The Parallelogram REMEMBER A parallelogram is a quadrilateral with opposite sides parallel. It has many special properties.

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1 ame: Date: The Parallelogram REMEMBER A parallelogram is a quadrilateral with opposite sides parallel. It has many special properties. If you are given parallelogram ABCD then: Property Meaning (1) opposite sides are parallel (1) AB ll DC, AD ll BC (2) opposite sides are congruent (2) AB DC, AD BC (3) opposite angles are congruent (3) <A <C, <D <B (4) consecutive angles are supplementary (4) m<a + m<b =180 m<b + m<c =180 m<c + m<d =180 m<d + m<a = 180 If the diagonals are drawn then: (5) the diagonals bisect each other (5) AE EC, DE EB 1. In parallelogram ABCD, m < B = 60. Find m < C. 4.In parallelogram ABCD, m<d=(6x+40) and m<b=(4x+70). Find the value of m<b. 2. In parallelogram PQRS, the ratio of measure of <Q to the measure of <R 1:5. Find m<q. 5. Which statement is not true for every given parallelogram PQRS? (1) PQ SR (3)PR is perpendicular to SQ (2)<P <R (4)m<P + m<s = In parallelogram CDEF, CD=(5x-6) and FE=(3x+8). Find the value of CD. 6. In parallelogram QRST, diagonals QS and RT intersect at point E. If is QE=4x+3 and ES=23, find the value of QE and x.

2 7. In parallelogram ABCD, the measures of angles A and B are in the ratio of 1:8. Find m<b. 12. In the accompanying diagram of parallelogram ABCD, DF is perpendicular to diagonal AC at point F. If m<cab=34, find m<cdf. 8.The measures of two opposite angles of a parallelogram are represented by 5x + 40 and 3x+50. Find x. 13. In parallelogram ABCD, m<a=(3x+40) and m<c=(7x-100). Find the measure in degrees of <D. 9. Which statement is always true? (1) The diagonals of a parallelogram are congruent (2) The diagonals of a parallelogram bisect the angle of the parallelogram (3) The diagonals of a parallelogram bisect each other. (4) The diagonals of a parallelogram are perpendicular to each other. 10. In parallelogram ABCD, diagonals AC and BD intersect at point E. (1) AC DB (2) ΔDEC ΔAEB (3) ΔABD ΔAED (4) ΔBEC ΔDEC 14. In parallelogram ABCD, diagonal AC is drawn. If m<d=110 and m<cad=50, find m<cab. 15. In parallelogram PQRS, diagonal PR is drawn. If m<s=100 and m<srp=40, then which of the following statements must be true? (1) SR>SP (2) SP>SR (3) SP=SR (4) PR is the smallest side of ΔPRS 11. In the accompanying diagram, side AB of parallelogram ABCD is extended to E. If m<cbe=35, find m<d.

3 Name: Date: Quadrilateral NOTES A is a polygon with 4 sides. The SUM of the 4 angles is 360. I) Properties of a : a) Opposite sides are. b) Opposite are congruent. c) Opposite are congruent. d) Consecutive angles are. e) The diagonals each other. Note: A diagonal divides a parallelogram into 2 congruent triangles. Examples: 1) If one angle of a parallelogram measures 55, then find the measures of the other 3 angles. 2) In parallelogram ABCD, m<a = 2x - 20 and m<c = 5x Find the value of <A. 3) In parallelogram FGHI, m<f:m<g = 2:7. Find the measure of <H.

4 Name: Date: Aim: To prove a quadrilateral is a parallelogram DO NOW Without looking at your notes or homework, try to list all the properties of a parallelogram (There are 5 Properties)

5 Proving a Quadrilateral is a Parallelogram. *You can be asked to prove a Parallelogram by either coordinate geometry or statement/reason. Six ways to prove a quadrilateral is a parallelogram: In a parallelogram, 1. (Coordinate Geometry: ) 2. (Coordinate Geometry: ) 3. (Coordinate Geometry: ) 4. (Coordinate Geometry: ) 5. (Coordinate Geometry: ) 6. (Coordinate Geometry: )

6 Name: Date: Proving Quadrilaterals are Parallelograms 1) The coordinates of three points in the coordinate plane are A(- 2,- 3), B(5,- 3) and C(2,2). a) Find the coordinates of D if ABCD is a parallelogram. b) By COORDINATE GEOMETRY, prove that ABCD is a parallelogram. 2) Given: PQRS is a parallelogram with PT RM Prove: TQMS is a parallelogram Statements Reasons

7 HOMEWORK 3) Quadrilateral ABCD has coordinates A(- 1,3), B(4,4), C(5,- 3) and D(- 2,- 2). Using coordinate geometry, prove: a) the diagonals are perpendicular. b) ABCD has a least one pair of congruent sides. c) ABCD is NOT a parallelogram. 4) Given: Quadrilateral ABCD. FGE, AGC, FG EG, AG CG and <B <D Prove: ABCD is a parallelogram Statements Reasons

8 Name: Date: Aim: To define the properties of a rectangle and a rhombus. To prove a quadrilateral is a rectangle and a rhombus. Rectangle Notes II) Rectangle Properties: a) b) c) To PROVE a quadrilateral is a rectangle: * 1). (Coordinate Geometry: ) 2). (Coordinate Geometry: )

9 Rhombus Notes III) Rhombus Properties: a) b) c) d) Proving that a Quadrilateral is a Rhombus * 1). (Coordinate Geometry: ) 2). (Coordinate Geometry: ) 3). (Coordinate Geometry: )

10 Name: Date: 1) The coordinates of vertices ABCD are A(-2,0), B(2,-2), C(5,4) and D(1,6). a) Graph the quadrilateral. b) Prove that ABCD is a rectangle.

11 2) The vertices of quadrilateral ABCD are A(-1,1), B(4,0), C(5,5) and D(0,6). a) Prove that ABCD is a parallelogram. b) Prove that the diagonals of ABCD are perpendicular. c) Is ABCD a rhombus? EXPLAIN!!

12 Name: Date: The RHOMBUS REMEMBER A rhombus is a parallelogram with adjacent sides equal. In addition to having all the properties of a parallelogram, the rhombus has several other properties. If you are given rhombus ABCD then: Property Meaning (1) all properties of the (1) Refer to notes parallelogram are true (2) all sides are congruent (2)AB BC CD AD (3) the diagonals bisect the opposite angles (3) m<dac=m<bac m<dca=m<bca m<adb=m<cdb m<abd=m<cbd (4) the diagonals are perpendicular (4) DB AC at E to each other 1. In rhombus ABCD, AB=(6x- 3) and BC=(4x+7). Find AB. X = 5 AB = If the lengths of the diagonals of a rhombus are 10 and 24, find the length of one side of the rhombus. 2.The lengths of the diagonals of a rhombus are 6 and 8. What is the length of a side of a rhombus? 6. In the accompanying diagram of rhombus ABCD, m<abd=50. Find m<a. Pythagorean Theorem 3. In the accompanying diagram of rhombus CDEF, diagonal FD is drawn and m<e=40. Find m<cfd Given a parallelogram, a rhombus and a rectangle. If one of these quadrilaterals is picked at random, what is the probability that its diagonals bisect each other? (1) 1 (2) 2/3 (3) 1/3 (4) 0 4. In rhombus ABCD, diagonal DB is congruent to side AD. What is the measure of <A? 60

13 Name: Date: The RECTANGLE REMEMBER A rectangle is a parallelogram with a right angle. In addition to having all the properties of a parallelogram, the rectangle has several other properties. If you are given rectangle ABCD then: Property Meaning (1) all the properties of the (1) Refer to notes parallelogram are true (2) all the angles are right (2) m<a=m<b=m<c=m<d=90 and are congruent (3) the diagonals are congruent (3) AC DB 1. In rectangle PQRS with diagonals PR and SQ, if PR=(4x- 10) and SQ=(7x- 40), find PR. 6. In rectangle ABCD, AD=6 and AB=8. What is the measure of diagonal AC? 2. In rectangle ABCD, AB=4 and BC=3. AC must be (1) 5 (2) 7 (3) 25 (4) 4 7. The diagonals of rectangle ABCD intersect at E. If DE=(2x+3) and AE=(x+6), find the value of AE. 3. In rectangle ABCD diagonal AC is drawn. In m<dca=30, find m<cad. 8. The diagonals of rectangle RSTV intersect at Q. If VQ=(3x- 3) and RT=(5x- 1) find the value of RT. 4. The diagonals of rectangle ABCD intersect at E. If DE=(3x+1) and EB=(2x+7), find the value of BE and ED. 9. The diagonals of a rectangle must always be perpendicular to each other. (True or False) 5. If the measure of one angle of a parallelogram is 90, what is the probability that the parallelogram is a rectangle? 10. In rectangle ABCD, BC=30 and AB=40. If diagonals AC and AB intersect at E, find the measure of BE.

14 Name: Date: Proofs using the Properties of a Rectangle and Rhombus E 1) Given: Rectangle ABCD with E the midpoint of DC Prove: <1 <2 Statements Reasons 2) Given: AECB is a rhombus, AED, FEC, <FAB <DCB Prove: FE DE Statements Reasons

15 Name: Date: 3) In the coordinate plane A, (-1,-2), B(1,2), C(-1,3), and D(-3,-1) are the vertices of a quadrilateral. a) Find AB and CD b) Find the slope of AB and the slope of CD c) What does the answer to part b tell you about AB and CD? d) Based on the answers to a and b, explain why ABCD is a parallelogram. e) Find the slope of AD. f) What do the answers to part b and e tell you about AB and AD? g) Based on the answers to part d and f, explain why ABCD is a rectangle.

16 4) In the coordinate plane, A(-1,0), B(3,-3), C(3,2), and D(-1,5) are the vertices of a quadrilateral. AC and BD intersect at M(1,1). a) Show that AC and BD bisect each other. b) Based on the answer to part a, explain why ABCD is a parallelogram. c) Show that AC BD. d) Based on the answers to parts b and c, explain why ABCD is a rhombus.

17 Name: Date: 5) Given: A(-2,2), B(6,5), C(4,0), D(-4,3) Prove: ABCD is a parallelogram but not a rectangle

18 6) The vertices of quadrilateral ABCD are A(2,3), B(11,6), C(10,9), and D(1,6). a) Using coordinate geometry, show that diagonals AC and BD bisect each other b) Using coordinate geometry, show that quadrilateral ABCD is a rectangle.

19 Name: Date: Aim: To define the properties of a square. To prove a quadrilateral is a square. Square Notes IV) SQUARE Properties: a) b) To PROVE a quadrilateral is a SQUARE: 1). 2).

20 Name: Date: The SQUARE REMEMBER A square is a rhombus with four angles or is a rectangle with four equal sides. Since it is both a rectangle and a rhombus, all the properties you have learned hold true for the square. Example: If the side of a square ABCD is 5, find the diagonal BD. Solution: AB=DA=5 sides are congruent x 2 = Pythagorean theorem x 2 =25+25 x 2 =50 x= 50= 25*2=5 2 ANSWER 1. In the accompanying diagram, quadrilateral ABCD is a square with diagonal DB. Which of the statements are NOT true? (1) AB CB (2) AD CB (3) DA AB (4) AD DB 2. Which statement is false? (1) a square is a rectangle 4. If the side of a square is 4, find the length of the diagonal. 5. In the accompanying diagram CDEF is a square with diagonal CE drawn. Which statement is NOT true? (2) a square is a rhombus (3) a rhombus is a square (4) a square is a parallelogram (1) ΔCFE is isosceles (2) ΔCFE is a right triangle (3) ΔCFE ΔCDE (4) ΔCDE is equilateral 3. In the accompanying diagram, PQRS is a square with diagonal SQ. Which statement is NOT true? 6. In square ABCD diagonal AC is drawn. How many degrees are there in the measure of <ACB? (1) <1 <2 (2) <2 <3 (3) <4 <P (4) <P <R 7. Find the diagonal of a square whose perimeter is 28.

21 1. 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.

22 2. The vertices of quadrilateral ABCD are A(-1,1), B(4,5), C(9,1), and D(4,-3). Using coordinate geometry, prove that a. ABCD is a rhombus b. ABCD is NOT a square

23 Name: Date: Trapezoid Notes V) Trapezoid *. (NOT ) The ANGLES of the trapezoid add up to. Isosceles Trapezoid -. 1). 2). 3).

24 Name: Date: The Trapezoid and Isosceles Trapezoid REMEMBER A trapezoid is a quadrilateral that has two and only two sides parallel. The parallel sides are called the bases and the non-parallel sides are called the legs. An isosceles trapezoid is a trapezoid, which has congruent legs. Example: Given isosceles trapezoid ABCD with AB ll CD, AB=4, CD=14, and AD=13. Find the length of an altitude of trapezoid ABCD. Solution: Draw in altitude AE and BF such that Rectangle ABFE is formed and EF=4. Since ABCD is isosceles, ΔADE can be proven to be congruent to ΔBFC and DE FC. Therefore, DE=FC=5. 1.In the accompanying diagram, isosceles trapezoid CDEF has bases of lengths 6 and 12 and an altitude of length 4. Find CD. By the Pythagorean theorem x =13 2 x 2 +25=169 x 2 =144 x=12 answer 4. In the accompanying diagram, isosceles trapezoid ABCD has bases AB and DC, and diagonals AC and BD are drawn. Which statement is NOT true? (1) AD BC (2) AC BD (3) AB DC (4) AB ll DC 2.In the accompanying diagram of trapezoid ABCD, CD=10, m<a=45, m<d=90, and base BC=3. Find the length of base AD. 5. ABCD is an isosceles trapezoid with bases AB and DC. If AD=3x+4 and BC=x+12, find AD. 3.In isosceles trapezoid ABCD, AB ll CD, AB=18, CD=6 and AD=10. Find the length of an altitude of ABCD. 6. CDEF is a trapezoid with CD // FE. If m<f and m<c are in the ratio 1:4, find the measure of <F.

25 In 7-12, answer true or false in each case. 7. In an isosceles trapezoid, nonparallel sides are congruent. 8.In a trapezoid, at least two sides must be congruent. 9. In a trapezoid, base angles are always congruent. 10. The diagonals of a trapezoid are congruent only if the nonparallel sides of the trapezoid are congruent. 11. A trapezoid is a special kind of parallelogram. 12. In a trapezoid, two consecutive angles that are not angles on the same base must be supplementary. In is a trapezoid with AB//DC. When ABCD is isosceles, sides AD and BC are marked as congruent. In each case, find the measures of the angles indicated by arcs in the diagram

26 17. Quadrilateral PQRS has vertices P(-3,-4), Q(9,5), R(-1,10), and S(-5,7). Prove that quadrilateral PQRS is an isosceles trapezoid.

27 19 18) Given: Isosceles Trapezoid ABCD with ABllDC and AD CB Prove: AC BD Statements Reasons 119) Given: Isosceles trapezoid GTHR with GR TH and diagonals GH and TR Prove: GH TR Statements Reasons