Name: __________________________

Geometry Period _______

Unit 7: Quadrilaterals

Part 1 of 2: Coordinate Geometry Proof and

Properties!

In this unit you must bring the following materials with you to class every day:

Calculator

Pencil

This Booklet

A device

Please note:

You may have random material checks in class

Some days you will have additional handouts to support your understanding of

the learning goals in that lesson. Keep these in a folder and bring to class every

day.

All homework for part one of this unit is in this booklet.

Answer keys will be posted as usual for each daily lesson on our website

After completion of this booklet there will be a quiz in class the next lesson!

Parallelogram Square Trapezoid Rectangle Isosceles Trapezoid

Rhombus

Sketch what it looks like here:

Sketch what it looks like here:

Sketch what it looks like here:

Sketch what it looks like here:

Sketch what it looks like here:

Sketch what it looks like here:

Fill in the blank boxes using your property sheet! Fill in wherever you see a box 1) 2 sets of

opposite sides are parallel

2) 2 sets of opposite sides are congruent

3) Diagonals bisect each other

4) One pair of opposite sides are congruent and parallel* *SOMETIMES USED IN MULTIPLE CHOICE!

1) 2 sets of opposite sides are parallel

2) 2 sets of opposite sides are congruent

3) Diagonals bisect each other

4) One pair of opposite sides are congruent and parallel

5) Diagonals are congruent

6) Adjacent sides are perpendicular

7) Adjacent sides are congruent

8) Diagonals are perpendicular

1) ONLY 1 set of opposite sides are parallel

1) 2 sets of opposite sides are parallel

2) 2 sets of opposite sides are congruent

3) Diagonals bisect each other

4) One pair of opposite sides are congruent and parallel

5) Diagonals are congruent

6) Adjacent sides are perpendicular

1) ONLY 1 set of opposite sides are parallel

2) ONLY 1 set of opposite sides are congruent (legs)

3) Diagonals are congruent

1) 2 sets of opposite sides are parallel

2) 2 sets of opposite sides are congruent

3) Diagonals bisect each other

4) One pair of opposite sides are congruent and parallel

5) Adjacent sides are congruent

6) Diagonals are perpendicular

* Be resourceful! If you are having a hard time sketching, do a search online to see what it should look like, or use your

packet over break!

Know your properties!

Finish PRACTICE FOR HW! CHECK THE KEY!

Part 1: Identify Properties

*Note: Starting tomorrow in class you will be taking property quizzes. You will be provided a specific quadrilateral and

will be quizzed on how well you know its properties.

1) What quadrilateral has only one set of parallel sides? __________________________ 2) Which quadrilateral(s) has all 4 sides congruent? __________________________

3) What quadrilateral(s) has diagonals which are congruent and bisect each other? ____________________________

4) What must a quadrilateral be if it has one set of sides both parallel and congruent? _______________

5) What quadrilateral has diagonals that are congruent, but have different midpoints? _______________ 6) Frank says all rhombuses are parallelograms. Is this true? _______________

7) Fiorella says all trapezoids are isosceles trapezoids. Do you agree?

8) Manuel says that a rectangle is a square, always. Is he correct? 9) What quadrilateral has opposite sides with equal slopes and adjacent sides with opposite reciprocal slopes? (Hint: Think about what the relationship is with these slope relationships)

QUICK! Go check your answers with the key before you continue on!

7-1 Practice

Part 2: Applying the Properties

Answer ALL parts of each question to complete.

1. In the accompanying diagram of parallelogram ABCD, diagonals and intersect at E, BE = 3x,

and ED = x+10.

a) What is the value of x?

b) State the property of a parallelogram that you used to answer this question. (use your property sheet)

2) As shown in the diagram of rectangle ABCD below, diagonals and intersect at E.

and

a) Solve for the length of

b) State the property/properties of a rectangle that you used to answer this question. (use your property sheet)

3) In the diagram below of rhombus ABCD, .

a) What do you know about the relationship between BC and CD? Why?

b) Mark the relationship you stated in a) into the diagram.

c) What type of triangle is DBC (classify by sides)? __________________________

d) What is ?

4) In the diagram below of isosceles trapezoid DEFG, , , , , and .

(Hint: There may be too much information)!

a) Find the value of x.

b) State the property/properties of an isosceles trapezoid that you used to answer this question. (use your

property sheet)

5) In the diagram below, and are bases of trapezoid ABCD.

a) If and , what is ?

b) State the property of a trapezoid that you used to answer this question. (use your property sheet)

6) a) The perimeter of a square is 24. In simplest radical form, find the length of the diagonal of the square.

b) State the property/properties of a square that you used to answer this question. (use your property sheet)

Lesson 7-2: Proving Parallelograms and Rectangles

Today's Goal: Can we complete a thorough coordinate geometry proof by verifying properties specific to the quadrilateral we are proving?

Let’s Prepare! What is an observation?

What is an inference?

What are the tools we can use to find them?

What does it mean to prove?

Proving Parallelograms

Properties of a Parallelogram: Shortcuts to proving quadrilaterals are parallelograms:

1) 2 sets of opposite sides are parallel

2) 2 sets of opposite sides are congruent

3) Diagonals bisect each other.

4) 1 set of opposite sides are

Congruent AND parallel

7-2 Notes

Proving Rectangles

Properties of a Rectangle: Shortcuts to proving quadrilaterals are rectangles:

1) All of the properties of a parallelogram. 2) Adjacent sides perpendicular. 3) Diagonals congruent.

What do quadrilateral proofs look like? Use the exemplar proofs to answer the follow-up questions with your groups. Be prepared to share out!

a) What proof got the higher grade? Which one got the lower grade?

b) What was the writer trying to prove? How was that represented in their work?

c) How did they use their work to defend their proof?

d) What was the structure of the proof? Can we add anything to the proof to make it even stronger?

Exemplars 1) Prove: A(-2,2), B(1,4),C(2,8) and D(-1,6) is a parallelogram. 2) Prove: A(2,3), B(11,6) C(10,9), D(1,6) is a rectangle.

Try the following two proofs in your groups. First focus on what property you want to use to prove, then show work to support your proof. Don’t forget your conclusions!

1) Quadrilateral ABCD has vertices A(-3, 6), B(6, 0), C(9, -9), and D(0, -3). Prove ABCD is a parallelogram.

2) A(1,1), B(5,1), C(5,3), D(1,3) Prove that ABCD is a rectangle.

7-2 Practice

Practice - Properties Questions

1) Decide whether the following statements are sometimes, always, or never true. Explain.

a) A square is a rhombus b) A rhombus is a parallelogram.

c) A parallelogram is a rectangle. d) A trapezoid is a parallelogram.

2) Which of the following is NOT a property of a rectangle?

a) Opposite sides parallel

b) Diagonals are perpendicular bisectors

c) Diagonals are congruent

d) Adjacent sides are perpendicular

3) Which of the following is NOT a property of a rhombus?

a) Opposite sides parallel

b) Diagonals are bisectors

c) Diagonals are perpendicular

d) Adjacent sides are perpendicular

4) Which choice best describes the properties of a trapezoid?

a) set of parallel sides

b) one set of parallel sides and one set of non-parallel sides.

c) one set of parallel sides and one set of congruent sides.

d) two sets of parallel sides.

5) In parallelogram JKLM, JK = 2b + 3, JM = 3a, ML = 45, KL = 21. Solve for a and b, and find the length of each side.

6) Carter and Melissa both wrote explanations describing ways to show that a quadrilateral is a parallelogram. Find

whose explanation is correct and explain your reasoning.

Carter: A quadrilateral is a parallelogram if 1 pair of opposite sides is congruent, and 1 pair of opposite sides is parallel.

Melissa: A quadrilateral is a parallelogram if one pair of opposite sides is congruent and parallel.

7-2 Homework

1. Do you know the properties for all quadrilaterals?! If not, review your study guide!!

2. Find x and y so that the quadrilateral is a parallelogram.

What property did you use?

3. Triangle ABC has vertices A(-2,-1), B(-1,1), C(3,-1).

a) Prove that this is a right triangle. – Hint: How would you show that it is a right triangle? Then show that it doesn’t work!

b) State the coordinates of point D such that quadrilateral ABCD is

a parallelogram.

c) Prove ABCD is a parallelogram.

5y

2y+ 12

2x-5 3x-18

4. A rectangular playground is surrounded by an 80 foot long fence. One side of the playground is 10 feet longer than

the other. Find s, the shorter side of the fence.

5. The diagonals of a rhombus are 12 centimeters and 16 centimeters long. Find the perimeter of the rhombus.

6. Given: Quadrilateral ABCD has vertices , , ,

and .

a) Prove: Quadrilateral ABCD is a parallelogram.

b) Prove: ABCD is NOT a rectangle – Hint: Prove as if it IS a rectangle, and then show that it doesn’t work!

Hint: What do we know about the

diagonals in a rhombus? They

_____and ________

How will that help us find the side

lengths of the rhombus?

Proving Squares and Rhombuses

Learning Goal: Can we complete a thorough coordinate geometry proof by verifying properties specific to the

quadrilateral we are proving?

Proving Rhombuses

Properties of a Rhombus: Shortcuts to proving quadrilaterals are rhombuses:

1) All the properties of a parallelogram.

2) Adjacent sides are congruent.

3) Diagonals are perpendicular.

Proving Squares

Properties of a Square: Shortcuts to proving quadrilaterals are squares:

1) All of the properties of a parallelogram.

2) The additional properties of a rectangle.

3) The additional properties of a rhombus.

7-3 Notes

Try the following two proofs in your groups. First focus on what property you want to use to prove, then show work to support your proof. Don’t forget your conclusions!

2) The vertices of quadrilateral GRID are G(4, 1), R(7, -3), I(11, 0), and D(8, 4). Using coordinate geometry, prove that

quadrilateral GRID is a square.

Make a plan! 3) Quadrilateral ABCD has coordinates A (-1,-3), B(4,2), C(3,9), and D(-2,4). Prove ABCD is a rhombus. Make a plan!

Method:

-

-

-

Method:

-

-

4) The coordinates of quadrilateral PRAT are R a b( , ),3 A a b( , ), 3 4 and T a b( , ). 6 2 Prove that RA is

parallel to PT.

5) Matthew is surveying a location for a new city park shaped like a parallelogram. When drawn on a coordinate axes, he knows that three of the vertices of parallelogram PARK are P(-2, 3), A(4, 6) and R(3, 2). Find the coordinates of point K and sketch parallelogram PARK on the set of axes given. {Justify mathematically that the figure you have drawn is a parallelogram.}

7-3 Homework

Do you know the properties for all quadrilaterals?! If not, review your study guide!!

1. Quadrilateral ABCD has vertices A(-2, 4), B(2, 8), C(6, 4) and D(2, 0). Prove that quadrilateral ABCD is a square.

2. The vertices of quadrilateral ABCD has vertices A(3, 0), B(4, 3), C(7, 3) and D(6, 0).

a) Prove that ABCD is a parallelogram.

b) Prove that ABCD is NOT a rhombus.

3. ( From a past regents Exam)

The vertices of quadrilateral MATH are M(r, s), A(0, 0), T(t, 0) and H(t + r, s). Using coordinate geometry, prove

that quadrilateral MATH is a parallelogram. (Hint: Use formulas as you normally would, sketch MATH so you

know how the sides/diagonals are labeled)

Trapezoid Proofs

Learning Goal: What is your method to prove a quadrilateral is a trapezoid? What is your method to prove a quadrilateral is an isosceles trapezoid?

PARTNER BRAINSTORM! SHARE OUT!!!

1) What do you think will be our method to prove a quadrilateral is a trapezoid?

2) What tool would you use and what would you have to show?

Proving Trapezoids

Property of a Trapezoid Method for proving quadrilaterals are trapezoids:

1 pair of opposite sides are parallel,

1 pair of opposite sides are not parallel

Proving Isosceles Trapezoids

Properties of Isosceles Trapezoids Shortcuts to be proving quadrilaterals are isosceles trapezoids:

1) All the properties of a trapezoid 2) Legs are congruent 3) Diagonals are congruent

7-4 Notes

Let’s Look at Bella’s Proof…

Given: A (1,1), B (2,5), C (5,7), D (7,5)

Prove: ABCD is a trapezoid

What do you notice about the following?

-Accuracy of Bella’s Calculations

-Organization of Bella’s Work

-Conclusion of Bella’s Proof

What feedback would you give Bella to help her improve her proof?

Practice!

1) Given:

Prove: ABCD is a trapezoid. [The use of the accompanying grid is

optional.]

2) The coordinates of quadrilateral JKLM are J (1,-2), K (13,4), L (6,8),

and M (-2,4). Prove that quadrilateral JKLM is a trapezoid but not an

isosceles trapezoid.

Check the box that describes you best with coordinate geometry proofs

This is hard I have to come to extra help!!!

I can do, it’s not too bad. Getting used to it

I don’t like to prove it, prove it!

I LOVE to prove it, prove it!

3) The coordinates of quadrilateral PRAT are P a b( , ), R a b( , ),3 A a b( , ), 3 4 and T a b( , ). 6 2 Prove that

RA is parallel to PT.

4) ABCD is an isosceles trapezoid, where ̅̅ ̅̅ is parallel to ̅̅ ̅̅ ̅ and ̅̅ ̅̅ ̅̅ ̅̅ . If AD = 2y – 5, and BC = y + 3, find AD.

5) Matthew is surveying a location for a new city park shaped like a parallelogram. When drawn on a coordinate axes, he knows that three of the vertices of parallelogram PARK are P(-2, 3), A(4, 6) and R(3, 2). Find the coordinates of point K and sketch parallelogram PARK on the set of axes given. {Justify mathematically that the figure you have drawn is a parallelogram.}

7-4 Homework

1. Given: ABC with vertices A(-6,-2), B(2,8) and C(6,-2)

a) ABhas midpoint D, BC has midpoint E, and AC has midpoint F. Plot and label on the grid.

b) Prove: ADEF is a parallelogram

ADEF is not a rhombus

2. In the diagram below, LATE is an isosceles trapezoid with , , , and . Altitudes and

are drawn.

What is the length of ?

3. ABCD is an isosceles trapezoid, where ̅̅ ̅̅ is parallel to ̅̅ ̅̅ ̅ and ̅̅ ̅̅ ̅̅ ̅̅ .

The perimeter of ABCD is 55 centimeters. If AD = DC = BC, and AB = 2AD, find the measure of each side of the trapezoid.

4. The coordinates of the vertices if ABCD are A(0,0), B(4, -1), C(5, 2), D(2, 6). a) Prove that ABCD is a trapezoid. b) Is ABCD an isosceles trapezoid? How do you know?

Today’s Goal: How can I discover the angle properties in quadrilaterals?

Before we explore...

FACT CHECK Consider trapezoid LMNO: State a pair of angles that are: 1) CONSECUTIVE ANGLES _____________ 2) OPPOSITE ANGLES _____________

Team Discovery Activity

We have examined the properties of parallelograms in terms of their sides and diagonals. Today we will make some

conjectures related to the interior angles of parallelograms.

Directions: Plot Parallelogram LOVE below. L( -3, 4) O(6 ,4) V(5, -2) E(-4, -2) Connect the points with a straight

edge and label all points.

Using these angle measures, answer the following questions on the next page.

7-5 Notes

Using the angles given on the

smartboard, find the degree measures of the

following angles.

1) Parallelogram Opposite Angles Conjecture-

Discuss with your partners and complete the sentence below:

Let’s say that m< L = 3x +40 and m< V = 9x -14

a. How can the conjecture from above help us set up an equation to solve for x?

b. Solve for x.

c. What would be the measure of angle L?

2) Parallelogram Consecutive Angles Conjecture-

Discuss with your partners and complete the sentence below:

Let’s say that m< E = x and m< V = x +80

a. How can the conjecture from above help us set up an equation to solve for x?

b. Solve for x.

c. What would be the measures of angle E and V?

The opposite angles of a parallelogram (Ex: <L and <V) are ____________________.

The consecutive angles in a parallelogram (Ex. <L and <E) are _______________________

L O

V E

L O

V E

1

2

Special Quadrilateral Angle Properties The Rhombus- We know that the Rhombus is a parallelogram, so the conjectures used in numbers 1 and 2 also apply, but the rhombus has an additional interior angle property. Examine

the following diagrams and make a conjecture about the vertex (Corner) angles to help you complete the sentences below.

Let’s say that m< DCA = 25 and the measure of < BCA = 2x -5.

a. What is the value of x?

b. What is the measure of < CDA?

The Trapezoid- The trapezoid also has some interesting interior angle relationships. Examine the diagrams below:

a. What relationship do you notice between the consecutive angles located between the bases?

b. In trapezoids, the bases are parallel. What type of special angle pair are angles <BAD and <CDA? (Consider AD a transversal).

Why does this explain what you wrote in part a?

Application: Solve for x and find the measure of <A and <D.

The Diagonals of a rhombus _______________________ the vertex (corner) angles.

35 55

BASE

3

4

Let’s Practice! Work with your teammates on the following questions based on the properties you discovered today.

1) Find x and y so that the quadrilateral is a parallelogram.

2) ABCD is a rhombus.

a. If m∠CBD = 58, find m∠ACB. b. If m∠BAD = 100 , find m∠BCD.

3) Given Isosceles Trapezoid ABCD, find the m< D

4) Find the m<L

7-5 Homework

1) The shape below is a square. Find x and y.

2) In the accompanying diagram of parallelogram ABCD, and . Find the number of

degrees in

3) The measures of two consecutive angles of a parallelogram are in the ratio 5:4. What is the

measure of an obtuse angle of the parallelogram?

4)

5) A parallelogram must also be a rectangle if its diagonals

1) Bisect each other

2) Bisect the angles to which they are drawn

3) Are perpendicular to each other

4) Are congruent

6) Given: Quadrilateral ABCD has vertices , , , and .

Prove: Quadrilateral ABCD is a parallelogram but not a rectangle

7) Which of the following is the equation of a line passing through and perpendicular to the line represented

by the equation ?

a) y =

x + 1

b) y = -

x

c) y = 2x + 1

d) y =

x -5