SWBAT… Agenda 1. Warm-up (10 min) 2. Review HW (15 min) 3. Notes on properties of parallelograms (20 min) Warm-Up: is 25% of what number? is what percent of 115? 3. Liz deposited $1,200 in a savings account that pays 5¼% annually. How much would Liz have in her account after one year? Thurs, 2/13 Pg. 364 – 365 #9 – 12, 14-16, 25-30, 38-40

SWBAT… use and apply the properties of parallelograms Agenda 1. Warm-up / Check HW (10 min) 2. Review HW (15 min) 3. 4 properties of parallelograms (20 min) Warm-Up: is 25% of what number? is what percent of 115? 3. As a realtor, you get a commission of 13% of your sales profit. If the sales profit is $256,808, what is your commission? 4. Liz deposited $1,200 in a savings account that pays 5¼% annually. How much would Liz have in her account after one year? Thurs, 2/13 HW: 2 Problems given at end of period

Properties of Parallelograms

Definition of a Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel. DC║AB and DA║CB. The symbol ABCD is read “parallelogram ABCD.”

Property # 1 If a quadrilateral is a parallelogram, then its opposite sides are congruent.  AD ≅ BC and AB ≅ DC

Property # 2 If a quadrilateral is a parallelogram, then its opposite angles are congruent.   P ≅  R and  Q ≅  S P Q R S

Property # 3 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.  m  P + m  Q = 180°  m  Q + m  R = 180°  m  R + m  S = 180°  m  S + m  P = 180° P Q R S Consecutive angles: Angles of a polygon that share a side (“next” to each other) Consecutive angles are supplementary because they are same-side interior angles

Property # 4 If a quadrilateral is a parallelogram, then its diagonals bisect each other.  E is the midpoint of AC and BD, therefore AE ≅ CE and DE ≅ BE

Examples Given HKLP. HK = _______ and HP = ________. m< K = m<______. m< L + m<______ = 180°. If m< P = 65 0, then m< H = ____, m< K = ____ and m< L =____. Given the diagonals with their point of intersection labeled M. If HM = 5, then ML = ____. If KM = 7, then KP = ____. If HL = 15, then ML = ____. If m<HPK = 36°, then m<PKL = _____. H K L P PLKL P P or K 115°65° 115° M 5 units 14 units 7.5 units 36° (Alternate interior angles are congruent.)

GUIDED PRACTICE What property can you use to show that the quadrilateral is a parallelogram? If two pairs of opposite sides are congruent, the quadrilateral is a parallelogram. ANSWER

GUIDED PRACTICE What property can you use to show that the quadrilateral is a parallelogram? If the opposite angles are congruent, the quadrilateral is a parallelogram. OR If the consecutive angles are supplementary, the quadrilateral is a parallelogram. ANSWER

Ex 1: Property #1

Ex Property #2

Ex. Property #3

m JML SOLUTION By Theorem 8.5, the consecutive angle pairs in JKLM are supplementary. So, m KJM + m JML = 180°. Because m KJM = 110°, m JML =180° –110° = 70°. Find the indicated measure in JKLM. Property #3

Ex 1: For what values of x and y must the figure be a parallelogram? Property #3

Ex. In the given figure, AD and BC are diagonals of parallelogram gram ABCD. A B C D O 1. If AO = 15 cm, how long is CO? Ans.( 15 cm ) 2. If DO is 18 cm, how long is BO? Ans. ( 18 cm ) Property #4

Solution: BS = TS 9x – 4 = 7x +2 9x – 7x = x = 6 x = 3 BS = 23, TS = 23 BT = 46 S B A TH Ex 1: BS = 9x – 4 and TS = 7x + 2. Find BT Property #4

Honors HW: For what values of x and y must the figure be a parallelogram?

Ex 1: Using properties of Parallelograms FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. a. JH b. JK Statement Reason a. JH = FG Opposite sides of a are ≅. JH = 5 Substitute 5 for FG. b. JK = GK Diagonals of a bisect each other. JK = 3 Substitute 3 for GK F G J H K 5 3

Ex 2: Using properties of parallelograms PQRS is a parallelogram. Find the angle measure. Explain your reasoning. a. m  R b. m  Q Statement Reason a. m  R = m  P Opposite angles of a are ≅. m  R = 70° Substitute 70° for m  P. b. m  Q + m  P = 180° Consecutive  s of a are supplementary. m  Q + 70° = 180° Substitute 70° for m  P. m  Q = 110° Subtract 70° from each side. P RQ 70° S

Ex. 3: Using Algebra with Parallelograms PQRS is a parallelogram. Find the value of x. Statement m  S + m  R = 180° 3x = 180 3x = 60 x = 20 Consecutive  s of a are supplementary. Substitute 3x for m  S and 120 for m  R. Subtract 120 from each side. Divide each side by 3. S Q P R 3x°120° Reason