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From HW # 5. In class we discussed that the sum of the measures of the interior angles of a convex quadrilateral is 360 o .” Using Geometer’s Sketchpad, determine whether this is true for a concave quadrilateral. Please display all appropriate

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In class we discussed that the sum of the measures of the interior angles of a

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#### Presentation Transcript

From HW # 5

• In class we discussed that the sum of the measures of the interior angles of a

• whether this is true for a concave quadrilateral. Please display all appropriate

• angle measures and sums.

• Another theorem that we stated in class said ”The sum of the measures of the

• exterior angles of any convex polygon, one angle at each vertex, is 360o.”

• Using Geometer’s Sketchpad, determine whether this is true for a concave

• 3. In the diagram at the right, all the vertices of quadrilateral

• ABCD lie on a circle (we say that quadrilateral ABCD is ,

• inscribed in the circle.).

• a. Construct the diagram using Geometer’s Sketchpad.

• b. Make a conjecture about how the measures of angle

• and angle BCD are related.

• c. Drag one or more of the quadrilateral’s vertices and

• verify your conjecture (or form a new conjecture if

• your first conjecture turns out to be incorrect).

B

A

C

F

P

D

E

From HW # 5

4. In the diagram, segments and are concurrent

(intersect at point P). What is the sum of the measures of the six “corner”

angles? (i.e. the sum of the measures of angles A, B, C, D, E, and F)

360°

5.

In the diagram, is parallel to , and is parallel to .

Prove that E F.

E

F

A

B

C

D

From HW # 5

In the diagram, four right triangles are shown, and and are

perpendicular. What is the sum of the measures of the four numbered angles?

P

1

2

3

4

R

Q

From HW # 5

6.

270°

4

3

2

x

1

5

A

B

C

From HW # 5

7. In the diagram, points A, B, and C are collinear. Express the sum of the

measures of the five numbered angles as a function of x.

x + 360°

B

C

A

D

E

F

From HW # 5

8. In the diagram, what is the sum of the measures of the six “corner”

angles (i.e. the sum of the measures of angles A, B, C, D, E, and F)?

360°

B

C

A

D

E

F

From HW # 5

8. In the diagram, what is the sum of the measures of the six “corner”

angles (i.e. the sum of the measures of angles A, B, C, D, E, and F)?

360°

B

C

A

D

E

F

From HW # 5

8. In the diagram, what is the sum of the measures of the six “corner”

angles (i.e. the sum of the measures of angles A, B, C, D, E, and F)?

360°

B

B

A

40

48

A

C

108

C

96

E

D

E

D

Practice Problems

• Find the sum of the measures of A, B, C, D, and E

• In the diagram, bisects ABC. What is the measure of BED?

B

B

A

40

48

A

C

108

C

96

E

D

E

D

Practice Problems

• Find the sum of the measures of A, B, C, D, and E

460

• In the diagram, bisects ABC. What is the measure of BED?

102

• In the diagram, compute the sum of the angles

• numbered 1 through 8.

1

7

5

2

6

8

4

3

• In the diagram, compute the sum of the angles

• numbered 1 through 8.

720

1

7

5

2

6

8

4

3

An additional method for proving triangles congruent

AAS

The three congruence postulates we have are SSS, SAS, and ASA.

leg

leg

base

The Isosceles Triangle Theorem (Base Angles Theorem)

If a triangle has two congruent sides, the angles opposite those sides are congruent.

The converse is also true: If a triangle has two congruent angles, the sides opposite those angles are congruent.

B

48

P

A

C

Application 1

• In the diagram, ABC is isosceles with AB = BC, and AP is an altitude. If the measure of angle B is 48, what is the measure PAC?

24

Another triangle congruence theorem:

hypotenuse - leg (HL)

Can two triangles be proven congruent by SSA ?

ASS

SSA

A counter-example to SSA

These triangles agree in SSA, yet they cannot be congruent.

X

Moral: If you use ASS to prove triangles congruent, you…

From HW # 3

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.

Conjecture:

A convex polygon with n-sides has diagonals.

A convex polygon with n-sides has .

Homework:

Theorem:

Every right angle is obtuse.

Here is the “proof” of the theorem:

E

F

1. Using a compass, construct points E and F on ray BA and ray CD so that .

A

D

C

B

Given: Right angle ABC and obtuse angle DCB

Prove:ABC  DCB

Question:

A

D

E

F

C

B

Suppose they were parallel.

Question:

A

D

E

F

C

B

Suppose they were parallel.

3.

A

D

E

F

C

B

A

D

E

F

C

B

4.

A

D

E

F

C

B

4.

5.

The perpendicular bisectors are not parallel.

 P

6. Call their point of intersection, P

A

D

E

N

F

C

B

M

4.

labeling the midpoints N and M, respectively.

7.

Construct segments EP, FP, BP, and CP

10. and

A

D

E

N

F

C

B

M

8. PEN  PFN

SAS

9. BPM  CPM

 P

SAS

A

D

E

N

F

C

B

M

 P

SSS

12. EBP  FCP

A

D

E

N

F

C

B

M

Therefore, right angle ABC is congruent to obtuse angle FCB.

Subtract the measures of the green angles from the measures of the red angles.

 P

Where is the flaw in the proof?