in class we discussed that the sum of the measures of the interior angles of a
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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 - PowerPoint PPT PresentationTRANSCRIPT
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).
From HW # 5
A
B
C
D
A
B
C
D
E
FP
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)
,, BEAD CF
360°
From HW # 5
A
E
B C D
F
In the diagram, is parallel to , and is parallel to .
Prove that E F.
AE BF DE FC5.
From HW # 5
P
Q R
1
2
3
4
In the diagram, four right triangles are shown, and and are
perpendicular. What is the sum of the measures of the four numbered angles?
PQ RQ
270°
6.
From HW # 5
1
23
4
5
x
AB C
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°
From HW # 5
A
BC
DEF
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°
From HW # 5
A
BC
DEF
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°
From HW # 5
A
BC
DEF
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°
From HW # 5
Practice Problems
1. Find the sum of the measures of A, B, C, D, and E
A
B
C
DE
40
2. In the diagram, bisects ABC. What is the measure of BED?
A
D
B
C
E
108
96
48
BE
Practice Problems
1. Find the sum of the measures of A, B, C, D, and E
A
B
C
DE
40
2. In the diagram, bisects ABC. What is the measure of BED?
A
D
B
C
E
108
96
48
BE
102
460
3. In the diagram, compute the sum of the angles numbered 1 through 8. 1
2
34
5 6
7
8
1
2
34
5 6
7
8
720
3. In the diagram, compute the sum of the angles numbered 1 through 8.
An additional method for proving triangles congruent
AAS
The three congruence postulates we have are SSS, SAS, and ASA.
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.
base
leg leg
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?
A
B
C
P
48
24
E
D
P
B
A C
Another triangle congruence theorem:
hypotenuse - leg (HL)
Can two triangles be proven congruent by SSA ?ASS SSA
•
X
These triangles agree in SSA, yet they cannot be congruent.
A counter-example to SSA
Moral: If you use ASS to prove triangles congruent, you…
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
Prove your conjecture.
A convex polygon with n-sides has diagonals.2
3)(n)(n
A convex polygon with n-sides has .2
3)(n)(n
Homework:
Download, print, and complete Homework # 6
Theorem: Every right angle is obtuse.
Here is the “proof” of the theorem:
Given: Right angle ABC and obtuse angle DCB
Prove: ABC DCB
A
B C
D
1. Using a compass, construct points E and F on ray BA and ray CD so that .
CFBE
EF
A
B C
D
EF
Question:
Suppose they were parallel.
A
B C
D
EF
Question:
Suppose they were parallel.
A
B C
D
EF
3.
A
B C
D
EF
4.
A
B C
D
EF
4.
A
B C
D
EF
N
M
P
6. Call their point of intersection, P
The perpendicular bisectors are not parallel.
5.
labeling the midpoints N and M, respectively.
4.
P
A
B C
D
EFN
M
Construct segments EP, FP, BP, and CP
7.
8. PEN PFN SAS
9. BPM CPM SAS
10. and FPEP CPBP
A
B C
D
EFN
M
P12. EBP FCP SSS
A
B C
D
EFN
M
P
Subtract the measures of the green angles from the measures of the red angles.
Where is the flaw in the proof?
Therefore, right angle ABC is congruent to obtuse angle FCB.