Sec 2.1 Geometry - Gwinnett County Public Schools...Sec 2.2 Geometry - Constructions Name: 1. [COPY...

Sec 2.1 Geometry – Parallel Lines and Angles Name: PARALLEL LINES

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 angles.

M. Winking Unit 2-1 page 19

● C

● A

● F

● D

These symbols imply the two

lines are parallel.

● E

● B

● H

● G

INTERIOR

EXTERIOR

EXTERIOR

1 2

3 4

5 6

7 8

WORD BANK:

Alternate Interior Angles

Alternate Exterior Angles Corresponding Angles

Vertical Angles

Consecutive Interior Angles ∡𝑮𝑩𝑪

Consecutive Exterior Angles

∡𝑫𝑬𝑯 ∡𝟒 Transversal

∡𝟓

∡𝑨𝑩𝑮

∡𝑯𝑬𝑭 Congruent

Supplementary

TRIANGLE’s INTERIOR ANGLE SUM

1. a. First, Create a random triangle on a piece of patty papers.

b. Using your pencil, write a number inside each interior angle a

label.

c. Next, cut out the triangle.

d. Finally, tear off or cut each of the angles from the triangle

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

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)

1

2

3

1

2

3

• Paste or Tape your 3 vertices here:

• Common Vertex


5. More formally, why do the 3 interior angles of any triangle sum to 180 ̊ ?

Consider ∆ABC. The segment 𝑨𝑩̅̅ ̅̅ 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 180 ̊.

( ) + 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 5 = N 6 = I 7 = E 8 = C

What type of Geometry is this?

1 2

3 4

5 6

7 8


16. Given lines p and q are parallel, find the value of x that makes each diagram true.

a. b.

17. Given lines p and q are parallel, find the value of x that makes each diagram true.

a. b.

18. Given lines m and n are parallel, find the value y of that makes each diagram true.

a. b.

40º

m

n

130º

yº

(6x+5)º

(2x+15)º

p

q

x = x =

40º

m

n 130º yº

y = y =

p

q (6x+5)º

(2x+45)º


x = x =

p

q

(x)º

140º

(x)º

155º p

q

19. ANGLE PUZZLE. Find 𝒎∡𝑨𝑬𝑭

𝑚∡𝐴𝐸𝐹 =

20. Converse of AIA, AEA, CIA, CEA. Which sets of lines are parallel and explain why?

a. b.

c. d.

●

●

●

●

A

B C

D

E

F

● G

Given:

𝒎∡𝑫𝑬𝑭 = 𝟖𝟓°

𝒎∡𝑨𝑩𝑮 = 𝟓𝟎° ∡𝑩𝑨𝑬 is a right angle ∡𝑪𝑮𝑬 and ∡𝑫𝑬𝑮 are supplementary

t

r 54º

28º

h

g 120º

p

q 70º

120º

r

n

m 75º

105º

p

q

j


Sec 2.2 Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB.

**Using a ruler measure the two lengths to make sure they have the same measure.

2. [COPY ANGLE] Construct an angle with ray 𝑯𝑰⃗⃗⃗⃗ ⃗ and congruent to the angle ∡𝑫𝑬𝑭

3.

4.

**Using a protractor measure the two angles to make sure they have the same measure.

A B

● C

● F E

● D

● H

● I

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.

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.


3. [Perpendicular Bisector] Construct a perpendicular bisector to the segment 𝑨𝑩̅̅ ̅̅ .

**Using a ruler measure the two halves of the segment to make sure they have the same measure.

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.

Step 3: Using a straight edge and a pencil, draw a line that passes through the intersection point of the two arcs.

A B


4. [Angle Bisector] Construct an angle bisector of the angle ∡𝑫𝑬𝑭

**Using a ruler measure the two halves of the segment to make sure they have the same measure.

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 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 3: Create an arc with the opened length determined in step 2 towards the interior of the angle.

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.

Step 5: Using a straight edge, draw ray from the vertex of the angle through were the two outer arcs intersect.

E

● D

● F


5. [Perpendicular to a Line Through a Point] Construct a perpendicular line to 𝑨𝑩⃡⃗ ⃗⃗ ⃗ through point C.

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 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 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 B ● ●


6. [Hexagon inscribed in a Circle] Construct a circle with radius 𝑨𝑩̅̅ ̅̅ 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 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 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.

A B

Step 4: The last mark should end right were you started with the needle. Then, connect each consecutive arc intersection with a segment.


7. [Triangle inscribed in a Circle] Construct a circle with radius 𝑨𝑩̅̅ ̅̅ and an inscribed regular triangle.


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 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.

A B

Step 4: The last mark should end right were you started with the needle. Then, connect every other consecutive arc intersection with a segment.


8. [Square inscribed in a Circle] Construct a circle with radius 𝑨𝑩̅̅ ̅̅ and an inscribed square.


Step 2: Line your straight edge up with the radius and extend the radius segment to create a diameter.

Step 3: Create a perpendicular bisector of the newly created diameter (see previous construction #3 if needed)

A B

Step 4: Connect the each endpoint of the diameter with each endpoint of where the perpendicular bisector intersects the circle.


9. [Construct a Parallel Line given a point and a line] Construct a parallel line to 𝑨𝑩⃡⃗ ⃗⃗ ⃗ through point C.

Step I: Start drawing a general line that passes through point O and

intersects line 𝑀𝑁⃡⃗ ⃗⃗ ⃗⃗ .

Step 2: Open the compass so that the needle is on the intersection of the new line just

created and line 𝑀𝑁⃡⃗ ⃗⃗ ⃗⃗ . 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 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 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

arc and line 𝑀𝑁⃡⃗ ⃗⃗ ⃗⃗ . Create a small arc to verify the intersection of the arcs mark the correct intersection.

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.

Step 6: D

A B

● C


Sec 2.3 Geometry – Dilations Name: [Example Dilation]: Dilate the ABCD by a factor of 𝟐.𝟎 from point E.

1. Dilate the ∆ ABC by a factor of 𝟑

𝟐 from point D.

a. Measure the length of

𝑨𝑩̅̅ ̅̅ in centimeters to

the nearest tenth.

AB =

b. Measure the length of

𝑨′𝑩′̅̅ ̅̅ ̅̅ in centimeters to

the nearest tenth.

A’B’ =

c. Determine the value of

A’B’ divided by AB.

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 = 𝒎∡𝑩′𝑨′𝑪′ =

f. What might you conclude about each pair of corresponding angles?

Step I: Measure the distance from the point of dilation to a point to be dilated (preferably using centimeters).

Step 2: Multiply the measured distance by the scale factor.

𝟐. 𝟓𝒄𝒎 × 𝟐. 𝟎 = 𝟓 𝒄𝒎

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).

● D

● A

● B

● C


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.

How could you characterize the lines 𝑨𝑩 ⃡ and 𝑨′𝑩′ ⃡ ?

● C

● A

● B


4. Dilate the line 𝑨𝑩 ⃡ by a factor of 𝟐. 𝟐 from point C.

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?

● A

● B

● C


Coordinate Dilations Name: 1. Plot the following points and connect the

consecutive points.

A(5,1)

B(8,2)

C(9,3)

D(9,7)

E(7,8)

F(8,7)

G(7,5)

H(6,4)

I(5,6)

J(6,5)

K(5,7)

L(4,6)

M(4,4)

N(3,5)

O(2,7)

P(3,8)

Q(1,7)

R(1,3)

S(2,2)

A(5,1)

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’( )

B’( )

C’( )

D’( )

E’( )

F’( )

G’( )

H’( )

I’( )

J’( )

K’( )

L’( )

M’( )

N’( )

O’( )

P’( )

Q’( )

R’( )

S’( )

A’( )

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)?


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?

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?

b. What is the ratio of the volumes of the two drinks?

c. If the large holds 30 ounces, how much does the medium hold?

• •

• • A

B

C

D

8 i

n.

6 i

n.


∆ABC ~ ∆DEF

𝑬𝑭

𝑩𝑪=

𝑫𝑬

𝑨𝑩=

𝑭𝑫

𝑪𝑩=

𝟑

𝟐= 𝟏. 𝟓

𝑭𝑬

𝑨𝑩=

𝑬𝑫

𝑩𝑪= 𝟐

∆ABC ~ ∆FED

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. 2) (SSS) Each pair of corresponding sides has the same ratio. 3) (SAS) Two pairs of corresponding sides have the same ratio and the angle

between the two corresponding pairs the angle is congruent.

Determine whether the following figures are similar. If so, write the similarity ratio and a similarity statement. If not, explain why not. 1. 2. 3.

∆ABC ~ ∆FDE

Notice that in the similarity

statement above that

corresponding angles must

match up.

∡𝑨𝑩𝑪 ≅ ∡𝑭𝑬𝑫


Assuming the following figures are similar use the properties of similar figures to find the unknown.

4. 5. 6.

7. 8.

9. Given the similarity statement ∆ABC ~ ∆DEF and the following measures, find the

requested measures. It may help to draw a picture.

AB = 8

AC = 10

DE = 20

EF = 30

𝒎∡𝑨𝑩𝑪 = 𝟒𝟎°

𝒎∡𝑬𝑭𝑫 = 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?

2x – 4

3

x + 3 4

x = y = n =

9

t

18

8

t =

3

x

7

x =

18


16. Explain why the reason the triangles are similar and find the measure of the requested side. A) B) Triangle mid-segment

17. Using (SSS, AA, SAS) which triangles can you determine must be similar? (explain why)

A) B) C)

D) E)

10

Similar? YES NO

SSS AA SAS

circle one

Similar? YES NO

SSS AA SAS

circle one

Similar? YES NO

SSS AA SAS

circle one

Similar? YES NO

SSS AA SAS

circle one

Similar? YES NO

SSS AA SAS

circle one

What is the measure of TU?

E


What is the measure of AU?

c

5 cm 6 cm

x 3 cm

6 cm

18. Using some type of similar figure find the unknown lengths.

A. B. C.

C. D.

22a.

x = 22b.

x =

22c.

y =

22c.

x = 22d.

x =

HINT:

E

M

N

O

P

Q

X

Y

Z

T

V


Prove the Pythagorean Theorem (a2 + b2 = c2) using similar right triangles:


72 in. 21 in.

62 i

n.

mirror

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?

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?

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?

13. Use your knowledge of special right triangles to measure

something that would otherwise be immeasurable.

35

f t

5 ft

3 ft

Height of Pyramid:

Distance from Shore:

Height of the Flag Pole:


14. Find the unknown area based on the pictures below.

15. If the small can holds 20 gallons how much will the big trashcan hold (assuming they

are similar shapes)

16. If the smaller spray bottle holds 37 fl. oz. , then how much does the larger one hold

assuming they are similar shapes?

17. The smaller of the two cars is a Matchbox car set at the usual 64

1th 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)

2 in. 3 in.

Scale Similarity Length Area Volume

Factors Ratios


Factors Ratios


Factors Ratios

36 in2

?? in2

5 in. 15 in.

Scale Length Area

Factors

2 ft

4 ft

Area of the small square:

Volume of the Large Trashcan:

Volume of the Large Spray Bottle:

2 in. 128 in.

Amount of Paint Needed:


18. If the following are similar determine the length of the unknown sides.

A.

B.

C.

D.

18a.

x =

Volume = 20 cm3

Volume = 4 cm3

Volume = 90 cm3 Volume = ??cm

3

18 b.

V =

18d.

V =

18c.

x =


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. 5. ONMABC ∆≅∆ 6. TORGEM ∆≅∆

42° 57°

=∠MNOm

(3x+20)°

50° (3x)°

=∠GEMmM. Winking Unit 2-5 page 45

Given the following congruencies find the requested unknown side.

7. ⊿TRI ≅ ⊿ANG 8. FUNMTH ∆≅∆

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. 11. 12.

8

17

=GN =UF

Circle which transformation(s) could be used to map ∆ABC onto ∆DEF.

Translation Reflection Rotation Dilation








Angle Puzzles. (angles are not drawn to scale)

9. Find 𝒎𝒎∡𝑩𝑩𝑩𝑩𝑩𝑩 10. Find 𝒎𝒎∡𝑮𝑮𝑮𝑮𝑮𝑮

Given: • 𝐴𝐴𝐴𝐴��⃗ bisects ∡𝐷𝐷𝐴𝐴𝐷𝐷 • 𝑚𝑚∡𝐷𝐷𝐴𝐴𝐷𝐷 = 50° • ∡𝐴𝐴𝐷𝐷𝐴𝐴 is a right angle

Given: • 𝐺𝐺𝐺𝐺�� bisects ∡𝐹𝐹𝐺𝐺𝐹𝐹 • 𝑚𝑚∡𝐹𝐹𝐺𝐺𝐹𝐹 = 60° • 𝑚𝑚∡𝐺𝐺𝐹𝐹𝐺𝐺 = 75° • ∡𝐹𝐹𝐺𝐺𝐹𝐹 is a right angle

I

F

H

G

●

●

●

● ●

●

●

●

E

A

B

●

C

D


𝒎𝒎∡𝑩𝑩𝑩𝑩𝑩𝑩 = 𝒎𝒎∡𝑮𝑮𝑮𝑮𝑮𝑮 =

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 Division Property of Equality: if a = b, then a

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.

SSS SAS ASA AAS HL Not Enough Information

Circle one of the following:

Congruence Statement if necessary:



































2. Prove which of the following triangles congruent if possible by filling in the missing blanks:

a. Given 𝑪𝑪𝑪𝑪�� ≅ 𝑨𝑨𝑨𝑨�� and 𝑪𝑪𝑪𝑪�⃖��⃗ ∥ 𝑨𝑨𝑨𝑨�⃖��⃗

b. Given 𝑷𝑷𝑷𝑷�� ≅ 𝒀𝒀𝑷𝑷�� and Point A is the

midpoint of 𝑷𝑷𝒀𝒀��

c. Given 𝑽𝑽𝑽𝑽�� ≅ 𝑹𝑹𝑽𝑽�� and 𝑷𝑷𝑹𝑹�⃖��⃗ ∥ 𝑽𝑽𝑽𝑽�⃖��⃗

Statements Reasons

1. 𝐶𝐶𝐶𝐶�� ≅ 𝐴𝐴𝐴𝐴��

2. 𝐶𝐶𝐶𝐶�⃖��⃗ ∥ 𝐴𝐴𝐴𝐴�⃖��⃗

3. ∡𝐶𝐶𝐶𝐶𝐴𝐴 ≅ ∡𝐴𝐴𝐴𝐴𝐶𝐶

4. 𝐶𝐶𝐴𝐴�� ≅ 𝐶𝐶𝐴𝐴��

5. ∆𝐶𝐶𝐶𝐶𝐴𝐴 ≅ ∆𝐴𝐴𝐴𝐴𝐶𝐶

Statements Reasons

1. Given

2. Given

3. Definition of Midpoint

4. Reflexive property of congruence

5. By steps 1,3,4 and SSS

Statements Reasons

1. Given

2. Given

3.

4.

5. ∆𝑃𝑃𝑃𝑃𝑃𝑃 ≅ ∆𝐸𝐸𝐸𝐸𝑃𝑃


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��

e. Given point I is the midpoint of 𝑿𝑿𝑿𝑿�� and

point I is the midpoint of 𝑨𝑨𝑽𝑽��

f. Given ∡𝐌𝐌𝐌𝐌𝐌𝐌 and ∡𝐌𝐌𝐇𝐇𝐌𝐌 are right angles

and 𝑿𝑿𝑨𝑨�� ≅ 𝑻𝑻𝑻𝑻��

Statements Reasons 1. ∆𝑊𝑊𝑃𝑃𝑊𝑊 is

isosceles

2. 𝑊𝑊𝑃𝑃�� ≅ 𝑊𝑊𝑃𝑃��

3. R is the midpoint of 𝑊𝑊𝑊𝑊��

4. 𝑊𝑊𝑃𝑃�� ≅ 𝑊𝑊𝑃𝑃��

5. 𝑃𝑃𝑃𝑃�� ≅ 𝑃𝑃𝑃𝑃��

6. ∆𝑊𝑊𝑃𝑃𝑃𝑃 ≅ ∆𝑊𝑊𝑃𝑃𝑃𝑃

7. ∡𝑃𝑃𝑊𝑊𝑃𝑃 ≅ ∡𝑃𝑃𝑊𝑊𝑃𝑃

Statements Reasons 1. I is the midpoint

of 𝑋𝑋𝑋𝑋��

2. Definition of Midpoint

3. I is the midpoint of 𝐴𝐴𝑃𝑃��

4.

5. ∡𝐴𝐴𝐴𝐴𝑋𝑋 ≅ ∡𝑃𝑃𝐴𝐴𝑋𝑋

6. ∆𝐴𝐴𝑋𝑋𝐴𝐴 ≅ ∆𝑃𝑃𝑋𝑋𝐴𝐴

Statements Reasons 1. ∡𝑋𝑋𝐴𝐴𝑀𝑀 & ∡𝑀𝑀𝐻𝐻𝑋𝑋

are right angles

2. 𝑋𝑋𝐴𝐴�� ≅ 𝐻𝐻𝑀𝑀��

3. 𝑋𝑋𝑀𝑀�� ≅ 𝑋𝑋𝑀𝑀��

4. ∆𝑋𝑋𝐴𝐴𝑀𝑀 ≅ ∆𝑀𝑀𝐻𝐻𝑋𝑋


Prove the suggested proofs by filling in the missing blanks.

g. Given 𝑮𝑮𝑪𝑪�⃖��⃗ ∥ 𝑷𝑷𝑷𝑷�⃖��⃗ and 𝑮𝑮𝑷𝑷�⃖��⃗ ∥ 𝑪𝑪𝑷𝑷�⃖��⃗

h. Given 𝑹𝑹𝑽𝑽�� ≅ 𝑻𝑻𝑽𝑽�� , ∡𝐇𝐇𝐌𝐌𝐖𝐖 ≅ ∡𝐌𝐌𝐌𝐌𝐌𝐌 and

𝑻𝑻𝑽𝑽�⃖��⃗ ∥ 𝑿𝑿𝑹𝑹�⃖��⃗

Statements Reasons

1. 𝐺𝐺𝐶𝐶�⃖��⃗ ∥ 𝑃𝑃𝑃𝑃�⃖��⃗

2.

3. 𝐺𝐺𝑃𝑃�⃖��⃗ ∥ 𝐶𝐶𝑃𝑃�⃖��⃗

4.

5.

6. ∆𝐺𝐺𝐶𝐶𝑃𝑃 ≅ ∆𝑃𝑃𝑃𝑃𝐶𝐶

Statements Reasons

1. 𝑀𝑀𝐸𝐸�� ≅ 𝑃𝑃𝑃𝑃��

2. 𝑹𝑹𝑽𝑽 = 𝑻𝑻𝑽𝑽

3. 𝑹𝑹𝑽𝑽 + 𝑃𝑃𝐸𝐸 = 𝑻𝑻𝑽𝑽 + 𝐸𝐸𝑃𝑃

4. 𝑹𝑹𝑽𝑽 + 𝑃𝑃𝐸𝐸 = 𝑃𝑃𝐸𝐸 and 𝑻𝑻𝑽𝑽 + 𝐸𝐸𝑃𝑃 = 𝑀𝑀𝑃𝑃

5. 𝑃𝑃𝐸𝐸 = 𝑀𝑀𝑃𝑃 Substitution Property

6. 𝑃𝑃𝐸𝐸�� ≅ 𝑃𝑃𝑃𝑃��

7. 𝐻𝐻𝑃𝑃�⃖��⃗ ∥ 𝑋𝑋𝑃𝑃�⃖��⃗

8. ∡𝑋𝑋𝑃𝑃𝐸𝐸 ≅ ∡𝐻𝐻𝑃𝑃𝑀𝑀

9. ∡𝐻𝐻𝑀𝑀𝑃𝑃 ≅ ∡𝑋𝑋𝐸𝐸𝑃𝑃

10. ∆𝐻𝐻𝑀𝑀𝑃𝑃 ≅ ∆𝑋𝑋𝐸𝐸𝑃𝑃


Prove the suggested proofs by filling in the missing blanks. i. Given ∡𝐌𝐌𝐇𝐇𝐌𝐌 ≅ ∡𝐇𝐇𝐌𝐌𝐌𝐌 , ∡𝐇𝐇𝐌𝐌𝐌𝐌 ≅ ∡𝐌𝐌𝐌𝐌𝐌𝐌,

and Point A is the midpoint of 𝑽𝑽𝑹𝑹�� ii.

j. Given that 𝑨𝑨𝑮𝑮��⃗ bisects ∡𝐍𝐍𝐌𝐌𝐌𝐌. Also, 𝑨𝑨𝑨𝑨�� and 𝑨𝑨𝑽𝑽�� are radii of the same circle with center A.

Statements Reasons

1. ∡𝐻𝐻𝑀𝑀𝐴𝐴 ≅ ∡𝑀𝑀𝐻𝐻𝐴𝐴

2. ∆𝑀𝑀𝐴𝐴𝐻𝐻 is an isosceles triangle

3. 𝑀𝑀𝐴𝐴�� ≅ 𝐻𝐻𝐴𝐴��

4. A is the midpoint of 𝐸𝐸𝑃𝑃��

5. 𝐸𝐸𝐴𝐴�� ≅ 𝐴𝐴𝑃𝑃��

6. ∡𝐻𝐻𝐴𝐴𝑃𝑃 ≅ ∡𝑀𝑀𝐴𝐴𝐸𝐸

7. ∆𝐻𝐻𝐴𝐴𝑃𝑃 ≅ ∆𝑀𝑀𝐴𝐴𝐸𝐸

Statements Reasons

1. 𝐴𝐴𝐺𝐺��⃗ bisects ∡𝑁𝑁𝐴𝐴𝐸𝐸

2. Definition of Angle Bisector

3. Radii of the same circle are congruent.

4. Reflexive property of congruence

5. ∆𝐴𝐴𝑁𝑁𝐺𝐺 ≅ ∆𝐴𝐴𝐸𝐸𝐺𝐺


Prove the suggested proofs by filling in the missing blanks. i. Given:

• ∡𝟏𝟏 ≅ ∡𝟗𝟗 • 𝑨𝑨𝑪𝑪�� ≅ 𝑮𝑮𝑮𝑮�� • 𝚫𝚫𝚫𝚫𝚫𝚫𝐌𝐌 forms an isosceles triangle

with base 𝑪𝑪𝑽𝑽�� Prove: ∆𝐴𝐴𝐶𝐶𝐶𝐶 ≅ ∆𝐺𝐺𝐹𝐹𝐸𝐸

Prove the suggested proofs by filling in the missing blanks. k. Given:

• The circle has a center at point C • 𝚫𝚫𝐌𝐌𝚫𝚫𝐌𝐌 forms an isosceles

triangle with base 𝑨𝑨𝑨𝑨��

Prove: ∆𝐴𝐴𝐶𝐶𝐶𝐶 ≅ ∆𝐴𝐴𝐶𝐶𝐶𝐶

Statements Reasons

1. ∡1 ≅ ∡9

2. 𝐴𝐴𝐶𝐶�� ≅ 𝐺𝐺𝐹𝐹��

3. ∡2 ≅ ∡4

4. ∆𝐶𝐶𝐴𝐴𝐸𝐸 is an isosceles triangle

5. Isosceles Triangle Theorem

6. ∡6 ≅ ∡7

7. ∡2 ≅ ∡7

8.

Statements Reasons 1. ∆𝐴𝐴𝐶𝐶𝐴𝐴 is an isosceles

triangle w/ base 𝐴𝐴𝐴𝐴��

2. 𝐴𝐴𝐸𝐸�� ≅ 𝐴𝐴𝐶𝐶��

3. The circle is centered at point C

4. 𝐴𝐴𝐶𝐶�� ≅ 𝐴𝐴𝐶𝐶��

5. Reflexive Property of Congruence

6. ∆𝐴𝐴𝐶𝐶𝐶𝐶 ≅ ∆𝐴𝐴𝐶𝐶𝐶𝐶


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)?

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.

SCALE

10 ft

●

●

Mr. Scitamehtam’s place of employment

Mrs. Scitamehtam’s place of employment


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


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.


Additional information about the Common Triangle Centers….

It is the INCENTER created by the ANGLE BISECTORS

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 CENTROID

created by the MEDIANS.

It is the ORTHOCENTER

and it is created by the

ALTITUDES of the

triangle.

It is the CIRCUMCENTER and it is created by the

PERPENDICULAR BISECTORS

It is the point that is equidistant from all of the triangle’s vertices

and the center of a circle that circumscribes the triangle.

p.34

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.

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.


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?


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.

4. 5. 6.

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. 9.

Circle one of the following: Not a

Polygon It is a

Polygon Name if Polygon:


Polygon It is a



Polygon It is a



Polygon It is a



Polygon It is a



Polygon It is a



CONCAVE CONVEX


CONCAVE CONVEX


CONCAVE CONVEX


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

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.

Convex Quadrilateral Concave Quadrilateral

Parallelograms

Rhombi. Squares . Rectangles.

Trapezoids .

Kites .


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. 1. 2.

3. 4.

5. 6.

7. 8.

Both dashed figures can be assumed to be circles.

∡𝐴𝐴𝐴𝐴𝐴𝐴 and ∡𝐶𝐶𝐴𝐴𝐴𝐴 are complimentary

Square

Parallelogram Kite

Rhombus Rectangle Trapezoid

Concave-Quadrilateral Convex-Quadrilateral

Please assume that different congruent

marks represent incongruent angles


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.

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

Robot 1

Robot 2 Robot 3

Robot 4

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

http://gwinnett.k12.ga.us/PhoenixHS/math/grade09/unit03/Unit%203%20-%20Robot%20Task.gsp

mailto:[email protected]

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:

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’:

+

TOTAL:


p.64




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:

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:


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) 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. Tear off or cut each of the angles from the triangle e. 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

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)

1

2

3

1

2

3

• Paste or Tape your 3 vertices here:

• Common Vertex


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) Exhibit B (Pentagon) Exhibit C (Hexagon) Exhibit D (Heptagon)

13. Look at the sums of the angles in each of the polygons, do you notice a pattern?

Can you briefly describe the pattern?

Interior Angle at Vertex ‘A’:

Interior Angle at Vertex ‘B’:

Interior Angle at Vertex ‘C’:

Interior Angle at Vertex ‘D’:

+

TOTAL:

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:

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:

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:


p.68


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.


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. All angles are right 2. Opposite Sides are parallel 3. All Sides are congruent 4. Diagonals bisect each other

5. Diagonals are perpendicular 6. Diagonals bisect angles 7. Diagonals are congruent


Find the value of x in each diagram below using properties of quadrilaterals.

1. 2.

3. 4.

5. 6.

(3x + 10)°

(8x +5 )°

50°

x°

AC = 9x + 4 BD = 6x + 16

1.

x =

2.

x =

3.

x =

4.

x =

5.

x =

6.

x =

8

x

x

2x

45


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).


EFHG? Support your answer mathematically using two

different methods.

6. Draw the diagonals of EFHG. Find the coordinates of


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?


Plot points P(4, 1), W(-2, 3), M(2,-5), and K(-6, -4).


PWKM? Support your answer mathematically.

10. Draw the diagonals of PWKM. Find the coordinates of


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.


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