one another.

4. Drag point Q and verify your conjecture or form a new conjecture.

5. Can you prove the conjecture?

Q

Conjecture: m2 = m1 + m3

Last class, we used Geometer’s Sketchpad to investigate the following problem.

BA

C

3

2

1D

E

F

2 = red + blue 1 + 3

Conjecture: m2 = m1 + m3

Q

Homework:

Download, print, and complete Homework # 4