Unit 6: Quadrilaterals

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1 Name: Geometry Period Unit 6: 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: Please note: Calculator Pencil This Booklet A device 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!

2 Parallelogram Square Trapezoid Rectangle Isosceles Trapezoid 1) 2 sets of opposite parallel 2) 2 sets of opposite congruent 3) Diagonals bisect each other 4) One pair of opposite congruent and parallel* *SOMETIMES USED IN MULTIPLE CHOICE and PROOF! 1) 2 sets of opposite parallel 2) 2 sets of opposite congruent 3) Diagonals bisect each other 4) One pair of opposite congruent and parallel 5) Diagonals are congruent 6) Adjacent perpendicular 1) ONLY 1 set of opposite parallel Know your properties! 1) 2 sets of opposite parallel 2) 2 sets of opposite congruent 3) Diagonals bisect each other 4) One pair of opposite congruent and parallel 5) Diagonals are congruent 6) Adjacent sides are perpendicular 1) ONLY 1 set of opposite parallel 2) ONLY 1 set of opposite congruent (legs) 3) Diagonals are congruent 6-1 LESSON Rhombus 1) 2 sets of opposite parallel 2) 2 sets of opposite congruent 3) Diagonals bisect each other 4) One pair of opposite congruent and parallel 5) Adjacent congruent 6) Diagonals are perpendicular 7) Adjacent congruent 8) Diagonals are perpendicular 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: Be resourceful! If you are having a hard time sketching, do a search online to see what it should look like.

6-1 Practice/HW Finish PRACTICE FOR HW! CHECK THE KEY! Part 1: Identify Properties *Note: You may be assessed at any time on any of these quadrilateral properties. YOU NEED TO MEMEORIZE THEM.

3 6-1 Practice/HW Finish PRACTICE FOR HW! CHECK THE KEY! Part 1: Identify Properties *Note: You may be assessed at any time on any of these quadrilateral properties. YOU NEED TO MEMEORIZE THEM. 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!

4 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. WRITE THIS INFORMATION IN THE DIAGRAM. a) Circle the property of a parallelogram that you will use to answer this question. (use your property sheet). 2 sets of opposite parallel 2 sets of opposite congruent Diagonals bisect each other One pair of opposite congruent and parallel b) Now write the algebraic equation that connects that property to diagram? c) Solve for the value of x? 2) As shown in the diagram of rectangle ABCD below, diagonals and intersect at E. and.solve for the length of a) State the property of a rectangle that you will use to answer this question. (use your property sheet). b) Now write the algebraic equation that connects that property to diagram? c) Solve for the length of

5 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) State the property of an isosceles trapezoid that you will use to answer this question. (use your property sheet). b) Now write the algebraic equation that connects that property to diagram? c) Solve for the value of x.

5) In the diagram below, and are bases of trapezoid ABCD. If and, what is? Before you solve! a) State the property of a trapezoid that you will use to answer this question. (use your property sheet).

6 5) In the diagram below, and are bases of trapezoid ABCD. If and, what is? Before you solve! a) State the property of a trapezoid that you will use to answer this question. (use your property sheet). b) Now solve for. 6) The perimeter of a square is 24. In simplest radical form, find the length of the diagonal of the square. Before you solve! a) State the property of a square that you will use to answer this question. (use your property sheet). b) What additional triangle theorem will you need to use?

7 7) The diagram below shows parallelogram ABCD with diagonals and intersecting at E. What additional information is sufficient to prove that parallelogram ABCD is also a rhombus? 1) bisects. 2) is parallel to. 3) is congruent to. 4) is perpendicular to. 8) In the following rhombus, the two diagonals measure 10 and 24. Find the perimeter of the rhombus. Let s take it one step at a time: a) Write the two properties of the diagonals of a rhombus (use property sheet) and MARK those properties on the diagram. b) Use the given lengths to label your diagram appropriately. c) Solve for one side of the rhombus. ( hint: what type of triangles do you see?) d) Solve for the perimeter (sum of the sides).

8 12) 9) In square, two sides have measures 3x + 2 and 5x 14. What is the perimeter of the square? SKETCH A DIAGRAM! a) State the property of a square that you will use to answer this question. (use your property sheet). b) Now write the algebraic equation that connects that property to diagram? c) Solve for the value of x and the perimeter. 10) For which quadrilaterals are the diagonals congruent? (select all that apply) a) Rhombus b) square c) rectangle d) isosceles trapezoid e) Parallelogram f) trapezoid 11) Parallelogram HAND is drawn below with diagonals and intersecting at S. Which statement is always true? 1) 2) 3) 4)

9 12) In rectangle ABCD, and. Find the length of. SKETCH IT FIRST. a) State the property of a rectangle that you will use to answer this question. (use your property sheet). b) Now write the algebraic equation that connects that property to diagram? c) Solve for. AC 13) A quadrilateral must be a parallelogram if one pair of opposite sides is a) Congruent only. b) Parallel only c) Both congruent and parallel d) Parallel and the other pair of opposite sides is congruent. Check your work carefully!

6-2 Notes Today's Goal: Can we complete a thorough coordinate geometry proof by verifying properties specific to the quadrilateral we are proving? What is an observation? Let s Prepare!

10 6-2 Notes Today's Goal: Can we complete a thorough coordinate geometry proof by verifying properties specific to the quadrilateral we are proving? What is an observation? Let s Prepare! What is an inference? What are the tools we can use to find them? What does it mean to prove? In part 1 of this unit, we will be writing proofs that require. This means we will be given coordinates, and use our,, and formulas to make inferences based on what we see. Proving Parallelograms Coming up with a plan: Properties of a Parallelogram: 1) 2 sets of opposite sides are parallel 2) 2 sets of opposite sides are congruent Shortcuts to proving quadrilaterals are parallelograms: 1) Pick you want to prove 2) (Fact WITH support from your work)! 3) Diagonals bisect each other. 4) 1 set of opposite Congruent AND parallel

11 Proving Rectangles Properties of a Rectangle: 1) All of the properties of a parallelogram. Shortcuts to proving quadrilaterals are rectangles: 1) Prove ONE PROPERTY of a 2) Adjacent sides perpendicular. 2) Prove ONE PROPERTY special to only the 3) Diagonals congruent. 3) What do quadrilateral proofs look like? Use the student example proofs to answer the follow-up questions with your groups. Be prepared to share out! a) What is the given in these types of proofs? b) What were the students trying to prove? c) What proof got the higher grade? Which one got the lower grade? Using the BETTER student response, d) How does the proof end? What details are included in the conclusion? e) How did they use their work (calculations) to defend their proof? f) What was the structure of the proof?

12 6-2 Practice 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.

13 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 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 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.

14 6-2 Homework CHECK THIS ACCURATELY 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. 5y 2x-5 3x-18 2y+ 12 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.

15 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. 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? 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!

16 6-3 Notes 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: 1) All the properties of a parallelogram. 2) Adjacent congruent. 3) Diagonals are perpendicular. Shortcuts to proving quadrilaterals are rhombuses: 1. Prove 1 property of a AND 2. Prove 1 property special to a 3. Conclusion Proving Squares Properties of a Square: 1) All of the properties of a parallelogram. 2) The additional properties of a rectangle. 3) The additional properties of a rhombus. Shortcuts to proving quadrilaterals are squares: 1. Prove 1 property of a 2. Prove 1 property of a 3. Prove 1 property of a 4. Conclusion (Using ALL EVIDENCE)! Use color and critique! Consider the following student work sample. With another color, add advise, details or inaccuracies that you believe should be fixed. Any advice for this student to improve their proof?

17 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! Method: ) 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: - -

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

19 6-3 Homework 1. Do you know the properties for all quadrilaterals?! If not, review your study guide!! 2. Quadrilateral ABCD has vertices A(-2, 4), B(2, 8), C(6, 4) and D(2, 0). Prove that quadrilateral ABCD is a square.

20 3. 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. 4. ( 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)

21 6-4 Notes 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 parallel, 1 pair of opposite 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

22 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?

23 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, it s not too bad. Getting used to it I don t like to prove it, prove it!

24 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, 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) ABCD is an isosceles trapezoid, where AB is parallel to DC and AD BC. If AD = 2y 5, and BC = y + 3, find AD. 4) Isosceles trapezoid ABCD has diagonals and. If and, a) what is the value of x? b) Explain the property that you used to solve for x. 5) The cross section of an attic is in the shape of an isosceles trapezoid, as shown in the accompanying figure. If the height of the attic is 9 feet, feet, and feet, find the length of to the nearest foot.

25 6-4 Homework 1. Given: ABC with vertices A( -6,-2), B( 2,8), andc( 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

26 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 AB is parallel to DC and AD BC. 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?

6-5 Notes Today s Goal: How can I discover the angle properties in quadrilaterals? FACT CHECK Consider trapezoid LMNO: State a pair of angles that are: Before we explore.

Today we will make some conjectures related to the interior angles of parallelograms. Directions: Plot Parallelogram LOVE below.

27 6-5 Notes Today s Goal: How can I discover the angle properties in quadrilaterals? FACT CHECK Consider trapezoid LMNO: State a pair of angles that are: Before we explore... 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 the angles given on the smartboard, find the degree measures of the following angles. Using these angle measures, answer the following questions on the next page.

28 1) Parallelogram Opposite Angles Conjecture- Discuss with your partners and complete the sentence below: The opposite angles of a parallelogram (Ex: <L and <V) are. L O Let s say that m< L = 3x +40 and m< V = 9x -14 E V 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: The consecutive angles in a parallelogram (Ex. <L and <E) are L O E V 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?

29 Special Quadrilateral Angle Properties 3. 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. The Diagonals of a rhombus the vertex (corner) angles. 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? 4. 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). BASE Why does this explain what you wrote in part a? Application: Solve for x and find the measure of <A and <D.

30 6-5 Practice 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 ABDC, find the m< D 4) Find the m<l

31 5) In the diagram below, and are bases of trapezoid ABCD. If and, what is? 6) In isosceles trapezoid QRST shown below, and are bases. If and, what is? 7) In the accompanying diagram of rectangle ABCD, and. What is?

32 6-5 Homework 1) The shape below is a square. Find x and y. 2) In the accompanying diagram of parallelogram ABCD, m A = (2x + 10) and m B = 3x. Find the number of degrees in m B. 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

33 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 = 1 x b) y = - 1 x 2 c) y = 2x + 1 d) y = 1 x -5 2

Unit 7: Quadrilaterals

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: Please note: Calculator