Download Application 1

From HW # 5
1. In class we discussed that the sum of the measures of the interior angles of a
convex quadrilateral is 360o.” Using Geometer’s Sketchpad, determine
whether this is true for a concave quadrilateral. Please display all appropriate
angle measures and sums.
2. 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
quadrilateral. Please display all appropriate angle measures and sums.
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).
A
D
B
C
From HW # 5
4. In the diagram, segments AD, BE , and CF 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)
B
A
360°
C
P
F
D
E
From HW # 5
5. In the diagram, AE is parallel to BF, and DE is parallel to FC.
Prove that E  F.
E
F
A
B
C
D
From HW # 5
6. In the diagram, four right triangles are shown, and PQ and RQ are
perpendicular. What is the sum of the measures of the four numbered angles?
270°
P
1
2
3
4
Q
R
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°
4
3
2
1
A
x
5
B
C
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)?
B
C
360°
A
E
F
D
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)?
B
C
360°
A
E
F
D
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)?
B
C
360°
A
E
F
D
Practice Problems
B
1. Find the sum of the measures of
A, B, C, D, and E
A
40
C
E
D
B
2. In the diagram, BE bisects ABC.
What is the measure of BED?
A
48
108
96
D
E
C
Practice Problems
B
1. Find the sum of the measures of
A, B, C, D, and E
A
40
460
C
E
D
B
2. In the diagram, BE bisects ABC.
What is the measure of BED?
A
48
108
102
96
D
E
C
3. In the diagram, compute the sum of the angles
numbered 1 through 8.
1
7
5
2
6
8
4
3
3. In the diagram, compute the sum of the angles
numbered 1 through 8.
1
720
7
5
2
6
8
4
3
The three congruence postulates we have are SSS, SAS, and ASA.
An additional method for proving triangles congruent
AAS
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.
leg
leg
base
Application 1
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?
B
48
P
A
24
C
B
P
E
A
D
C
Another triangle congruence theorem:
hypotenuse - leg (HL)
Can two triangles be proven congruent by SSA
ASS ?
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
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. ItProve
is notyour
necessary
to prove your conjecture.
conjecture.
Conjecture: AC  CD
B
D
A
C
(n  3)(n)
A convex polygon with n-sides has
diagonals.
2
(n  3)(n)
A convex polygon with n-sides has
.
2
Homework:
Download, print, and complete Homework # 6
Theorem: Every right angle is obtuse.
Here is the “proof” of the theorem:
A
D
E
B
Given:
Right angle ABC and obtuse angle DCB
Prove:
ABC  DCB

F
C
1. Using a compass, construct points E and F on ray BA and ray CD so that BE  CF.
A
D
E

C
B
Question:
Suppose they were parallel.
F
A
D
E

C
B
Question:
Suppose they were parallel.
F
A
D
E
B
3.

C
F
A
E
B
4.
D


C
F
A
D
E
B
4.



C
F
A
D
E
B

M

N

F
C
5. The perpendicular bisectors are
not parallel.
4.
P
labeling the midpoints N and
M, respectively.
6. Call their point of intersection, P
A
D
E
B

M

P
7. Construct segments EP, FP,
BP, and CP
N

F
C
8. PEN  PFN
SAS
9. BPM  CPM
SAS
10. EP  FP and BP  CP
A
D
E
B

M

P
N

F
C
12. EBP  FCP
SSS
A
D
E
B

M

N

F
C
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?