Properties of Quadrilaterals

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Wilfred Goodman
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1 Properties of Quadrilaterals 1

2 Proving Properties of Parallelograms Given: ABC is a parallelogram Prove: AB C and A BC A B C Statements Reasons Given 2. AB C and A BC AB CB and CB AB B B AB CB AB C and A CB 6. Property 1: Opposite sides of a parallelogram are. Guided Practice: Solve for the missing variable. 1. Solve for x. 2. Find VU T x + 1 W J K 2x M L U x V 2

3 Given: ABC is a parallelogram Prove: B and A C A B (Hint: to make things easier, you may want to draw your triangles out to the side) Statements Reasons C Given 2. AB C and A BC B B and AC AC ABC CA and AB CB CPCTC Property 2: Opposite angles of a parallelogram are. Guided Practice Solve for the missing variable. 1. Solve for x. 2. Solve for x. M x L N 58 K 3

4 Find the measure indicated in each parallelogram. 3. Find m G 4. Find m V F E 3x + 11 T 8x + 6 S G 5x - 9 U 10x - 6 V Given: ABC is a parallelogram Prove: AP PC and P PB A P B C Statements Reasons 1. ABC is a parallelogram AB C AB C and A BC AB CB and CAB AC APB CP AP CP and P BP 6. Property 3: iagonals of a parallelogram each other. 4

5 Guided Practice Solve for the missing variable or measure indicated. 1. Solve for x. YR = 19 and RW = 3x 2 2. Solve for x. NG = 19 and EG = 4x + 6 W Z F G R N X Y E 3. SA = x + 11 and AU = 2x + 1, Find SU U R A T S Given: ABC is a parallelogram Prove: m A + m = 180 and m C + m = 180 A B C Statements Reasons 1. ABC is a parallelogram AB C and A BC m A + m = m C + m =

6 Property 4: Consecutive angles of a parallelogram are. Guided Practice: Solve for the missing variable or measurement indicated. 1. Find x. 2. m W 8x 2 and m X 8 26x Find m X U T Z 20x - 5 8x + 2 W V 65 S Y x X 3. Find m N N K 19x + 5 M 15x +5 L 6

Proving Properties of Rectangles P Q T S R Statements Reasons 1. PQRS is a rectangle. Given 2. PQRS is a parallelogram. A rectangle is a parallelogram. 3. QR SP 4. RS SR 5.

7 Proving Properties of Rectangles P Q T S R Statements Reasons 1. PQRS is a rectangle. Given 2. PQRS is a parallelogram. A rectangle is a parallelogram. 3. QR SP 4. RS SR 5. m PSR = 90 m QRS = 90 efinition of a rectangle 6. m PSR = m QRS Transitive property 7. PRS PRS 8. PR QS Property: iagonals of a rectangle are. Guided Practice: Solve for x. 1. AC= 15 and B= 2x AR= 3x, RC= 4, and B= 25 7

8 Proving Properties of Rhombi E H F Given: EFG is a rhombus Prove: EFH GFH and EH FEH G Statements Reasons 1. EFG is a rhombus EF FG G E EFG is a parallelogram EH HG and H HF EHF GHF EFH GFH EH FEH EH FEH 8. Property: iagonals of a rhombus opposite angles. 8

9 E H F Given: EFG is a rhombus Prove: EG F G Statements Reasons 1. EFG is a rhombus EF FG G E EFG is a parallelogram EH HG and H HF EHF GHF EHF GHF m EHF = m GHF 7. efinition of Congruence 8. m EHF + m GHF = efinition of Linear Pair 9. m EHF + m EHF = Substitution 10. 2(m EHF) = m EHF = m GHF = efinition of perpendicular Property: iagonals of a rhombus are. Since a square is a regular quadrilateral, what properties apply to a square? 9

10 Skills Practice State the most specific name for each figure. 1) 2) 3) 4) Solve for x. Each figure is a parallelogram. 5) 6) F C R U 3x + 4 E S 13x T 7) E 8) F 5x + 5 F 33x - 1 G 6x - 3 G E 115 H 9) W 7 V 10) E F 2x - 11 x X 4x - 1 U H G 10

11 11) FH = 38, YH = 2x ) YW = 28, VW = 2x - 8 H E W Z Y V G F X Y Find the measurement indicated in each parallelogram. 13) Find m L L K W x 14) Find m W T 12x + 7 M 9x + 5 J V 8x - 7 U 15) Find CB 16) VJ = x + 11, JT = x. Find VJ A 2x - 11 T W B x + 1 C U J V 11

12 Midpoint = x 1+x 2, y 1+y Skills Review Formulas: Slope = y 2 y 1 x 2 x 1 distance = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Find the slope between the two points given. Leave your answers in simplified fractional form. 1. (6,-12) and (-16,-13) 2. (9,17) and (-12,7) 3. (6,5) and (20, -10) 4. (17,-7) and (-8,11) 5. (-8, 1) and (6,8) 6. (-7,-16) and (11,-16) Find the distance between the two given points. Leave your answers in simplified radical form. 7. (0,3) and (-4, 6) 8. (-7,3) 9. (-3,-6) and (-2,1) 10. (4,7) and (-2,-5) 11. (7,-6) and (-4,-7) 12. (0,4) and (9,-2) 12

13 Find the midpoint for each set of given points. 13. (-10, -5) and (4,-1) 14. (-4, -9) and (-10,-11) 15. (3,-5) and (0,-2) 16. (9,-1) and (10,-11) 17. (-9, -4) and (3,-3) 18. (-12, -3) and (7,9) Find the slope of the lines that would be parallel and perpendicular to the points. 19. (6,-12) and (-16, -13) Slope of parallel line: Slope of perpendicular line: 20. (9,17) and (-12, 7) Slope of parallel line: Slope of perpendicular line: 21. (6,5) and (20, -10) Slope of parallel line: Slope of perpendicular line: 22. (-7,-16) and (11, -16) Slope of parallel line: Slope of perpendicular line: 13

14 Proving Quadrilaterals on a Coordinate Plane Recall definitions and properties of Special Parallelograms: Parallelogram Rectangle- 1. Rhombus Square Guided Practice: 1. Plot points A(-3, -1), B(-1, 2), C(4, 2), and (2, -1). a. Find the length of all four sides. b. Find the slope of all four sides. c. What specialized geometric figure is quadrilateral ABC? How do you know? 14

15 2. Plot points E(1, 2), F(2, 5), G(4, 3) and H(5, 6). a. Find the length of all four sides. b. Find the slope of all four sides. c. What specialized geometric figure is quadrilateral EFHG? How do you know? d. escribe another way that we could have shown that this figure was a rhombus? 3. Plot points A(1, 0), B(-1, 2), C(2, 5), and (4,3). a. Find the length of all four sides. b. Find the slope of all four sides. c. What specialized geometric figure is quadrilateral ABC? How do you know? d. escribe another way that we could have shown that this figure was a rectangle? 15

16 Classwork 1. Plot the points W(2, -1), X(1, 3), Y(6, 5), and Z(7, 1). a. What properties do you need to prove WXYZ is a parallelogram? b. Show that WXYZ is a parallelogram. 2. Plot the points P(3, 1), Q(3, -3), R(-2, -3), and S(-2, 1). a. What properties do you need to prove PQRS is a rectangle.? b. Show that PQRS is a rectangle. 16

17 3. Plot the points P(5, 2), Q(1, 9), R(-3, 2), and S(1, -5). a. What properties do you need to prove PQRS is a rhombus? b. Show that PQRS is a rhombus. 4. Plot the points P(5, 2), Q(2, 5), R(-1, 2), and S(2, -1). a. What properties do you need to prove to show PQRS is a square? b. Show that PQRS is a square. 17

18 Homework etermine whether the given points represent the vertices of a parallelogram, rectangle, rhombus, or square. Justify your answer mathematically. 1. A(-2, 8), B(5, 8), C(2, 0), (-5, 0) 2. P(2, 5), Q(-4, 5), R(2, -7), S(-4, -7) 18

19 3. J(-5, 6), K(-4, -2), L(4, -1), M(3, 7) 4. P(5, 1), Q(9, 6), R(5, 11), S(1, 6) 19

20 irections: Question 5 is multiple choice. Choose the correct answer. 5. Three vertices of a rectangle on the coordinate plane are (-2, -1), (6, -1), and (-2, 1). Which of the following is the coordinate of the fourth vertex? A. (6,1) B. (6, -1) C. (-7, 1). (2, 1) 6. Ashanti is surveying for a new parking lot shaped like a parallelogram. She knows that three of the vertices of parallelogram ABC are (0,0), (5,2), and (6,5). Find the coordinates of point. 7. A parallelogram has vertices L(-2, 5), M(3, 3), and N(1, 0). What are possible coordinates for its fourth vertex? Explain. 20

2. A diagonal of a parallelogram divides it into two congruent triangles. 5. Diagonals of a rectangle bisect each other and are equal and vice-versa.

2. A diagonal of a parallelogram divides it into two congruent triangles. 5. Diagonals of a rectangle bisect each other and are equal and vice-versa. QURILTERLS 1. Sum of the angles of a quadrilateral is 360. 2. diagonal of a parallelogram divides it into two congruent triangles. 3. In a parallelogram, (i) opposite sides are equal (ii) opposite angles