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
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
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 q? 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.
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?
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.
Given: ABCD is a parallelogram
Prove: ∆ABD @ ∆ CDB
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?
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.
a. EXPLORE: Add the other diagonal. What is the relationship between the two diagonals?
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:
Prove: ∆ACD @ ∆BCD
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?
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?
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 ______
are ______.
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 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.
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 and What is special about and What about and
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 Given: PQRS is a rhombus
Prove: and bisect Prove:
each other
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.
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 ______
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: and
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
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.