Download All of Unit 2

Sec 2.1 Geometry – Parallel Lines and Angles
Name:
PARALLEL LINES
G
●
EXTERIOR
1
A
●
3
B
●
2
●
4
C
These symbols
imply the two
lines are parallel.
INTERIOR
5
●
D
7
●
EXTERIOR
WORD BANK:
●E
6
●
8
H
∡𝑮𝑩𝑪
Consecutive Interior Angles
∡𝟓
Alternate Interior Angles
∡𝑫𝑬𝑯
F
Transversal
∡𝑯𝑬𝑭
Congruent
∡𝟒
Vertical Angles
Alternate Exterior Angles
Corresponding Angles
Consecutive Exterior Angles
Supplementary
∡𝑨𝑩𝑮
1. Give an alternate name for angle ∡𝟐 using 3 points:
2. Angles ∡𝑨𝑩𝑬 and ∡𝑪𝑩𝑮 can best be described as:
3. Angles ∡𝟔 and ∡𝟑 can best be described as:
4. The line ⃡𝑮𝑯 can best be described as a:
5. Which angle corresponds to ∡𝑫𝑬𝑩 :
6. Angles ∡𝑭𝑬𝑩 and ∡𝑪𝑩𝑬 can best be described as:
7. Angles ∡𝟏 and ∡𝟖 can best be described as:
8. Which angle is an alternate interior angle with ∡𝑪𝑩𝑬 :
9. Angles ∡𝑮𝑩𝑪 and ∡𝑩𝑬𝑭 can best be described as:
10. Angles ∡𝟐 and ∡𝟖 can best be described as:
11. Which angle is an alternate exterior angle with ∡𝑨𝑩𝑮 :
12. Which angle is a vertical angle to ∡𝑨𝑩𝑮 :
13. Which angle can be described as consecutive exterior angle with ∡𝟏 :
14. Any two angles that sum to 180 ̊ can be described as
M. Winking
Unit 2-1
angles.
page 19
TRIANGLE’s INTERIOR ANGLE SUM
1. a. First, Create a random triangle on a piece of patty
papers.
2
b. Using your pencil, write a number inside each interior angle a
label.
3
1
2
c.
Next, cut out the triangle.
d.
Finally, tear off or cut each of the angles from
the triangle
3
1
e. Using tape, carefully put all 3 angles next to one another so that
they all have the same vertex and the edges are touching but they
aren’t overlapping
Paste or Tape your 3 vertices here:
•
•
Common Vertex
2. What is the measure of a straight angle or the angle that creates a line by using two
opposite rays from a common vertex?
3. Collectively does the sum of your 3 interior angles of a triangle form a straight angle?
What about others in your class?
4. Make a conjecture about the sum of the interior angles of a triangle. Do you think your
conjecture will always be true? (please explain using complete sentences)
M. Winking
Unit 2-1
page 20
5. More formally, why do the 3 interior angles of any triangle sum to 180 ̊ ?
Consider ∆ABC. The segment ̅̅̅̅
𝑨𝑩 is extended into a line and a parallel line is constructed
⃡ .
through the opposite vertex. So, ⃡𝑨𝑩 ∥ 𝑪𝑫
a. Why is ∡𝟏 ≅ ∡𝟐 ?
b. Why is ∡𝟓 ≅ ∡𝟒 ?
c. Why is 𝒎∡𝟐 + 𝒎∡𝟑 + 𝒎∡𝟒 = 𝟏𝟖𝟎° ?
d. Using substitution we can replace 𝒎∡𝟐 with 𝒎∡𝟏 and 𝒎∡𝟒 with 𝒎∡𝟓 to show that the
interior angles of a triangle must always sum to 180̊.
(
) + 𝒎∡𝟑 + (
) = 𝟏𝟖𝟎°
Write the angle number in the
and then write the letter that corresponds with the number
based on the code at the bottom in the box.
7. Angle 2 and Angle
are alternate exterior angles.
8. Angle 7 and Angle
are alternate exterior angles.
9. Angle 4 and Angle
are corresponding angles.
10. Angle 5 and Angle
are consecutive interior angles.
11. Angle 3 and Angle
are alternate interior angles.
12. Angle 7 and Angle
are consecutive exterior angles.
13. Angle 6 and Angle
are vertical angles.
14. Angle 2 and Angle
are a linear pair and on the same side of the transversal.
15. Angle 1 and Angle
are corresponding angles.
1=D
2=U
3= L
4=A
1 2
3 4
5=N
6=I
Unit 2-1
page 21
What type of Geometry is this?
M. Winking
5 6
7 8
7=E
8=C
16. Given lines p and q are parallel, find the value of x that makes each diagram true.
a.
b.
(x)º
q
(x)º
q
p
140º
p
155º
x=
x=
17. Given lines p and q are parallel, find the value of x that makes each diagram true.
a.
b.
(6x+5)º
q
q
(6x+5)º
(2x+45)º
(2x+15)º
p
p
x=
x=
18. Given lines m and n are parallel, find the value y of that makes each diagram true.
a.
40º
b.
m
m
130º
n
yº
130º
40º
n
yº
y=
y=
M. Winking
Unit 2-1
page 22
19. ANGLE PUZZLE. Find 𝒎∡𝑨𝑬𝑭
Given:
 𝒎∡𝑫𝑬𝑭 = 𝟖𝟓°
 𝒎∡𝑨𝑩𝑮 = 𝟓𝟎°


C●
B
D●
G
●
∡𝑩𝑨𝑬 is a right angle
∡𝑪𝑮𝑬 and ∡𝑫𝑬𝑮 are supplementary
E
●
𝑚∡𝐴𝐸𝐹 =
A
F
●
20. Converse of AIA, AEA, CIA, CEA. Which sets of lines are parallel and explain why?
q
a.
b.
r
54º
m
75º
t
105º
n
28º
p
j
c.
70º
120º
d.
120º
q
g
h
p
r
M. Winking
Unit 2-1
page 23
Sec 2.2 Geometry - Constructions
1.
Name:
[COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB.
A
C
B
●
**Using a ruler measure the two lengths to make sure they have the same measure.
⃗⃗⃗⃗⃗ and congruent to the angle ∡𝑫𝑬𝑭
2. [COPY ANGLE] Construct an angle with ray 𝑯𝑰
Step I: Open the compass to
any length and create an arc on
the original angle with the
needle at the vertex
Step 2: Leave the compass set
to the exact same length and
create a new arc on the new
ray with the needle at the end
point of the ray.
Step 3: Open the compass to
the length of the intercepted
arc in the original angle and
draw a small arc to be sure the
created arcs intersect on the
ray of the angle.
Step 4: Leave the compass open to
the exact same length as it was
from step 3. Then with the needle
at the intersection of the second arc
and the copied ray, create an arc
that intersects the first arc.
3.
4.
Step 5: Finally, use a straight
edge to draw a ray from the
endpoint of the original bottom
ray through the intersection of
the two created arcs.
D●
E
H
F
●
I
●
●
**Using a protractor measure the two angles to make sure they have the same measure.
M. Winking
Unit 2-2
page 24
3. [Perpendicular Bisector] Construct a perpendicular bisector to the segment ̅̅̅̅
𝑨𝑩.
Step I: Open the compass to a length that
is certainly larger than half of the segment
length. Draw an arc with the needle at the
left endpoint of the segment.
Step 2: Leave the compass open to the
same length from step 1 and create
another arc only this time starting with
the needle at the right endpoint.
A
Step 3: Using a straight edge and a pencil,
draw a line that passes through the
intersection point of the two arcs.
B
**Using a ruler measure the two halves of the segment to make sure they have the same measure.
M. Winking
Unit 2-2
page 25
4. [Angle Bisector] Construct an angle bisector of the angle ∡𝑫𝑬𝑭
Step 3: Create an arc with the opened
length determined in step 2 towards the
interior of the angle.
Step 2: Open the compass so that the
needle is on one intersection of the arc
and the angle and the pencil is at the
other intersection of the arc and the
angle.
Step I: Open the compass to any length
and create an arc with the needle of the
compass at the vertex of the angle.
Step 5: Using a straight edge, draw ray
from the vertex of the angle through were
the two outer arcs intersect.
Step 4: Move the compass so that the
needle and pencil are on the opposite
intersection points that they were on in
step 2 and create another arc to the
exterior of the angle.
D
●
●
E
F
**Using a ruler measure the two halves of the segment to make sure they have the same measure.
M. Winking
Unit 2-2
page 26
5. [Perpendicular to a Line Through a Point] Construct a perpendicular line to ⃡⃗⃗⃗⃗
𝑨𝑩 through point C.
Step 2: Open the compass so that the
needle is on one intersection of the arc
and the pencil is on the other intersection
of the arc. Then, create a arc above and
below the line.
Step I: Start by placing the needle on the
point not on the line. Then, open your
compass to a length such that when you
draw an arc it will pass through the line
twice and draw the arc.
Step 3: Opening the compass to the same
length as the previous step and switching
the needle and pencil to the opposite
intersections of arcs that they were on in
step 2, create another arc above and
below the line.
Step 3: Using a straight edge, draw a line
passing through the intersection of the
two most recent arc intersections. The line
should incidentally also pass through the
point not on the line.
C
●
●A
M. Winking
●B
Unit 2-2
page 27
6. [Hexagon inscribed in a Circle] Construct a circle with radius ̅̅̅̅
𝑨𝑩 and an inscribed regular hexagon.
Step I: Start by placing the needle on the
point that is to be the center of the circle
and the pencil on the other endpoint of
the radius. Create the entire circle with
the compass.
Step 3: Leaving the compass still open to
the same length move the needle to the
new intersection point and mark another
intersection
point on the
circle
again.
Continue
repeating
this
process
until
you
have made
it all the
way around
the circle.
Step 2: With the compass still open to the
exact length of the radius place the needle
on the endpoint of the radius that is on
the circle and with the pencil mark an
intersection on the circle with a new arc.
Step 4: The last mark should end right
were you started with the needle. Then,
connect each consecutive arc intersection
with a segment.
A
M. Winking
Unit 2-2
page 28
B
7. [Triangle inscribed in a Circle] Construct a circle with radius ̅̅̅̅
𝑨𝑩 and an inscribed regular triangle.
Step I: Start by placing the needle on the
point that is to be the center of the circle
and the pencil on the other endpoint of
the radius. Create the entire circle with
the compass.
Step 3: Leaving the compass still open to
the same length move the needle to the
new intersection point and mark another
intersection
point on the
circle
again.
Continue
repeating
this
process
until
you
have made
it all the
way around
the circle.
Step 2: With the compass still open to the
exact length of the radius place the needle
on the endpoint of the radius that is on
the circle and with the pencil mark an
intersection on the circle with a new arc.
Step 4: The last mark should end right
were you started with the needle. Then,
connect every other consecutive arc
intersection with a segment.
A
M. Winking
Unit 2-2
page 29
B
8. [Square inscribed in a Circle] Construct a circle with radius ̅̅̅̅
𝑨𝑩 and an inscribed square.
Step I: Start by placing the needle on the
point that is to be the center of the circle
and the pencil on the other endpoint of
the radius. Create the entire circle with
the compass.
Step 3: Create a perpendicular bisector of
the newly created diameter (see previous
construction #3 if needed)
Step 2: Line your straight edge up with the
radius and extend the radius segment to
create a diameter.
Step 4: Connect the each endpoint of the
diameter with each endpoint of where the
perpendicular bisector intersects the
circle.
A
M. Winking
Unit 2-2
page 30
B
9. [Construct a Parallel Line given a point and a line] Construct a parallel line to ⃡⃗⃗⃗⃗
𝑨𝑩 through point C.
Step I: Start drawing a general line
that passes through point O and
intersects line ⃡⃗⃗⃗⃗⃗
𝑀𝑁.
Step 4: Put the compass needle on the
intersection of the transversal line and first
arc that you created as shown below and
open the pen to the intersection of the first
⃡⃗⃗⃗⃗⃗ . Create a small arc to verify
arc and line 𝑀𝑁
the intersection of the arcs mark
the correct intersection.
Step 2: Open the compass so that the needle
is on the intersection of the new line just
created and line ⃡⃗⃗⃗⃗⃗
𝑀𝑁. Then, open the
compass a little (it needs to be less than the
distance to reach point O ). Create an arc as
shown below.
Step 5: Leave the compass open to the same
length as the previous step and put the
compass needle on the intersection of the
transversal line and the second arc that you
created. Then, create an arc of the same
radius to intersect the second arc created as
shown below.
C
●
A
B
M. Winking
Unit 2-2
page 31
Step 3: Leave the compass set to the same
opening and recreate another arc of the same
radius but with the needle at point O.
Step 6: D
Sec 2.3 Geometry – Dilations
Name:
[Example Dilation]: Dilate the ABCD by a factor of 𝟐. 𝟎 from point E.
Step I: Measure the distance from the
point of dilation to a point to be dilated
(preferably using centimeters).
Step I: With the ruler in the same place as it was in
step #1, mark a point at the measured distance
determined in step #2 as the
image of the original point.
Repeat the process for all
points that are to be dilated.
Usually, denoted with
the same letter but
a single quote after
it (referred
to as a
prime).
Step 2: Multiply
the measured
distance by the
scale factor.
𝟐. 𝟓𝒄𝒎 × 𝟐. 𝟎 = 𝟓 𝒄𝒎
1.
Dilate the ∆ ABC by a factor of
𝟑
𝟐
from point D.
B
D
●
A
●
●
C
a. Measure the length of
̅̅̅̅
𝑨𝑩 in centimeters to
the nearest tenth.
AB =
●
b. Measure the length of
̅̅̅̅̅̅
𝑨′𝑩′ in centimeters to
the nearest tenth.
c. Determine the value of
A’B’ divided by AB.
𝑨’𝑩’
A’B’ =
𝑨𝑩
=
d. What might you conclude about the scale factor and the ratio of dilated segment
measure to its pre-image?
e. Measure angle ∡𝑩𝑨𝑪 and the angle ∡𝑩′𝑨′𝑪′ using a protractor.
𝒎∡𝑩𝑨𝑪 =
f.
𝒎∡𝑩′𝑨′𝑪′ =
What might you conclude about each pair of corresponding angles?
M. Winking
Unit 2-3
page 32
2.
Consider the following picture in which BCDE has been dilated from point A.
a. What is the scale factor
of the dilation based on
the sides?
b. What is the area of
BCDE?
c. What is the area of
B’C’D’E’?
d. What is the value of the area of B’C’D’E’ divided by area of BCDE?
e. What might you conclude about the ratio of two dilated shapes sides compared to the
ratio of their areas?
3.
Dilate the line ⃡𝑨𝑩 by a factor of 𝟎. 𝟓 from point C.
A
●
B
C●
How could you characterize the lines ⃡𝑨𝑩 and ⃡𝑨′𝑩′ ?
M. Winking
Unit 2-3
page 33
●
4.
Dilate the line ⃡𝑨𝑩 by a factor of 𝟐. 𝟐 from point C.
B
●
●
C
●
A
⃡
How could you characterize the lines ⃡𝑨𝑩 and 𝑨′𝑩′
?
5.
Consider the following picture in which rectangular prism A has been dilated from point G.
a. What is the scale factor
of the dilation based on
the sides?
b. What is the volume of
rectangular prism A?
c. What is the volume of
rectangular prism A’?
d. What is the value of the volume of prism A’ divided by volume of prism A?
e. What might you conclude about the ratio of two dilated solids sides compared to the ratio
of their volumes?
M. Winking
Unit 2-3
page 34
Coordinate Dilations
1. Plot the following points and connect the
consecutive points.
A(5,1)
H(6,4)
O(2,7)
B(8,2)
I(5,6)
P(3,8)
C(9,3)
J(6,5)
Q(1,7)
D(9,7)
K(5,7)
R(1,3)
E(7,8)
L(4,6)
S(2,2)
F(8,7)
M(4,4)
A(5,1)
G(7,5)
N(3,5)
2. First use the dilation rule D:(x,y) → (0.5x, 0.5y)
using the points from the previous problem, plot
the newly created points, and connect the
consecutive points.
A’(
)
H’(
)
O’(
)
B’(
)
I’(
)
P’(
)
C’(
)
J’(
)
Q’(
)
D’(
)
K’(
)
R’(
)
E’(
)
L’(
)
S’(
)
F’(
)
M’(
)
A’(
)
G’(
)
N’(
)
3. What would happen with the rule: D:(x,y) → (3x, 3y)?
4. What would happen with the rule: D:(x,y) → (0.5x, 1y)?
5. What would happen with the rule: R:(x,y) → (– 1x, 1y)?
6. What would happen with the rule: R:(x,y) → (1x, – 1y)?
M. Winking
Unit 2-3
page 35
Name:
7. Figure FCDE has been dilated to create to create F’C’D’E’.
a. What is the dilation scale
factor?
b. What is the location of the
center of dilation?
c. What is the ratio of the areas?
8. Which point would be the center of dilation?
C
A
•
B
•
•
D
•
9. The different sizes of soft drink cups at a movie theater are created by using dilations. If
the large is 8 inches tall and the medium is 6 inches tall, answer the following.
a. What is the scale factor of the dilation from a large to
a medium drink?
c. If the large holds 30 ounces, how much does the
medium hold?
M. Winking
Unit 2-3
page 36
8 in.
6 in.
b. What is the ratio of the volumes of the two drinks?
Sec 2.4 Geometry – Similar Figures
Name:
Two figures are considered to be SIMILAR if the two figures have the same shape but may differ in
size. To be similar by definition, all corresponding sides have the same ratio OR all corresponding
angles are congruent. Alternately, if one figure can be considered a transformation (rotating,
reflection, translation, or dilation) of the other then they are also similar.
Two triangles are similar if one of the following is true:
1) (AA) Two corresponding pairs of angles are congruent.
∆ABC ~ ∆FDE
Notice that in the similarity
statement above that
corresponding angles must
match up.
2) (SSS) Each pair of corresponding sides has the same ratio.
∆ABC ~ ∆DEF
𝑬𝑭
𝑩𝑪
𝑫𝑬
𝑭𝑫
𝟑
= 𝑨𝑩 = 𝑪𝑩 = 𝟐 = 𝟏. 𝟓
3) (SAS) Two pairs of corresponding sides have the same ratio and the angle
between the two corresponding pairs the angle is congruent.
𝑭𝑬
∆ABC ~ ∆FED
𝑬𝑫
= 𝑩𝑪 = 𝟐
𝑨𝑩
∡𝑨𝑩𝑪 ≅ ∡𝑭𝑬𝑫
Determine whether the following figures are similar. If so, write the similarity ratio and a
similarity statement. If not, explain why not.
1.
2.
M. Winking
3.
Unit 2-4
page 37
Assuming the following figures are similar use the properties of similar figures to find the unknown.
4.
5.
n=
x=
6.
y=
2x – 4
3
4
x+3
7.
x=
8.
t=
x
t
3
7
9
8
18
18
9. Given the similarity statement ∆ABC ~ ∆DEF and the following measures, find the
requested measures. It may help to draw a picture.
 𝒎∡𝑨𝑩𝑪 = 𝟒𝟎°
 AB = 8
 DE = 20
 𝒎∡𝑬𝑭𝑫 = 𝟑𝟎°
 AC = 10
 EF = 30
a. Find the measure of DF =
b. Find the measure of BC =
c. Find the measure of 𝒎∡𝑫𝑬𝑭 =
d. Find the measure of 𝒎∡𝑩𝑪𝑨 =
e. Find the measure of 𝒎∡𝑪𝑨𝑩 =
f.
Which angles are ACUTE?
g. Which angles are OBTUSE?
M. Winking
Unit 2-4
page 38
A)
16. Explain why the reason the triangles are similar and find the measure of the requested side.
B) Triangle mid-segment
10
What is the measure of TU?
What is the measure of AU?
17. Using (SSS, AA, SAS) which triangles can you determine must be similar? (explain why)
A)
B)
C)
E
circle one
circle one
Similar?
SSS
YES
AA
Similar?
NO
SSS
SAS
D)
YES
AA
circle one
NO
Similar?
SAS
SSS
E)
circle one
circle one
Similar?
SSS
YES
AA
Similar?
NO
SSS
SAS
M. Winking
Unit 2-4
page 39
YES
AA
NO
SAS
YES
AA
NO
SAS
18. Using some type of similar figure find the unknown lengths.
Z
O
A.
B.
N
C.
x
3 cm
P
c
Y
5 cm
M
Q
6 cm
T
X
22a.
x=
C.
E
V
22c.
y=
22b.
x=
D.
6 cm
HINT:
22c.
x=
22d.
x=
M. Winking
Unit 2-4
page 40
Prove the Pythagorean Theorem (a2 + b2 = c2) using similar right triangles:
M. Winking
Unit 2-4
page 41
10. Thales was one of the first to see the power of the property of ratios and similar figures. He
realized that he could use this property to measure heights and distances over immeasurable
surfaces. Once, he was asked by a great Egyptian Pharaoh if he knew of a way to measure the
height of the Great Pyramids. He looked at the Sun, the shadow that the
pyramid cast, and his 6 foot staff. By the drawing below can you figure
out how he found the height of the pyramid?
Height of Pyramid:
3 ft
5 ft
35 f t
11. Using similar devices he was
able to measure ships
distances off shore. This
proved to be a great advantage
in war at the time. How far from
the shore is the ship in the
diagram?
Distance from Shore:
62 in.
12. Using a mirror you can also create similar triangles (Thanks to the
properties of reflection similar triangles are created). Can you find
the height of the flag pole?
Height of the Flag Pole:
21 in.
13. Use your knowledge of special right triangles to measure
something that would otherwise be immeasurable.
M. Winking
Unit 2-4
page 42
mirror
72 in.
14. Find the unknown area based on the pictures below.
Scale
Factors
36 in2
2
?? in
5 in.
15 in.
Length
Area
Area of the small square:
15. If the small can holds 20 gallons how much will the big trashcan hold (assuming they
are similar shapes)
Scale
Factors
Similarity
Ratios
Length
Area
Volume
4 ft
2 ft
Volume of the Large Trashcan:
16. If the smaller spray bottle holds 37 fl. oz. , then how much does the larger one hold
assuming they are similar shapes?
Scale
Factors
Similarity
Ratios
Length
Area
Volume
Volume of the Large Spray Bottle:
2 in.
3 in.
1
17. The smaller of the two cars is a Matchbox car set at the usual 64 th scale (the length)
and it takes 0.003 fluid ounces to paint the car. If the smaller is a perfect scale of
the actual car and the ratios of the paint remains the same then how many gallons
of paint will be needed for the real car? (128 fl. oz = 1 gallon)
Scale
Factors
Similarity
Ratios
2 in.
Length
Area
Volume
128 in.
Amount of Paint Needed:
M. Winking
Unit 2-4
page 43
18. If the following are similar determine the length of the unknown sides.
A.
18a.
x=
B.
18 b.
V=
C.
18c.
x=
Volume = 4 cm3
Volume = 20 cm3
D.
Volume = 90 cm3
Volume = ??cm3
M. Winking
18d.
V=
Unit 2-4
page 44
Sec 2.5 Geometry – Congruence
Name:
Any two congruent figures can be mapped onto one another using a series of rigid or isometric
transformation (reflections, rotations, and translations). – See GSP Lab (Transformations) –
Each of the following pairs of figures shown below are congruent. Write a congruence statement for each
and tell whether or not a reflection would be needed to map the pre-image onto the image.
1.
2.
3.
4.
Given the following congruencies find the requested unknown angle.
6. ∆GEM ≅ ∆TOR
5. ∆ABC ≅ ∆ONM
42°
57°
(3x+20)°
50°
m∠MNO =
M. Winking
(3x)°
m∠GEM =
Unit 2-5
page 45
Given the following congruencies find the requested unknown side.
7.
⊿TRI ≅ ⊿ANG
8.
∆MTH ≅ ∆FUN
17
8
UF =
GN =
The following pairs of triangles are congruent. Provide a suggested transformation or series of transformations that can map
one triangle onto the other congruent triangle. (In each diagram ∆ABC ≅ ∆DEF )
9.
10.
Circle which transformation(s) could
be used to map ∆ABC onto ∆DEF.
Circle which transformation(s) could
be used to map ∆ABC onto ∆DEF.
Translation
Reflection
Translation
Reflection
Dilation
Rotation
Dilation
Rotation
11.
12.
Circle which transformation(s) could
be used to map ∆ABC onto ∆DEF.
Circle which transformation(s) could
be used to map ∆ABC onto ∆DEF.
Translation
Reflection
Translation
Reflection
Rotation
Dilation
Rotation
Dilation
M. Winking
Unit 2-5
page 46
Angle Puzzles. (angles are not drawn to scale)
10. Find 𝒎𝒎∡𝑮𝑮𝑮𝑮𝑮𝑮
9. Find 𝒎𝒎∡𝑩𝑩𝑩𝑩𝑩𝑩
Given:
���� bisects ∡𝐹𝐹𝐹𝐹𝐹𝐹
• 𝐺𝐺𝐺𝐺
• 𝑚𝑚∡𝐹𝐹𝐹𝐹𝐹𝐹 = 60°
• 𝑚𝑚∡𝐺𝐺𝐺𝐺𝐺𝐺 = 75°
• ∡𝐹𝐹𝐹𝐹𝐹𝐹 is a right angle
Given:
�����⃗ bisects ∡𝐷𝐷𝐷𝐷𝐷𝐷
• 𝐴𝐴𝐴𝐴
• 𝑚𝑚∡𝐷𝐷𝐷𝐷𝐷𝐷 = 50°
• ∡𝐴𝐴𝐴𝐴𝐴𝐴 is a right angle
G
●
B
●
F
●
E
●
●
C
A ●
●
H●
D
𝒎𝒎∡𝑩𝑩𝑩𝑩𝑩𝑩 =
𝒎𝒎∡𝑮𝑮𝑮𝑮𝑮𝑮 =
M. Winking
Unit 2-5
page 47
●
I
Sec 2.6 Geometry – Triangle Proofs
Name:
COMMON POTENTIAL REASONS FOR PROOFS
Definition of Congruence: Having the exact same size and shape and there by having the exact same measures.
Definition of Midpoint: The point that divides a segment into two congruent segments.
Definition of Angle Bisector: The ray that divides an angle into two congruent angles.
Definition of Perpendicular Lines: Lines that intersect to form right angles or 90°
Definition of Supplementary Angles: Any two angles that have a sum of 180°
Definition of a Straight Line: An undefined term in geometry, a line is a straight path that has no thickness and
extends forever. It also forms a straight angle which measures 180°
Reflexive Property of Equality: any measure is equal to itself (a = a)
Reflexive Property of Congruence: any figure is congruent to itself (𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐴𝐴 ≅ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐴𝐴)
Addition Property of Equality: if a = b, then a + c = b + c
Subtraction Property of Equality: if a = b, then a – c = b – c
Multiplication Property of Equality: if a = b, then ac = bc
a
Division Property of Equality: if a = b, then =
c
b
c
Transitive Property: if a = b & b =c then a = c OR if a ≅ b & b ≅ c then a ≅ c.
Segment Addition Postulate: If point B is between Point A and C then AB + BC = AC
Angle Addition Postulate: If point S is in the interior of ∠PQR, then m∠PQS + m∠SQR = m∠PQR
Side – Side – Side Postulate (SSS) : If three sides of one triangle are congruent to three sides of another triangle,
then the triangles are congruent.
Side – Angle – Side Postulate (SAS): If two sides and the included angle of one triangle are congruent to two sides
and the included angle of another triangle, then the triangles are congruent.
Angle – Side – Angle Postulate (ASA): If two angles and the included side of one triangle are congruent to two
angles and the included side of another triangle, then the triangles are congruent.
Angle – Angle – Side Postulate (AAS) : If two angles and the non-included side of one triangle are congruent to
two angles and the non-included side of another triangle, then the triangles are congruent
Hypotenuse – Leg Postulate (HL): If a hypotenuse and a leg of one right triangle are congruent to a hypotenuse
and a leg of another right triangle, then the triangles are congruent
Right Angle Theorem (R.A.T.): All right angles are congruent.
Vertical Angle Theorem (V.A.T.): Vertical angles are congruent.
Triangle Sum Theorem: The three angles of a triangle sum to 180°
Linear Pair Theorem: If two angles form a linear pair then they are adjacent and are supplementary.
Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third
pair of angles are congruent.
Alternate Interior Angle Theorem (and converse): Alternate interior angles are congruent if and only if the
transversal that passes through two lines that are parallel.
Alternate Exterior Angle Theorem (and converse): Alternate exterior angles are congruent if and only if the
transversal that passes through two lines that are parallel.
Corresponding Angle Theorem (and converse) : Corresponding angles are congruent if and only if the
transversal that passes through two lines that are parallel.
Same-Side Interior Angles Theorem (and converse) : Same Side Interior Angles are supplementary if and only if
the transversal that passes through two lines that are parallel.
Pythagorean Theorem (and converse): A triangle is right triangle if and only if the given the length of the legs a
and b and hypotenuse c have the relationship a2+b2 = c2
Isosceles Triangle Theorem (and converse): A triangle is isosceles if and only if its base angles are congruent.
Triangle Mid-segment Theorem: A mid-segment of a triangle is parallel to a side of the triangle, and its length is
half the length of that side.
CPCTC: Corresponding Parts of Congruent Triangles are Congruent by definition of congruence.
1. Tell which of the following triangle provide enough information to show that they must be congruent.
If they are congruent, state which theorem suggests they are congruent (SAS, ASA, SSS, AAS, HL)
and write a congruence statement.
Circle one of the following:
SSS
SAS
ASA AAS
HL
Not Enough
Information
SSS
SAS
ASA AAS
HL
Not Enough
Information
SSS
SAS
ASA AAS
HL
Not Enough
Information
SSS
SAS
ASA AAS
Congruence Statement
if necessary:
HL
ASA AAS
HL
Circle one of the following:
SSS
SAS
ASA AAS
HL
SAS
Not Enough
Information
ASA AAS
HL
SAS
Not Enough
Information
Not Enough
Information
ASA AAS
HL
Not Enough
Information
Congruence Statement
if necessary:
M. Winking
SAS
ASA AAS
HL
Not Enough
Information
HL
Not Enough
Information
HL
Not Enough
Information
Circle one of the following:
SSS
SAS
ASA AAS
Circle one of the following:
SSS
SAS
ASA AAS
Congruence Statement
if necessary:
Circle one of the following:
SSS
SSS
Congruence Statement
if necessary:
Circle one of the following:
SSS
Circle one of the following:
Congruence Statement
if necessary:
Congruence Statement
if necessary:
Congruence Statement
if necessary:
Circle one of the following:
SAS
Congruence Statement
if necessary:
Congruence Statement
if necessary:
Circle one of the following:
SSS
Not Enough
Information
Congruence Statement
if necessary:
Congruence Statement
if necessary:
Circle one of the following:
Circle one of the following:
Circle one of the following:
SSS
SAS
ASA AAS
Congruence Statement
if necessary:
Unit 2-6
page 49
HL
Not Enough
Information
2. Prove which of the following triangles congruent if possible by filling in the missing blanks:
���� and ⃖����⃗
a. Given ����
𝑪𝑪𝑪𝑪 ≅ 𝑨𝑨𝑨𝑨
𝑪𝑪𝑪𝑪 ∥ ⃖����⃗
𝑨𝑨𝑨𝑨
Statements
Reasons
���� ≅ 𝐴𝐴𝐴𝐴
����
1. 𝐶𝐶𝐶𝐶
⃖����⃗ ∥ 𝐴𝐴𝐴𝐴
⃖����⃗
2. 𝐶𝐶𝐶𝐶
3. ∡𝐶𝐶𝐶𝐶𝐶𝐶 ≅ ∡𝐴𝐴𝐴𝐴𝐴𝐴
4. ����
𝐵𝐵𝐵𝐵 ≅ ����
𝐵𝐵𝐵𝐵
5. ∆𝐵𝐵𝐵𝐵𝐵𝐵 ≅ ∆𝐷𝐷𝐷𝐷𝐷𝐷
���� ≅ 𝒀𝒀𝒀𝒀
���� and Point A is the
b. Given 𝑷𝑷𝑷𝑷
midpoint of ����
𝑷𝑷𝑷𝑷
Statements
���� ≅ ����
⃖����⃗ ∥ ⃖����⃗
c. Given 𝑽𝑽𝑽𝑽
𝑹𝑹𝑹𝑹 and 𝑷𝑷𝑷𝑷
𝑽𝑽𝑽𝑽
Reasons
1.
Given
2.
Given
3.
Definition of Midpoint
4.
Reflexive property of
congruence
5.
By steps 1,3,4 and SSS
Statements
1.
Given
2.
Given
3.
4.
5. ∆𝑃𝑃𝑃𝑃𝑃𝑃 ≅ ∆𝐸𝐸𝐸𝐸𝐸𝐸
M. Winking
Reasons
Unit 2-6
page 50
Prove the Isosceles Triangle Theorem and the rest of the suggested proofs.
d. Given ∆𝐖𝐖𝐖𝐖𝐖𝐖 is isosceles and point R is
the midpoint of �����
𝐖𝐖𝐖𝐖
Statements
1. ∆𝑊𝑊𝑊𝑊𝑊𝑊 is
isosceles
Reasons
2. �����
𝑊𝑊𝑊𝑊 ≅ ����
𝐾𝐾𝐾𝐾
3. R is the midpoint
�����
of 𝑊𝑊𝑊𝑊
4. �����
𝑊𝑊𝑊𝑊 ≅ ����
𝐾𝐾𝐾𝐾
5. ����
𝑂𝑂𝑂𝑂 ≅ ����
𝑂𝑂𝑂𝑂
6. ∆𝑊𝑊𝑊𝑊𝑊𝑊 ≅ ∆𝐾𝐾𝐾𝐾𝐾𝐾
7. ∡𝑂𝑂𝑂𝑂𝑂𝑂 ≅ ∡𝑂𝑂𝑂𝑂𝑂𝑂
����� and
e. Given point I is the midpoint of 𝑿𝑿𝑿𝑿
����
point I is the midpoint of 𝑨𝑨𝑨𝑨
Statements
1. I is the midpoint
�����
of 𝑋𝑋𝑋𝑋
2.
Reasons
Definition of Midpoint
3. I is the midpoint
of ����
𝐴𝐴𝐴𝐴
4.
5. ∡𝐴𝐴𝐴𝐴𝐴𝐴 ≅ ∡𝑂𝑂𝑂𝑂𝑂𝑂
6. ∆𝐴𝐴𝐴𝐴𝐴𝐴 ≅ ∆𝑂𝑂𝑂𝑂𝑂𝑂
f.
Given ∡𝐌𝐌𝐌𝐌𝐌𝐌 and ∡𝐇𝐇𝐇𝐇𝐇𝐇 are right angles
����
and �����
𝑴𝑴𝑴𝑴 ≅ 𝑻𝑻𝑻𝑻
Statements
1. ∡𝑀𝑀𝑀𝑀𝑀𝑀 & ∡𝐻𝐻𝐻𝐻𝐻𝐻
are right angles
2. �����
𝑀𝑀𝑀𝑀 ≅ ����
𝑇𝑇𝑇𝑇
3. �����
𝑀𝑀𝑀𝑀 ≅ �����
𝑀𝑀𝑀𝑀
4. ∆𝑀𝑀𝑀𝑀𝑀𝑀 ≅ ∆𝐻𝐻𝐻𝐻𝐻𝐻
M. Winking
Unit 2-6
page 51
Reasons
Prove the suggested proofs by filling in the missing blanks.
⃖����⃗ ∥ 𝑷𝑷𝑷𝑷
⃖����⃗ and 𝑮𝑮𝑮𝑮
⃖����⃗ ∥ ⃖����⃗
g. Given 𝑮𝑮𝑮𝑮
𝑪𝑪𝑪𝑪
Statements
Reasons
1. ⃖���⃗
𝐺𝐺𝐺𝐺 ∥ ⃖���⃗
𝑃𝑃𝑃𝑃
2.
3. ⃖���⃗
𝐺𝐺𝐺𝐺 ∥ ⃖���⃗
𝐶𝐶𝐶𝐶
4.
5.
6. ∆𝐺𝐺𝐺𝐺𝐺𝐺 ≅ ∆𝑃𝑃𝑃𝑃𝑃𝑃
���� ≅ 𝑯𝑯𝑯𝑯
����� , ∡𝐓𝐓𝐓𝐓𝐓𝐓 ≅ ∡𝐌𝐌𝐌𝐌𝐌𝐌 and
h. Given 𝑹𝑹𝑹𝑹
⃖����⃗
𝑴𝑴𝑴𝑴
𝑻𝑻𝑻𝑻 ∥ ⃖�����⃗
Statements
Reasons
����
���� ≅ 𝑂𝑂𝑂𝑂
1. 𝐻𝐻𝐻𝐻
2. 𝑹𝑹𝑹𝑹 = 𝑯𝑯𝑯𝑯
3.
4.
𝑹𝑹𝑹𝑹 + 𝑂𝑂𝑂𝑂 = 𝑯𝑯𝑯𝑯 + 𝐸𝐸𝐸𝐸
𝑹𝑹𝑹𝑹 + 𝑂𝑂𝑂𝑂 = 𝑅𝑅𝑅𝑅 and
𝑯𝑯𝑯𝑯 + 𝐸𝐸𝐸𝐸 = 𝐻𝐻𝐻𝐻
5. 𝑅𝑅𝑅𝑅 = 𝐻𝐻𝐻𝐻
����
���� ≅ 𝑂𝑂𝑂𝑂
6. 𝑅𝑅𝑅𝑅
⃖����⃗ ∥ ⃖�����⃗
7. 𝑇𝑇𝑇𝑇
𝑀𝑀𝑀𝑀
8. ∡𝑀𝑀𝑀𝑀𝑀𝑀 ≅ ∡𝑇𝑇𝑇𝑇𝑇𝑇
9. ∡𝑇𝑇𝑇𝑇𝑇𝑇 ≅ ∡𝑀𝑀𝑀𝑀𝑀𝑀
10. ∆𝑇𝑇𝑇𝑇𝑇𝑇 ≅ ∆𝑀𝑀𝑀𝑀𝑀𝑀
M. Winking
Unit 2-6
page 52
Substitution Property
Prove the suggested proofs by filling in the missing blanks.
i.
Given ∡𝐇𝐇𝐇𝐇𝐇𝐇 ≅ ∡𝐓𝐓𝐓𝐓𝐓𝐓 , ∡𝐓𝐓𝐓𝐓𝐓𝐓 ≅ ∡𝐇𝐇𝐇𝐇𝐇𝐇,
and Point A is the midpoint of ����
𝑬𝑬𝑬𝑬
ii.
Statements
Reasons
1. ∡𝑇𝑇𝑇𝑇𝑇𝑇 ≅ ∡𝐻𝐻𝐻𝐻𝐻𝐻
2. ∆𝐻𝐻𝐻𝐻𝐻𝐻 is an
isosceles triangle
3. ����
𝐻𝐻𝐻𝐻 ≅ ����
𝑇𝑇𝑇𝑇
4. A is the midpoint
of ����
𝐸𝐸𝐸𝐸
���� ≅ ����
5. 𝐸𝐸𝐸𝐸
𝐴𝐴𝐴𝐴
6. ∡𝑇𝑇𝑇𝑇𝑇𝑇 ≅ ∡𝐻𝐻𝐻𝐻𝐻𝐻
7. ∆𝑇𝑇𝑇𝑇𝑇𝑇 ≅ ∆𝐻𝐻𝐻𝐻𝐻𝐻
j.
�����⃗ bisects ∡𝐍𝐍𝐍𝐍𝐍𝐍.
Given that 𝑨𝑨𝑨𝑨
����
𝑨𝑨𝑨𝑨 are radii of the
Also, 𝑨𝑨𝑨𝑨 and ����
same circle with center A.
Statements
�����⃗ bisects ∡𝑁𝑁𝑁𝑁𝑁𝑁
1. 𝐴𝐴𝐴𝐴
2.
Definition of Angle Bisector
3.
Radii of the same circle are
congruent.
4.
Reflexive property of
congruence
5. ∆𝐴𝐴𝐴𝐴𝐴𝐴 ≅ ∆𝐴𝐴𝐴𝐴𝐴𝐴
M. Winking
Reasons
Unit 2-6
page 53
Prove the suggested proofs by filling in the missing blanks.
i.
Given:
• ∡𝟏𝟏 ≅ ∡𝟗𝟗
����
• ����
𝑨𝑨𝑨𝑨 ≅ 𝑮𝑮𝑮𝑮
• 𝚫𝚫𝚫𝚫𝚫𝚫𝚫𝚫 forms an isosceles triangle
with base ����
𝑪𝑪𝑪𝑪
Prove: ∆𝐴𝐴𝐴𝐴𝐴𝐴 ≅ ∆𝐺𝐺𝐺𝐺𝐺𝐺
Statements
Reasons
1. ∡1 ≅ ∡9
2. ����
𝐴𝐴𝐴𝐴 ≅ ����
𝐺𝐺𝐺𝐺
3. ∡2 ≅ ∡4
4. ∆𝐶𝐶𝐶𝐶𝐶𝐶 is an
isosceles triangle
5.
Isosceles Triangle Theorem
6. ∡6 ≅ ∡7
7. ∡2 ≅ ∡7
8.
Prove the suggested proofs by filling in the missing blanks.
k. Given:
• The circle has a center at point C
• 𝚫𝚫𝚫𝚫𝚫𝚫𝚫𝚫 forms an isosceles
triangle with base ����
𝑨𝑨𝑨𝑨
Prove: ∆𝐴𝐴𝐴𝐴𝐴𝐴 ≅ ∆𝐷𝐷𝐷𝐷𝐷𝐷
Statements
1. ∆𝐴𝐴𝐴𝐴𝐴𝐴 is an isosceles
triangle w/ base ����
𝐴𝐴𝐴𝐴
Reasons
���� ≅ ����
2. 𝐴𝐴𝐴𝐴
𝐷𝐷𝐷𝐷
3. The circle is centered at
point C
���� ≅ ����
4. 𝐴𝐴𝐴𝐴
𝐷𝐷𝐷𝐷
Reflexive Property of
Congruence
5.
6. ∆𝐴𝐴𝐴𝐴𝐴𝐴 ≅ ∆𝐷𝐷𝐷𝐷𝐷𝐷
M. Winking
Unit 2-6
page 54
Sec 2.7 Geometry – Locus of Points & Triangle Centers
Name:
1. Anna is stuck out on a swimming dock in the lake. If the picture below is a scale drawing, how far
must she swim to get to shore (convert using the scale provided)?
SCALE
10 ft
2. Measure the distance (cm) from the point to the line in each of the following problems
a.
b.
●
●
3. Describe how you attempted to measure the distance from a point to a line??
Locus of Points.
1. Mrs. and Mr. Scitamehtam were looking to purchase a new house. The map below shows where
both of the two work. They both wish to be completely fair to each other in selecting a new house
that is equal in distance from each of their respective places of work. Using a ruler or other
construction utility can you show all such possible locations.
Mrs. Scitamehtam’s
place of employment
Mr. Scitamehtam’s
place of employment
M. Winking
Unit 2-7
page 55
DEFINITIONS:
1. MEDIAN of a triangle: All triangle’s have 3 medians. Each median
is a segment with one endpoint on the midpoint of a side of a
triangle and the other endpoint at the opposite vertex of the triangle.
2. ALTITUDE of a triangle: All triangle’s have 3 altitudes. Each
altitude is a line that passes through a vertex of the triangle and is
perpendicular to the opposite side.
CONSTRUCTIONS:
1. Construct all 3 medians of the triangle below using either a straight edge & compass or a
MIRA
2. Construct all 3 altitudes of the triangle
below using either a compass
compass and straight edge or a mira
M. Winking
Unit 2-7
page 56
3. Construct all 3 perpendicular bisectors of each side of the triangle below using either a compass
compass and straight edge or a mira
4. Construct all 3 angle bisectors of the triangle below using either a
compass and straight edge or a mira
Each one of these constructions should create a common center of a triangle.
1. Circumcenter: Is the center of a circle that perfectly passes through each vertex.
2. Incenter: Is the center of largest possible circle that still completely fits inside the circle.
3. Centroid: Is the center that is the center of gravity of the triangle (balancing point) and there is a
constant ratio between the distance from the centroid to the midpoint and centroid to the vertex.
4. Orthocenter: Is the fourth common center but has no unique properties other than it is
on the EULER line.
CAN YOU GUESS WHICH CENTER IS WHICH?
(Hint: LOOK at the BOLD letters in each prompt.
M. Winking
Unit 2-7
page 57
Additional information about the Common Triangle Centers….
It is the CENTROID
created by the MEDIANS.
It is the
INCENTER
created by the
ANGLE
BISECTORS
If you wanted to balance a
cut out of this triangle on
one finger, you would
hold the triangle at the
centroid. It represents the
center of gravity.
It is the point that all
sides are equidistant
from and the center of a
circle that is
circumscribed by the
triangle.
It is the ORTHOCENTER
and it is created by the
It is the CIRCUMCENTER
and it is created by the
PERPENDICULAR
BISECTORS
ALTITUDES of the
triangle.
The only additional interesting
fact about the orthocenter is
that is a point on Euler’s Line
(the line also includes the
Centroid and Circumcenter.
It is the point that is equidistant
from all of the triangle’s vertices
and the center of a circle that
circumscribes the triangle.
M. Winking
Unit 2-7
page 58
p.34
There are some interesting uses of some of the common triangle centers. Identify which center
would be best for each situation.
1. A person wishes to make a triangular table with a single post holding
up the table. Which triangle center should the post connect to that
would help keep the table the most balanced and stable?
2. A person has a triangular piece of scrap ornate fabric.
The person wants to create a circular table cloth.
Which center might help her cut out the biggest
possible circle from the fabric?
3. The space shuttle is monitoring 3 GPS satellites and has positioned itself so that it is
equidistant from each of the satellites. Which triangle center might have helped the pilot
determine this particular location?
M. Winking
Unit 2-7
page 59
Sec 2.8 Geometry – Polygons & Quadrilaterals
Name:
Polygon: A closed plane figure formed by three or more segments such that each segment
intersects or connects end to end to form a closed shape.
Simple Polygon: A polygon in which sides only share each endpoint with one other side.
Regular Polygon: A polygon that is both equilateral and equiangular
Determine whether each figure below is a polygon or not a polygon.
1.
2.
3.
Circle one of the following:
Not a
It is a
Polygon
Polygon
Circle one of the following:
Not a
It is a
Polygon
Polygon
Circle one of the following:
Not a
It is a
Polygon
Polygon
Name if
Polygon:
Name if
Polygon:
Name if
Polygon:
4.
5.
6.
Circle one of the following:
Not a
It is a
Polygon
Polygon
Circle one of the following:
Not a
It is a
Polygon
Polygon
Circle one of the following:
Not a
It is a
Polygon
Polygon
Name if
Polygon:
Name if
Polygon:
Name if
Polygon:
Concave Polygon: A polygon in which a diagonal can be drawn such that part of one of the
diagonals contains point in the exterior of the polygon.
Convex Polygon: A polygon in which no diagonal contains points in the exterior of the polygon.
Determine whether each figure below is a Convex or Concave.
7.
8.
Circle one of the following:
CONVEX
CONCAVE
9.
Circle one of the following:
Circle one of the following:
CONVEX
M. Winking
CONVEX
CONCAVE
Unit 2-8
page 60
CONCAVE
Quadrilateral: A four-sided polygon.
Parallelogram: A quadrilateral with two pairs of parallel sides.
Trapezoid: A quadrilateral with exactly one pair of parallel sides.
Rectangle: A quadrilateral with four right angles.
Rhombus: A quadrilateral with four congruent sides.
Square: A quadrilateral with four congruent sides and four right angles.
Kite: A quadrilateral with exactly two pairs of congruent consecutive sides
but opposite sides are not congruent.
The Hierarchy of Quadrilaterals
Convex Quadrilateral
Trapezoids .
Concave
Quadrilateral
Parallelograms
Rhombi.
Squares .
Rectangles.
Kites .
Answer each of the following with ALWAYS, SOMETIMES, or NEVER
____________________10.
A square is a rhombus.
____________________11.
A rhombus is a square.
____________________12.
A trapezoid is a parallelogram.
____________________13.
A rectangle is a rhombus.
____________________14.
A kite is a concave quadrilateral.
____________________15.
A parallelogram is a rectangle.
____________________16.
A rhombus is a trapezoid.
____________________17.
A convex quadrilateral is a trapezoid.
____________________18.
A rectangle is a parallelogram.
M. Winking
Unit 2-8
page 61
Using your knowledge of congruent triangles and parallel triangles determine the highest level and
most appropriate definition of a quadrilateral for each ABCD quadrilateral below. Each definition in
the word wall should be used exactly once. Briefly explain why for each quadrilateral.
Square
Kite
Convex-Quadrilateral
Rhombus
Rectangle
1.
Please assume that
different congruent
marks represent
incongruent angles
3.
5.
Parallelogram
2.
4.
∡𝐴𝐴𝐴𝐴𝐴𝐴 and ∡𝐶𝐶𝐶𝐶𝐶𝐶
are complimentary
7.
Concave-Quadrilateral
Trapezoid
Both dashed figures can
be assumed to be circles.
6.
8.
M. Winking
Unit 2-8
page 62
Sec 2.9 Geometry – Polygon Angles
Name:
The Asimov Museum has contracted with a company that provides Robotic Security Squads to guard the exhibits
during the hours the museum is closed. The robots are designed to patrol the hallways around the exhibits and are
equipped with cameras and sensors that detect motion.
Each robot is assigned to patrol the area around a specific exhibit. They are designed to maintain a consistent
distance from the wall of the exhibits. Since the shape of the exhibits change over time, the museum staff must
program the robots to turn the corners of the exhibit.
Below, you will find a map of the museum's current exhibits and the path to be followed by the robot. One robot is
assigned to patrol each exhibit.
When a robot reaches a corner, it will stop, turn through a programmed angle, and then continue its patrol. Your job
is to determine the angles that R1, R2, R3, and R4 will need to turn as they patrol their area. Keep in mind the
direction in which the robot is traveling and make sure it always faces forward as it moves around the exhibits.
Robot 4
Robot 1
Robot 2
Robot 3
Try measuring the angles on the following pages using a protractor or using the Geometer’s Sketchpad animation found at
http://gwinnett.k12.ga.us/PhoenixHS/math/grade09/unit03/Unit%203%20-%20Robot%20Task.gsp
Original Task developed by Janet Davis at Georgia Department of Education 2016 ©
Modified, Updated, and Edited by M. Winking e-mail: [email protected]
p.63
1. What angles will Robot 1 need to turn? What is the total of these turns?
Exterior Angle at Vertex ‘A’:
Exterior Angle at Vertex ‘B’:
Exterior Angle at Vertex ‘C’:
Exterior Angle at Vertex ‘D’:
+
TOTAL:
2. What angles will Robot 2 need to turn? What is the total of these turns?
Exterior Angle at Vertex ‘E’:
Exterior Angle at Vertex ‘F’:
Exterior Angle at Vertex ‘G’:
Exterior Angle at Vertex ‘H’:
Exterior Angle at Vertex ‘I’:
+
Original Task developed by Janet Davis at Georgia Department of Education 2016 ©
Modified, Updated, and Edited by M. Winking e-mail: [email protected]
p.64
TOTAL:
3. What angles will Robot 3 need to turn? What is the total of these turns?
Exterior Angle at Vertex ‘J’:
Exterior Angle at Vertex ‘K’:
Exterior Angle at Vertex ‘L’:
Exterior Angle at Vertex ‘M’:
Exterior Angle at Vertex ‘N’:
Exterior Angle at Vertex ‘O’:
+
TOTAL:
4. What angles will Robot 4 need to turn? What is the total of these turns?
Exterior Angle at Vertex ‘P’:
Exterior Angle at Vertex ‘Q’:
Exterior Angle at Vertex ‘R’:
Exterior Angle at Vertex ‘S’:
Exterior Angle at Vertex ‘T’:
Exterior Angle at Vertex ‘U’:
Exterior Angle at Vertex ‘V’:
+
TOTAL:
Or Original Task developed by Janet Davis at Georgia Department of Education 2016 ©
Modified, Updated, and Edited by M. Winking e-mail: [email protected]
p.65
5. What do you notice about the sum of the angles? Do you think this will always be true? (please explain
using complete sentences)
6. The museum also requested a captain robot, CR, to patrol the entire exhibit area. What angles will
CR need to turn? What is the total of these turns?
Exterior Angle at Vertex ‘A’:
Exterior Angle at Vertex ‘B’:
Exterior Angle at Vertex ‘C’:
Exterior Angle at Vertex ‘D’:
Exterior Angle at Vertex ‘E’:
Exterior Angle at Vertex ‘F’:
Exterior Angle at Vertex ‘G’:
Exterior Angle at Vertex ‘H’:
+
TOTAL:
Original Task developed by Janet Davis at Georgia Department of Education 2016 ©
Modified, Updated, and Edited by M. Winking e-mail: [email protected]
p.66
7. What makes Captain Robot’s path different from the other robot’s paths (other than the number of
sides being different)? (please explain using complete sentences)
(RECALL FROM EARLIER IN THIS UNIT)
2
8. a. Create a random triangle on a piece of patty papers.
b.
Write a number inside each interior angle
c.
Cut out the triangle
d.
e.
3
1
2
3
1
Tear off or cut each of the angles from the triangle
Paste all 3 angles next to one another so that they
all have the same vertex and the edges are touching
but they aren’t overlapping
•
Paste or Tape your 3 vertices here:
•
Common Vertex
9. What is the measure of a straight angle or the angle that creates a line by using two opposite rays
from a common vertex?
10. Collectively does the sum of your 3 interior angles of a triangle form a straight angle? What about
others in your class?
11. Make a conjecture about the sum of the interior angles of a triangle. Do you think your conjecture
will always be true? (please explain using complete sentences)
Original Task developed by Janet Davis at Georgia Department of Education 2016 ©
Modified, Updated, and Edited by M. Winking e-mail: [email protected]
p.67
12. Determine the measure of the interior angles of the polygons of the polygons formed by Exhibits A through D.
Exhibit A
(Quadrilateral)
Interior Angle
at Vertex ‘A’:
Interior Angle
at Vertex ‘B’:
Interior Angle
at Vertex ‘C’:
Interior Angle
at Vertex ‘D’:
TOTAL:
+
Exhibit B
(Pentagon)
Interior Angle
at Vertex ‘E’:
Interior Angle
at Vertex ‘F’:
Interior Angle
at Vertex ‘G’:
Interior Angle
at Vertex ‘H’:
Interior Angle
at Vertex ‘I’:
TOTAL:
+
Exhibit C
Exhibit D
(Hexagon)
Interior Angle
at Vertex ‘J’:
Interior Angle
at Vertex ‘K’:
Interior Angle
at Vertex ‘L’:
Interior Angle
at Vertex ‘M’:
Interior Angle
at Vertex ‘N’
Interior Angle
at Vertex ‘O’:
+
TOTAL:
13. Look at the sums of the angles in each of the polygons, do you notice a pattern?
Can you briefly describe the pattern?
Original Task developed by Janet Davis at Georgia Department of Education 2016 ©
Modified, Updated, and Edited by M. Winking e-mail: [email protected]
p.68
(Heptagon)
Interior Angle
at Vertex ‘P’:
Interior Angle
at Vertex ‘Q’:
Interior Angle
at Vertex ‘R’:
Interior Angle
at Vertex ‘S’:
Interior Angle
at Vertex ‘T’:
Interior Angle
at Vertex ‘U’
Interior Angle
at Vertex ‘V’:
TOTAL:
+
14. Using your hypothesis from the previous problem can you determine the sum of the interior angles of the following:
A. Octagon (8-Sides)
B. Dodecagon (12-sides)
C. Icosagon (20 sides)
D. A polygon with n-sides
15. Using the information from above what would be the measure of a single interior angle each were regular polygons?
A. Regular Octagon
B. Regular Dodecagon
C. Regular Icosagon
D. A regular polygon with n-sides
16. The museum intends to create regular polygons for its 6th exhibition. For regular polygons the robot will make the same ‘turn’ at each
vertex. Can you determine the angle of each turn for the robot for a regular Pentagon?
a. A regular Pentagon?
b. A regular Hexagon?
c. A regular Nonagon?
17. A sixth exhibit was added to the museum. The robot patrolling this exhibit will make 15º turns at each vertex. How many sides must
the exhibit have and what is the name of the polygon?
What makes it possible for the robot to make the same turn each time (what type of polygon must the exhibit be)?
18. Find the value of x in each diagram below.
a.
b.
c.
Original Task developed by Janet Davis at Georgia Department of Education 2016 ©
Modified, Updated, and Edited by M. Winking e-mail: [email protected]
p.69
Sec 2.10 Geometry – Quadrilateral Properties
Name:
Name all of the properties of a parallelogram and its diagonals.
1. Opposite Sides are parallel
2. Opposite Sides are congruent
3. Opposite Angles are congruent
4. Consecutive Angles are supplementary
5. Diagonals bisect each other
The properties of a rectangle and its diagonals:
1. All angles are right
2. Opposite Sides are parallel
3. Opposite Sides are congruent
4. Diagonals bisect each other
5. Diagonals are congruent
The properties of a rhombus and its diagonals:
1. Opposite angles are congruent
2. Opposite Sides are parallel
3. All Sides are congruent
4. Consecutive angles are supplementary
5. Diagonals are perpendicular
6. Diagonals bisect interior angles.
The properties of a square and its diagonals:
1.
2.
3.
4.
All angles are right
Opposite Sides are parallel
All Sides are congruent
Diagonals bisect each other
M. Winking
5. Diagonals are perpendicular
6. Diagonals bisect angles
7. Diagonals are congruent
Unit 2-10
page 70
Find the value of x in each diagram below using properties of quadrilaterals.
1.
2.
(8x +5 )°
(3x + 10)°
1.
2.
x =
3.
x =
4.
x°
AC = 9x + 4
50°
BD = 6x + 16
4.
3.
x =
x =
5.
x
6.
45
2x
x
8
6.
5.
x =
x =
M. Winking
Unit 2-10
page 71
Plot points A(-3, -1), B(-1, 2), C(4, 2), and D(2, -1).
1. What specialized geometric figure is quadrilateral
ABCD? Support your answer mathematically.
2. Draw the diagonals of ABCD. Find the coordinates of
the midpoint of each diagonal. What do you notice?
3. Find the slopes of the diagonals of ABCD. What do
you notice?
4. The diagonals of ABCD create four small triangles.
Are any of these triangles congruent to any of the
others? Why or why not?
Plot points E(1, 2), F(2, 5), G(4, 3) and H(5, 6).
5. What specialized geometric figure is quadrilateral
EFHG? Support your answer mathematically using two
different methods.
6. Draw the diagonals of EFHG. Find the coordinates of
the midpoint of each diagonal. What do you notice?
7. Find the slopes of the diagonals of EFHG. What do you
notice?
8. The diagonals of EFHG create four small triangles. Are
any of these triangles congruent to any of the others?
Why or why not?
M. Winking
Unit 2-10
page 72
Plot points P(4, 1), W(-2, 3), M(2,-5), and K(-6, -4).
9. What specialized geometric figure is quadrilateral
PWKM? Support your answer mathematically.
10. Draw the diagonals of PWKM. Find the coordinates of
the midpoint of each diagonal. What do you notice?
11. Find the lengths of the diagonals of PWKM. What do
you notice?
12. Find the slopes of the diagonals of PWKM. What do
you notice?
13. The diagonals of ABCD create four small triangles.
Are any of these triangles congruent to any of the
others? Why or why not?
Plot points A(1, 0), B(-1, 2), and C(2, 5).
14. Find the coordinates of a fourth point D that would
make ABCD a rectangle. Justify that ABCD is a
rectangle.
15. Find the coordinates of a fourth point D that would
make ABCD a parallelogram that is not also a
rectangle. Justify that ABCD is a parallelogram but is
not a rectangle.
M. Winking
Unit 2-10
page 73