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Problem of the Day
(Monday)
Alternate Interior Angles?
a. 3 & 6, 4 & 5
b. 1 & 4, 2 & 3, 5 & 8, 7 & 6
c. 1 & 5, 2 & 6, 3 & 7, 4 & 8
d. 1 & 8, 2 & 7
Angles Review
Corresponding Angles?
a. 1 & 2, 2 & 4, 3 & 4, 1 & 3
6 & 5, 6 & 8, 7 & 8, 5 & 7
1 2
3 4
5
6
7 8
b. 1 & 4, 2 & 3, 5 & 8, 7 & 6
c. 1 & 5, 2 & 6, 3 & 7, 4 & 8
d. 1 & 8, 2 & 7
Supplementary Angles?
a. 1 & 2, 2 & 4, 3 & 4, 1 & 3
6 & 5, 6 & 8, 7 & 8, 5 & 7
b. 1 & 4, 2 & 3, 5 & 8, 7 & 6
c. 1 & 5, 2 & 6, 3 & 7, 4 & 8
d. 1 & 8, 2 & 7
Alternate Exterior Angles?
a. 1 & 2, 2 & 4, 3 & 4, 1 & 3
6 & 5, 6 & 8, 7 & 8, 5 & 7
b. 1 & 4, 2 & 3, 5 & 8, 7 & 6
c. 1 & 5, 2 & 6, 3 & 7, 4 & 8
d. 1 & 8, 2 & 7
Vertical Angles?
a. 1 & 2, 2 & 4, 3 & 4, 1 & 3
6 & 5, 6 & 8, 7 & 8, 5 & 7
b. 1 & 4, 2 & 3, 5 & 8, 7 & 6
c. 1 & 5, 2 & 6, 3 & 7, 4 & 8
d. 1 & 8, 2 & 7
Polygon:
Closed figure
Regular Polygon:
All sides are e q ual
Triangles Everywhere
Problem of the Day (Tuesday)
What is the measure of
angle x?
a. 720°
b. 120°
c. 360°
d. 60°
x
A regular pentagon:
1. What is the sum of the
interior angles?
2. What is the measure of each
interior angle?
3. What is the sum of the
exterior angles?
4. What is the measure of each
exterior angle?
An irregular pentagon
What is the measure of the missing angle?
90˚
130˚
?
90˚
130˚
A regular hexagon:
1. What is the sum of the
interior angles?
What is the measure of the
missing angle?
90˚
120˚
170˚
100˚
130˚
?
A regular octagon:
1. What is the sum of the
interior angles?
2. What is the measure of each
interior angle?
3. What is the sum of the
exterior angles?
4. What is the measure of each
exterior angle?
A regular decagon:
1. What is the sum of the
interior angles?
2. What is the measure of each
interior angle?
3. What is the sum of the
exterior angles?
4. What is the measure of each
exterior angle?
Name__________________
Missing Angles Challenge
*You may only assume that the pentagon and
hexagon are regular.
l
m
h
g b
j
k
c
n
q
s t
i
a
d
e
f
45°
o
110°
p r
u v
x
a = _____
b = _____
c = _____
d = _____
e = _____
f = _____
g = _____
h = _____
i = _____
j = _____
k = _____
l = _____
m = _____
n = _____
o = _____
p = _____
q = _____
r = _____
s = _____
t = _____
u = _____
v = _____
w = _____
x = _____
Learning Target
I can…
Find the exterior angles of any polygon
Exterior Angles of a Polygon
Level Five – Find the missing angles
1.
3.
b
a
2.
4.
34˚
a
b
Missing Angles
m
The lines m and n are
parallel
What is the measure
angle b?
What is the measure of
angle a?
What is the measure of
angle g?
What is the measure of
angle f?
n
a
98°
b c
d
f
e
g
Create your own puzzle
Rules:
1. You must use at least 2 regular
polygons
2. Angle measures must be close (they do
not have to be perfectly drawn to scale)
3. You must use a ruler
4. You must include 3 of the following:
complementary angles, supplementary
angles, interior angles, exterior angles,
corresponding angles, vertical angles
Problem of the Day (Tuesday)
What is the measure of
angle x?
a. 720°
b. 120°
c. 360°
d. 60°
x
A regular pentagon:
1. What is the sum of the
interior angles?
2. What is the measure of each
interior angle?
3. What is the sum of the
exterior angles?
4. What is the measure of each
exterior angle?
Congruent Triangles
• All sides are congruent and all angles are
congruent
H
L
K
J
G
I
Congruent Triangles
SSS
B
A
E
C
D
F
SAS
H
L
K
J
G
I
ASA
AAS
Congruent Polygons
Show that each pair of triangles is congruent.
a.
b.
Q
QP
Q
QPR
E
EY
Y
Angle
Side
Angle
EYT by ASA.
SQ
VT
Q
T
QR
TU
SQR
Side
Angle
Side
VTU by SAS.
Does AAA guarantee that two
triangles are congruent? Why or
why not?
No
Same angles, different sizes
Example:
80°
80°
50°
50°
50°
50°
Congruent Polygons
A surveyor drew the diagram below to find the distance from
J to I across the canyon. Show that GHI
KJI. Then find JK.
J
JI
KIJ
H
Both are right angles.
HI
Both measure 48 ft.
GIH
They are vertical angles.
So ∆GHI
∆ JI by ASA
Corresponding parts of congruent triangles are congruent.
JK corresponds to HG, so JK is 36 ft.
Congruent Polygons
Use
ABC and
XYZ to answer the questions.
1.Suppose AC= XZ, AB = XY, and BC = YZ.
Write a congruence statement for the figures.
∆ABC  ∆XYZ
2.Suppose ABC and XYZ are congruent. If AB = 5 cm,
BC = 8 cm, and AC = 10 cm, find XZ.
10 cm
3. Suppose
ABC
ASA
B
Y,
XYZ?
A
X, and AB
XY. Why is
Congruent Polygons
4. Let AB = XY = 9 inches; BC = YZ = 24 inches; and
m B = 85°, m Z = 35°, and m Y = 85°. Prove that
the triangles are congruent and find m C.
∆ABC  ∆XYZ by SAS; m C =35°
Problem of the Day (Wednesday)
Are the two triangles
congruent? Why?
a.
b.
c.
d.
Yes, because of SAS
Yes, because of ASA
Yes, because of SSS
No, because two side are
congruent, no angles
B
A
D
C
a. ABC is congruent to DCB because
of SAS
b. ABC is congruent to DCB because
of SAS
c. ABC is congruent to BCD because
of AAS
d. The triangles are not congruent because
they only have a side and angle that are
congruent
Are the two triangles congruent?
Why or why not?
a.
b.
c.
d.
Yes, because of SAS
Yes, because of ASA
Yes, because of SSA
No, because SSA is not a congruent
triangle rule
Review
Are the two triangles
congruent? Why?
a.
b.
c.
d.
Yes, because of SAS
Yes, because of ASA
Yes, because of SSS
No, because two side are
congruent, no angles
Are the two triangles
congruent? Why?
a.
b.
c.
d.
Yes, because of SAS
Yes, because of ASA
Yes, because of SSS
No, because two side are
congruent, no angles
Are the two triangles
congruent? Why?
a.
b.
c.
d.
Yes, because of AAS
Yes, because of ASA
Yes, because of SSS
No, because two sides
are congruent, no angles
Is
a.
b.
c.
d.
? Why?
Yes, because of SAS
Yes, because of ASA
Yes, because of SSS
No, because two side are
congruent, no angles
Corresponding parts
If two polygons are congruent, then their
corresponding parts are congruent.
For example:
Since QRS = HGJ
That means
Similar Triangles
Triangles are similar if:
1. All angles are congruent
2. Corresponding sides are proportional
Symbol: ABC ~ DEF
Example:
E
B
2
80°
50°
A
80°
2
6
6
50°
5
C
50°
50°
D
15
F
Corresponding Sides
B
• Sides that match
ABC ~ XBY
AC corresponds to ____
AB corresponds to _____
BY corresponds to _____
X
A
Y
C
Are the triangles similar?
E
B
4
3
A
1
C
D
No, ¾ is not proportional to ½
2
F
Are the triangles similar? Why or
why not?
2
1.5
If ABC ~ DEF, find x
E
B
9
A
40.5
11
C
D
F
x
Other Polygons Can Be Similar Too
*They still must have congruent angles and sides must be proportional
B
C
F
G
2
5
E
A
16
D
If the two figures are similar, what is the
measure of side EH?
H
You try!
Which polygons are always
similar?
(hint: they always have congruent angles and
they will always have proportional sides)
a. Rhombuses
b. Squares
c. Triangles
d. Pentagons
CPS Learning Series Questions!
Problem of the Day (Thursday)
If the two figures are similar, what is the
value of x?
52
8
x
a.11
b.5.81
c.464.75
d.14
71.5
Right Triangles
hypotenuse
leg
leg
Pythagorean Theorem
1. Use when you know 2 sides of a right
triangle and you need to figure out the 3rd
2. a² + b² = c²
3. a and b are lengths of sides and c is
the length of the hypotenuse
Example
a²+ b² = c²
a = 3, b = 4
3² + 4² = c²
9 + 16 = c²
25 = c²
5=c
3
x
4
You try
x
5
Find the measure of
the hypotenuse x
12
You try again!
You also can find the length
of a side
Find the missing length
a² + b² = c²
8² + x² = 10²
64 + x² = 100
-64
x² = 36
x=6
10
8
x
You try!
Pam is making a new sail for her sailboat
pictured below. What is the height of the
sail?
a² + b² = c²
a² + 10² = 26²
a² + 100 = 676
-100
a² = 576
a
= 24
The Pythagorean Theorem
A ladder, placed 4 ft from a wall, touches the wall 11.3 ft
above the ground. What is the approximate length of the ladder?
Draw a diagram to illustrate the problem.
Use the Pythagorean Theorem.
Use the Pythagorean
c2 = a2 + b2
Theorem.
c2 = 42 + 11.32
Substitute.
c2 = 16 + 127.69
Square 4 and 11.3.
c2 = 143.69
Add.
Take the square root of
each side.
Use a calculator.
c2 =
143.69
c = 11.98708
The length of the ladder is about 12 ft.
a² + b² = c²
6² + 8² = 9² ?
36 + 64 = 81?
100 = 81?
No, she cannot use these boards
Problem of the Day (Friday)
You Try!
Draw a picture!