Download Chapter 1: Shapes and Transformations

This chapter opens with a set of explorations designed to introduce you to new geometric topics that you will
explore further in Chapters 8 through 11. You will learn about the special properties of a quadrilaterals and use
a hinged mirror to learn more about regular polygons.
The second half of this chapter builds upon your work from Chapters 3 through 6. Using congruent triangles,
you will explore the relationships of the sides and diagonals of a parallelogram, kite, trapezoid, rectangle, and
rhombus. As you explore new geometric properties, you will formalize your understanding of proof. The
chapter ends with an exploration of coordinate geometry.
In this chapter, you will learn:
 the relationships of the sides, angles, and diagonals of special quadrilaterals, such as parallelograms,
rectangles, kites and rhombi (plural of rhombus)
 how to write a convincing proof in a variety of formats
 how to find the midpoint of a line segment
 how to use algebraic tools to explore quadrilaterals on coordinate axes
pg.1
7.1 – How Can I Create It?_________________________________
Using Symmetry to Study Polygons
In Chapter 1, you used a hinged mirror to study the special angles associated with regular polygons. In
particular, you investigated what happens as the angle formed by the sides of the mirror is changed. Today, you
will use a hinged mirror to determine if there is more than one way to build each regular polygon using the
principals of symmetry. And what about other types of polygons? What can a hinged mirror help you
understand about them?
7.1 – THE HINGED MIRROR TEAM CHALLENGE
Obtain a hinged mirror, a piece of unlined color paper, and a protractor from your teacher. With your team
review how to use the mirror to create regular polygons. (Remember that a regular polygon has equal sides
and angles). Once everyone remembers how the hinged mirror works, select a team member to read the task
below.
Your Task: Below are four challenges for your team. Each requires you to find a creative
way to position the mirror in relation to the colored paper. You can tackle the challenges in
any order, but you must work together as a team on each. Whenever you successfully
create a shape, do not forget to measure the angle formed by the mirror, as well as draw a
diagram on your paper of the core region in front of the mirror. If your team decides that a
shape is impossible to create with a hinged mirror, explain why.
a. Create a regular hexagon (6 sides)
b. Create an equilateral triangle at least two
different ways
c. Create a rhombus that is not a square
d. Create a circle
pg.2
7.2 – ANALYSIS
How can symmetry help you to learn more about shapes? Discuss each question below with the class.
a. One way to create a regular hexagon with a hinged mirror is with six triangles, as
shown in the diagram at right. (Note: the gray lines represent reflections of the bottom
edges of the mirrors and the edge of the paper, while the core region is shaded.)
What is special about each of the triangles in your diagram? What is the relationship
between the triangles? Support your conclusions. Would it be possible to create a
regular hexagon with 12 triangles? Explain.
b. What special type of triangle is the core region? Can all regular polygons be created with a right triangle in a
similar fashion?
c. What if it is a quadrilateral? What is the measure of the central angle?
7.3 – PRACTICE
Use what you learned today to answer the questions below.
a. Examine the regular octagon at right. What is the measure of the central angle ? Explain
how you know.
b. Quadrilateral ABCD is a right rhombus. If BD = 6 units and AC = 18 units,
then what is the perimeter of ABCD? Show all work.
pg.3
7.4 – CONCLUSIONS
What does the central angle tell you about a regular polygon? Can you break all regular polygons into
triangles? How does knowing the definition of shapes help you to create them with the mirrors?
pg.4
7.2 – What can Congruent Triangles Tell Me?____________________
Special Quadrilaterals and Proof
In earlier chapters you studied the relationships between the sides and angles of a triangle, and solved problems
involving congruent and similar triangles. Now you are going to expand your study of shapes to quadrilaterals.
What can triangles tell you about parallelograms and other special quadrilaterals?
7.5 – PARALLELOGRAMS
Carla is thinking about parallelograms (quadrilateral with both opposite sides parallel), and wondering if there
are as many special properties for parallelograms as there are for triangles. She remembers that it is possible to
create a shape that looks like a parallelogram by rotating a triangle about the midpoint of one of its sides.
a. Carefully trace the triangle at right onto tracing paper. Be sure to copy the angle
markings as well. Then rotate the triangle to make a shape that looks like a
parallelogram.
b. Is Carla's shape truly a parallelogram? Use the angles to convince your teammates that both of the opposite
sides must be parallel.
c. What else can the congruent triangles tell you about a parallelogram? Look for any relationships you can
find between the angles and sides of a parallelogram.
d. Does this work for all parallelograms? That is, does the diagonal of a parallelogram always split the shape
into two congruent triangles? Knowing only that the opposite sides of a parallelogram are parallel, create a
proof to show that the triangles are congruent. Hint: there should be 7 statements and reasons.
A
Given: ABCD is a parallelogram
Prove: ∆ABD  ∆ CDB
D
B
C
e. Now that you proved the triangles will always be congruent in a parallelogram, what can you say is
true about both of the opposite sides? Why?
pg.5
7.6 – ANOTHER WAY
Kip is confused. He put his two triangles from the last problem together as shown at
right, but he didn't get a parallelogram.
a. What shape did it make? Mark any of the equal sides in the picture. What
transformation did Kip use to form his shape?
b. What do the congruent triangles tell you about the angles of the shape?
7.7 – KITES
Kip shared his findings about his kite with his teammmate, Carla, who wants to learn more
about the diagonals of a kite. Carla quickly sketched the kite at right onto her paper with a
diagonal showing the two congruent triangles.
C
A
B
a. EXPLORE: Add the other diagonal. What is the relationship between the two diagonals?
E
b. CONJECTURE: Complete the conditional statement below.
If a quadrilateral is a kite, then its diagonals are _____________ and one ______________ the other.
c. PROVE: When she drew the second diagonal, Carla noticed that four new triangles appeared. "If any of
these triangles are congruent, then they may be able to help us prove our conjecture from part (b)," she said.
Examine triangle ∆ABC below. Are ∆ACD and ∆BCD congruent? Create a proof to justify your conclusion.
Hint: There are 4 steps.
Given: AC  BC; ACD  BCD
Prove: ∆ACD  ∆BCD
D
E
d. Since the triangles are congruent, can you justify your conjecture? Why or why not?
e. Could you have proven both conjectures if you proved ∆ACD  ∆ADE? Why or why not?
pg.6
7.8 – CONGRUENT TRIANGLES
When you have proven that two triangles are congruent, what can you say about their corresponding parts?
a. Examine the two triangles at right and the proof below. What is the given? What are you trying to prove?
1.
2.
3.
4.
5.
6.
Statements
ABCD is a kite
AD = AB
DC = BC
AC = AC
∆ABC  ∆ADC
ADC  ABC
A
Reasons
1.
2.
3.
4.
5.
6.
Given
Definition of Kite
Definition of Kite
Reflexive
SSS
D
B
C
b. Notice there is no reason given for Statement #6. Why do you know those angles will be congruent
based on this proof?
c. This reason is called "Corresponding Parts of Congruent Triangles Are Congruent." It can be
shortened to CPCTC. Or you can write an arrow diagram to show the meaning by stating:  ∆   parts.
Complete the reason for the proof above.
7.9 – CONCLUSIONS
When does a proof end with the reason of
CPCTC? Why?
7.10 – WHAT I KNOW FOR SURE
Use the boxes below to fill in what you have proven about parallelograms and kites, including their definitions.
Parallelogram:
 Both opposite _________
are __________________________.
 Both opposite _________
Kite:
 Both adjacent ________
are ________.
 Diagonals are _________.
are __________________________.
 One pair of opposite
________ are _________.
pg.7
7.3 – What Is Special about a Rhombus?____________________
Propterites of Rhombi
In the previous lesson, you learned that congruent triangles can be a useful tool to discover new information
about parallelograms and kites. But what about other quadrilaterals? Today you will use congruent triangles to
investigate and prove special properties of rhombi (the plural of rhombus). At the same time you will continue
to develop your ability to make conjectures and prove them convincingly.
7.11 – RHOMBUS VS. PARALLELOGRAM
Audrey has a favorite quadrilateral – the rhombus. Even though a rhombus is defined as having four
congruent sides, she suspects that the sides of a rhombus has other special properties as well.
a. EXPLORE: Mark the side lengths equal at right. What appears to be true about the
sides of the rhombus?
b. CONJECTURE: Complete the conditional statement below.
If a quadrilateral is a rhombus, then both of its opposite sides are _________________.
c. PROVE: Audrey knows congruent triangles can help prove other properties about quadrilaterals. She starts
by adding a diagonal PR to her diagram so that two triangles are formed. Add this diagonal to your diagram
and prove that the triangles are congruent. Hint: you will need 5 statements.
Given: PQRS is a rhombus
Prove: ∆PRS  ∆ RPQ
d. How can the triangles from part (c) help you prove your conjecture from part (b) above? Return to
your proof in part (c) and include the missing 4 statements that will prove the opposite sides are parallel.
e. The definition of a parallelogram is that both pairs of opposite sides are parallel. Based on this
definition, is a rhombus also a parallelogram? Add this remaining statement to the proof.
pg.8
7.12 – DIAGONALS OF A RHOMBUS
Now that you know the opposite sides of a rhombus are parallel, what else can you prove about a rhombus?
Consider this as you answer the questions below.
a. EXPLORE: Remember than in lesson 7.1, you explored the shapes that could be formed
with a hinged mirror. During this activity, you used symmetry to form a rhombus. Think
about what you know about the reflected triangles in the diagram. What do you think is true
about the diagonals SQ and PR ? What is special about ST and QT ? What about PT and
RT ?
b. CONJECTURE: Complete the conditional statements below.
If a quadrilateral is a rhombus, then its diagonals _______________ each other.
If a quadrilateral is a rhombus, then its diagonals are _______________ to each other.
c. PROVE: Complete the TWO proofs that proves your conjecture from part (b). As you work, think
about what triangles you need to prove are congruent to prove your conjectures. Be sure to mark each
equal statement that leads to proving triangles are congruent.
Given: PQRS is a rhombus
Prove: PR and QS bisect
each other
T
Statements
Reasons
Given: PQRS is a rhombus
Prove: AB  CD
Statements
T
Reasons
1. PQRS is a rhombus
1.
1.
1. given
2.
2. def. of rhombus
2. PS  PQ
2.
3.
3.
3. Opposite sides of a
rhombus are parallel
3. Diagonals of a rhombus
bisect each other
4. PQT = RST
4.
4. Reflexive
4.
5. vertical angles are 
5. ∆PTS  ∆PTQ
5.
5.
6. ∆PQT  ∆RST
6. PTS = PTQ
6.
6.
7. PT  RT
7. PTS + PTQ = 180
7. Def. of straight angle
7.
8. cpctc
8. PTS = 90
8. Subtraction
8.
9.
9. Def. of perpendicular
lines
9.
9. Def. of segment bisector
pg.9
7.13 – OPPOSITE ANGLES OF A PARALLELOGRAM
There are often many ways to prove a conjecture. You have rotated triangles to create parallelograms and
use congruent parts of congruent triangles to justify that opposite sides are parallel. But is there another
way?
Ansel wants to prove the conjecture, "If a quadrilateral is a parallelogram, then opposite angles are
congruent." He started by drawing parallelogram TUVW at right. Complete his flowchart. Make sure each
statement has a reason.
TUVW is a parallelogram
given
TU WV
Def. of parallelogram
Def. of parallelogram
b=c
alt. int. s
substitution
7.14 – WHAT I KNOW FOR SURE
Use the boxes below to fill in what you have proven about parallelograms and rhombi.
Parallelogram:
 Both opposite angles are __________
Rhombus:
 ________ congruent sides
 Diagonals ______________ each other
 Diagonals are ______________
pg.10
7.4 – What Else Can be Proved?__________________________
More Proof with Congruent Triangles
In the previous lessons, you used congruent triangles to learn more about parallelograms, kites, and rhombi.
You now possess the tools to do the work of a geometrician: to discover and prove new properties about the
side and angles of shapes.
As you investigate these shapes, focus on proving your ideas. Remember to ask yourself and your teammmates
questions like, "Why does that work?" and "Is it always true?" Decide whether your argument is convincing
and work with your team to provide all of the necessary justification.
7.15 – RECTANGLES
Carla decided to turn her attention to rectangles. Knowing that a rectangle is
defined as a quadrilateral with four right angles, she drew the diagram at right.
After some exploration, she conjectured that all rectangles are also
parallelograms. Help her prove that her rectangle ABCD must be a parallelogram.
That is, prove that the opposite sides must be parallel.
Given: ABCD is a rectangle
Prove: AD BC and AB DC
7.16 – DIAGONALS OF A RECTANGLE
What can congruent triangles tell us about the diagonals of a rectangle? Using the
fact that a rectangle is a parallelogram, prove that the diagonals of the rectangle are
congruent. It might be helpful to draw the triangles separately.
Given: ABCD is a rectangle
Prove: AC = BD
pg.11
7.17 – DIAGONALS OF A RHOMBUS
What can congruent triangles tell us about the diagonals of and angles of a rhombus? Prove that the
diagonals of a rhombus bisect the angles.
Given: ABCD is a rhombus
Prove: ABD = CBD
.
7.11
.
7.16
7.18 – WHAT I KNOW FOR SURE
Use the boxes below to fill in what you have proven about rhombi and rectangles.
Rectangle:
Rhombus:



pg.12
7.5 – What Else Can I Prove?_____________________________
More Properties of Quadrilaterals
Today you will work with your team to apply what you have learned to other shapes. Remember to ask
yourself and your teammates questions like, "Why does that work?" and "Is it always true?"
7.19 – TRAPEZOIDS
The definition of a trapezoid is to have one pair of parallel sides.
a. Is a trapezoid a parallelogram? Why or why not?
b. Examine the trapezoid at right. What do the angles tell you about this
quadrilateral? What lines can you prove are parallel to prove this shape is
a trapezoid? Write a proof below.
A
Given: A and D are right angles
Prove: ABCD is a trapezoid
D
B
C
c. This type of trapezoid is called a right trapezoid. Are all quadrilaterals with two right angles right
trapezoids?
W
d. In your proof for part (b) you showed that since A + D = 180,
the lines are parallel. Consider this when you look at the following
trapezoid. What angles add to 180? Why?
Z
X
Y
M
N
e. What angles in a parallelogram add to 180? List as many angles that
add to 180 in the diagram as possible.
P
O
pg.13
7.20 – ISOSCELES TRAPEZOIDS
An isosceles trapezoid is a trapezoid with a pair of congruent sides.
E
a. Examine the trapezoid at right. What sides are equal? These sides are
called legs of the trapezoid. What sides are not congruent, but parallel?
These sides are called bases.
F
H
G
b. How are isosceles trapezoids and isosceles triangles alike? How are they different?
c. What appears to be the relationship between the angles in an isosceles trapezoid? Complete the
conjecture below.
If a trapezoid is isosceles, then its ___________ angles are ______________.
d. Prove your conjecture in part (c) below. Start by drawing in the height of the trapezoid through points
E and F. Name the intersections to help in your proof.
E
F
Given: EFGH is an isosceles trapezoid
Prove: H = G
H
G
pg.14
7.21 – DIAGONALS OF AN ISOSCELES TRAPEZOIDS
Draw in the diagonals of the trapezoid at right.
E
F
a. What appears to be the relationship between the diagonals?
H
b. Complete the conjecture below.
G
If a trapezoid is isosceles, then its diagonals angles are ______________.
c. Write a proof below proving your conjecture from part (b). Use the fact that H = G, that you proved
in the previous problem to help you. It might also help to draw the triangles separately.
E
Given: EFGH is an isosceles trapezoid
Prove: EG  FH
H
F
G
7.22 – WHAT ELSE IS A PARALLELOGRAM
So far in this chapter you have shown that rectangles and rhombi are special types of parallelograms. Consider
the Venn diagram below.
a. Place each shape in the
Venn diagram to the right.
A.
B.
C.
D.
E.
Rhombus
Rectangle
Parallelogram
Square
Trapezoid
parallelograms
#1: Four
right
angles
#2: Four
equal sides
b. What does this tell you about
squares?
pg.15
pg.16
7.6 – What Can I Prove On a Grid?_________________________
Midsegments of Triangles
Today you will return to your work from chapter 3 in proving triangles are similar. You will use similar
triangles to make statements about triangles. You will then apply reasoning with your algebra skills.
7.23 – TRIANGLE MIDSEGMENT THEOREM
As Sergio was drawing shapes on his paper, he drew a line segment that connected the midpoints of two
sides of a triangle. (This is called the midsegment of a triangle.) "I wonder what we can find out about
this midsegment," he said to his team. Examine his drawing at right.
a. How are DE and AB related?
b. Complete the conjectures below.
If a segment is a midsegment, then it is __________ to the third side.
If a segment is a midsegment, then it is __________ the length of the third side.
c. Sergio wants to prove that AB = 2DE. However, he does not see any congruent triangles in the
diagram. How are the triangles related? Prove your conclusion in a flowchart.
d. What is the common ratio between side lengths in the similar triangles?
e. Now Sergio wants to prove that DE AB. Use similar triangles to find all the pairs of equal angles you
can in the diagram. Then use your relationship to prove the segments are parallel.
pg.17
7.24 – TRIANGLE MIDSEGMENT THEOREM ON A GRID
Cassie wants to confirm what Sergio proved using a coordinate grid. She started with ∆ABC, with A(0, 0);
B(2, 6); and C(7, 0).
a. Graph ∆ABC on the grid.
b. With your team, find the coordinates of P, the midpoint of AB.
Likewise, find the coordinates of Q, the midpoint of BC.
c. Verify that the length of the midsegment, PQ , is half the length
of AC.
d. Verify that PQ and AC are parallel by finding their slopes.
7.25 – MIDPOINT FORMULA
As Cassie worked on the previous problem, Esther, had difficulty
finding the midpoint of BC. The study team decided to try to find
another way to find the midpoint of a line segment.
a. As part of her team, Cassie wants you to draw AM , with A(3, 4)
and B(8, 11) on graph paper. Then extend the line segment to find a
point B so that M is the midpoint of AB. Justify your location of
point B by drawing and writing numbers on the graph.
b. Esther thinks she understands how to find the midpoint on the
graph. "I always look for the middle of the line segment. But what
if the coordinates are not easy to graph?" she asks. With your team,
find the midpoint of KL if K(2, 125) and L(98, 15). Be ready to
share your method with the class.
c. Test your team's method by verifying that the midpoint between  5,7  and  9,4  is  2,5.5 .
pg.18
7.26 – MIDPOINT FORMULA
Compare your formula you created with the
one below? How are they alike? How are
they different?
 x1  x2 y1  y2 
,


2 
 2
7.27 – MIDSEGMENT REVIEW
What are the two relationships formed when
you draw in a midsegment of a triangle?
7.28 – ALGEBRA REVIEW
What algebra skills are important to know to prove things in shapes. Review the following list below.
Match the algebra calculation with the goals.
1. Find parallel lines
A. Pythagorean thm.
2. Find perpendicular lines
B. Slopes are opposite AND reciprocal
3. Find the midpoint
C. Slopes are the same
4. Find the length of a side
D. Use the formula, or find the point in the middle
pg.19
7.7 – What Makes a Quadrilateral Special?____________________
Studying Quadrilaterals on a Coordinate Grid
In this chapter you have investigated special types of quadrilaterals, such as parallelograms, kites, and
rhombi. Each of these quadrilaterals has special properties you have proved: parallel sides, sides of equal
length, equal opposite angles, bisected diagonals, etc.
But not all quadrilaterals have a special name. How can you tell if a quadrilateral belongs to one of these
types? And if a quadrilateral doesn't have a special name, can it still have special properties?
7.29 – ALGEBRA REVIEW
Review some of the algebra tools you already have. On graph paper, draw AB given A(0, 8)and B(9, 2).
Then graph CD given C(1, 3) and D(9, 15).
a. Find the length of each segment. Are they equal?
b. Is AB CD ? Is AB  CD ? Justify your answer.
7.30 – AM I SPECIAL?
Shayla just drew quadrilateral SHAY, shown at right.
a. Shayla thinks her quadrilateral is a trapezoid. Is she correct? Use
algebra to justify your reasoning.
b. Does Shayla's quadrilateral look like it is one of the other kinds of special
quadrilaterals you have studied? If so, which one?
c. Even if Shayla's quadrilateral doesn't have a special name, it may still have some special properties. If
you haven't already done so, find all the lengths of the sides and all the slopes. If you find any special
properties, be ready to justify your claim.
pg.20
7.31 – WHAT'S MY NAME?
Randy has decided to study the triangle graphed at right.
a. Consider all the special properties this triangle can have. Without using any
tools yet, predict the best name for this triangle.
b. For your answer to part (a) to be correct, what is the minimum amount of
information that must be true about ∆RND?
c. Use your algebra tools to verify each of the properties you listed in part (b). If you need, you may
change your prediction of the shape of ∆RND.
7.32 – RHOMBUS
Tommy remembers that the diagonals of a rhombus are perpendicular to each other. Consider the points
A(1, 4); B(6, 6) ; C(4, 1) and D(-1, -1).
a. Graph the points above to form quadrilateral ABCD. Is ABCD a rhombus?
Show how you know.
b. Compare the slopes of AC and BD. What do you notice?
pg.21
7.33 – THE MUST BE/COULD BE GAME
Mr. Quincey plays a game with his class. He says, "My quadrilateral has four right angles." His students say,
"Then it must be a rectangle" and "It could be a square." For each description of a quadrilateral below, say what
special type the quadrilateral must be and/or what special type the quadrilateral could be. Look out: Some
descriptions may not have any "must be"s and some may have many "could be"s!
a. "My quadrilateral has four equal sides."
Must Be
Could Be
c. "My quadrilateral has two consecutive right
angles."
Must Be
Could Be
b. "My quadrilateral has two pairs of opposite
parallel sides"
Must Be
Could Be
d. "My quadrilateral has two pairs of equal sides."
Must Be
Could Be
7.34 – CONCLUSIONS
How does algebra help you determine
what shape is being formed? How can
slope tell you if lines are parallel or
perpendicular?
pg.22
7.7 – What Kind of Quadrilateral Is It?____________________
Quadrilaterals on a Coordinate Plane
Today you will continue to use algebra tools to investigate the properties of a quadrilateral and then use
those properties to identify the type of quadrilateral it is.
7.35 –MUST BE/COULD BE
Mr. Quincey has some new challenges for you! For each description below, decide what special type the
quadrilateral must be and/or could be. Look out: Some descriptions may not have any "must be"s and
some may have many "could be"s!
a. "My quadrilateral has three right angles."
Must Be
Could Be
c. "My quadrilateral has two consecutive equal
angles."
Must Be
b. "My quadrilateral has a pair of parallel sides"
Must Be
Could Be
d. "My quadrilateral has diagonals that are
perpendicular."
Could Be
Must Be
Could Be
7.36 –THE SHAPE FACTORY
You just got a job in the Quadrilateral Division of your uncle's Shape Factory. In the old days, customers
called up your uncle and described the quadrilaterals they wanted over the phone: "I'd like a
parallelogram with....". "But nowadays," your uncle says, "customers using computers have been
emailing orders in lots of different ways." Your uncle needs your team to help analyze the most recent
orders listed below to identify the quadrilaterals and help the shape-makers know what to produce.
Your Task: For each of the quadrilaterals on the next page,



Create a diagram of the quadrilateral on graph paper
Decide if the quadrilateral ordered has a special name. To help the shape-makers, your name must
be as specific as possible. (don't just call a shape a rectangle when it is also a square!)
Record and be ready to present a justification that the quadrilateral ordered must be the kind you
say it is. It is not enough to say that a quadrilateral looks like it is of a certain type or looks like it
has a certain property. Customers will want to be sure they get the type of quadrilateral they
ordered!
pg.23
7.37 –THE SHAPE FACTORY, CONTINUED
Here are the orders.
a. A quadrilateral formed by the intersection of these lines:
3
y   x3
2
y
3
x3
2
3
y   x9
2
y
3
x3
2
Slopes: _________
_________
_________
_________
Distance: _________
_________
_________
_________
b. A quadrilateral with vertices at these points:
A(0, 2), B(1, 0), C(7, 3), D(4, 4)
Slopes: _________
_________
_________
_________
Distance: _________
_________
_________
_________
c. A quadrilateral with vertices at these points:
A(0, -1), B(1, 4), C(4, 3), D(3, -2)
Slopes: _________
_________
_________
_________
Distance: _________
_________
_________
_________
pg.24
Chapter 7 Closure
What have I learned?
Reflection and Synthesis
7.38 – TOPICS
What have you studied in this chapter? What ideas and words were important in what you learned? Be as
detailed as possible.
7.39 – CONNECTIONS
How are the topics, ideas, and words that you learned in previous courses connected to the new ideas in this
chapter? Again, make your list as long as you can.
7.40 – VOCABULARY
The following is a list of vocabulary used in this chapter. Make sure that you are familiar with all of these
words and know what they mean.
Bisect
Conjecture
Diagonal
Midsegment
Parallelogram
Quadrilateral
Regular polygon
Theorem
Central angle
Coordinate geometry
Kite
Opposite
Perpendicular
Rectangle
Rhombus
Trapezoid
Congruent
CPCTC
Midpoint
Parallel
Proof
Reflexive property
Square
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