(2x+10)°

(3x – 13)°

•

•

•

•A B

C

G

•

•E

F

•

•

D

H

In the diagram, AB CD. The measures of GED and GFB are shown.

1. Find the value of the variable x.

2. Find m(BFH)

A

B

C

DP

Q

In the diagram, AB CD, and PQ bisects CPB.

If m(PCQ) = 38°, what is the measure of PQD?•

•

•

••

•

x = 23

124

108

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.

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.  

ax

b

From HW # 2

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. 

In problems 1 - 4 below, parallel lines are indicated. Find the measure of the angle marked x in each. 1. 2. 3. 4.

50

150 x

5x

4x

x

x

49

42

x

Name__________________________ Math 3395 HW 2 Koppelman

x = 131x = 48

x = 20

x = 22.5

From HW # 2

m

S

Paper and pencil constructions

5. Construct a parallel through point S to line m.

From HW # 2

A

Paper and pencil constructions

6. Construct a triangle two of whose three angles have the same measure as A.

From HW # 2

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.

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.  

ax

bA B

D CE F

P

From HW # 2

Conjecture: a + b – x = 180

A

C D

B

a. Construct the diagram below, in which is parallel to . 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?

AC BD

8. Using Geometer’s Sketchpad

From HW # 2

A

C D

B

a. Construct the diagram below, in which is parallel to . 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?

AC BD

8. Using Geometer’s Sketchpad

From HW # 2 Conjecture: The measures have a sum of 180

1 2

3

4

5 6

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.

Converse

If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

If two parallel lines are cut by a transversal, then corresponding angles are congruent.

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

B

P

QA

R

l

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 as one of its sides.

Conclusion: is parallel to PR AB

PQ

A

B

C

P

Q

N

Basic Construction 4: Steps for 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

Using Geometer’s Sketchpad, construct the diagram shown, in which

AB is parallel to CD, MP bisects BMN, and NP bisects DNM.

P

DC

M

N

AB

Using Geometer’s Sketchpad, construct the diagram shown, in which

AB is parallel to CD, MP bisects BMN, and NP bisects DNM.

Make a conjecture about the measure of P.Explain why your conjecture will always be

true.

P

DC

M

N

AB

Basic Construction 2: Copying a given angle ABC:

1. “Construct” a ray .

2. Construct a circle of convenient radius with point B as center. Call the

intersection of the circle with point M and the intersection of the circle

with point N.

1. Construct a congruent circle with point P as center. Call its intersection with

point R.

4. Construct a circle with center M and radius .

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” .

Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.

PQ

BA

BC

MN

PQ

PS

Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.

S

R

M

N PQ

B

A

C

How can we be sure that our conclusion is correct? (in other words, why does this process guarantee

that SPR is congruent to ABC?

Congruent triangles are triangles with all pairs of corresponding sides and all pairs of corresponding angles congruent.

F

E

B

A

C

D

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 .

2. Construct a circle of convenient radius with point B as center. Call the

intersection of the circle with point M and the intersection of the circle

with point N.

1. Construct a congruent circle with point P as center. Call its intersection with

point R.

4. Construct a circle with center M and radius .

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” .

Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.

PQ

BA

BC

MN

PQ

PS

Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.

S

R

M

N PQ

B

A

C

Conclusion: Angle SPR is congruent to (is a copy of) angle ABC.

S

R

M

N PQ

B

A

C

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).

Proof of the construction

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

Homework:

Download, print, and complete Homework # 3

(Constructions Continued)