Notes: Parallelograms. A parallelogram is a quadrilateral with opposite parallel sides. Property BLANK Meaning
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1 Notes: Parallelograms A parallelogram is a quadrilateral with opposite parallel sides. If you are given parallelogram ABCD then: Property BLANK Meaning (1) opposite sides are parallel (1) AB DC and AD BC (2) opposite sides are congruent (2) AB = DC and AD = BC (3) opposite angles are congruent (3) A = C and 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: consecutive angles are supplementary BLANK (5) the diagonals bisect each other (5) AE = EC and DE = EB 1
2 Practice Problems: Parallelograms Use the properties in the notes section to help you solve the following problems. (1) In parallelogram P QRS, the ratio of measure of Q to the measure of R is 1:5. Find m Q. They don t give us a diagram but that s okay! You may draw your own parallelogram however you want as long as you label P,Q,R,S in the given order. All of the diagrams below are correct, they are labeled clockwise but you may label them counter-clockwise as well: The following diagrams are NOT correct: Now to actually solve the problem, Q and R are consecutive angles and property (4) says that consecutive angles are supplementary 1x + 5x = 180 6x = 180 x = 30 m Q = 30 2
3 Practice Problems: Parallelograms Use the properties in the notes to help you solve the following problems. (2) In parallelogram ABCD, m B = 60. Find m C. (3) In parallelogram CDEF, CD = (5x 6) and F E = (3x + 8). Find the value of x. (4) In parallelogram ABCD, m D = (6x + 40) and m B = (4x + 70). Find the value of x. (5) Which statement is not true for every given parallelogram P QRS? (a) P Q = SR (c) P R SQ (b) m P = m R (d) m P + m S = 180 (6) In parallelogram QRST, diagonals QS and RT intersect and point E. If QE = 4x+3 and ES = 23, find the value of x. 3
4 (7) In parallelogram ABCD, the measures of angles A and B are in the ratio of 1:8. Find m B. (8) The measures of two opposite angles of a parallelogram are represented 5x + 40 and 3x Find x. (9) Which statement is always true? The diagonals of a parallelogram... (a) are congruent. (b) bisect the angles of the parallelogram. (c) bisect each other. (d) are perpendicular to each other. (10) In parallelogram ABCD, diagonals AC and BD intersect at point E. Which statement is always true? (a) AC DB (b) DEC = AEB (c) ABD = AED (d) BEC = DEC (11) In the accompanying diagram, side AB of parallelogram ABCD is extended to E. If m CBE = 35, find m D. 4
5 (12) In the accompanying diagram of parallelogram ABCD, DF is perpendicular to diagonal AC at point F. If m CAB = 34, find m CDF. (13) In parallelogram ABCD, m A = (3x + 40) and m C = (7x 100). Find the measure in degrees of D. (14) In parallelogram ABCD, diagonal AC is drawn. If m D = 110 and m CAD = 50, find the m CAB. (15) In parallelogram P QRS, diagonal P R is drawn. If m S = 100 and m SRP = 40, then which of the following statements must be true? (a) SR > SP (b) SP > SR (c) SP = SR (d) P R is the smallest side of P RS 5
6 Notes: Rectangles 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 properties of the parallelogram are true (1) see properties on page 1 (2) all the angles are right (2) m A = m B = m C = m D = 90 (3) the diagonals are congruent (3) AC = DB 6
7 Practice Problems: Rectangles Use the properties in the notes section to help you solve the following problems. (1) In rectangle ABCD, AB = 4 and BC = 3. AC must be (a) 5 (b) 7 (c) 25 (d) 4 (2) The diagonals of a rectangle must always be perpendicular to each other. (True or False) (3) In rectangle P QRS with diagonals P R and SQ, if P R = (4x 10) and SQ = (7x 40), find x. (4) In rectangle ABCD diagonal AC is drawn. If m DCA = 30, find m CAD. (5) The diagonals of rectangle ABCD intersect at E. If DE = (3x + 1) and EB = (2x + 7), find the value of x. 7
8 (6) If the rectangle ABCD, AD = 6 and AB = 8. What is the measure of diagonal AC? (7) The diagonals of rectangle ABCD intersect at E. If DE = (2x+3) and AE = (x+6), find the value of x. (8) The diagonals of a rectangle RST V intersect at Q. If V Q = (3x 3) and RT = (5x 1) find the value of x. (9) In rectangle ABCD, BC = 30 and AB = 40. If diagonals AC and BD intersect at E, find the measure of BE. 8
9 Notes: Rhombus 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 a rhombus ABCD then: Property Meaning (1) all properties of the parallelogram are true (1) see properties on page 1 (2) all sides are congruent (2) AB = BC = DC = 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 to each other (4) DB AC at E Practice Problems: Rhombus Use the properties in the notes section to help you solve the following problems. (1) In the accompanying diagram of a rhombus CDEF, diagonal F D is drawn and m E = 40. Find m CF D. 9
10 (2) In rhombus ABCD, AB = (6x 3) and BC = (4x + 7). Find x. (3) The lengths of the diagonals of a rhombus are 6 and 8. What is the length of a side of a rhombus? (4) In rhombus ABCD, diagonal DB is congruent to side AD. What is the measure of A? (5) If the lengths of the diagonals of a rhombus are 10 and 24, find the length of one side of the rhombus. (6) In the accompanying diagram of rhombus ABCD, m ABD = 50. Find m A. 10
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