5-1 Properties of Parallelograms. Objectives Apply the definition of a. parallelogram,

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1 hapter 5 Quadrilaterals 5-1 Properties of Parallelograms Quadrilaterals pply the definition of a Prove that certain quadrilaterals are s pply the theorems and definitions about the special quadrilaterals pply the definition of a List the other properties of a through new theorems ny 4 sided figure He s ack. efinition of a Parallelogram ( ) Naming a Parallelogram Parallelograms are special types of quadrilaterals with unique properties If you know you have a, then you can prove that these unique properties exist With each property we learn, say the following to yourself.. If a quadrilateral is a, then. What If a quadrilateral do you think is the a definition, is based then on opposite the diagram? sides are parallel. Partners: What do we know about the angles of a b/c it has parallel sides? *consecutive angles = 180 Use the symbol for and name using the 4 vertices in order either clockwise or counter clockwise. heorem heorem he fact that we know opposite sides are parallel, we can deduce addition properties through theorems What Opposite did we angles discuss of a at the beginning of are the congruent. lesson about s-s int. angles? Opposite sides of a are congruent. What would be our plan for solving this theorem? 1

2 heorem he diagonals of a bisect each other. What is another name for and? Parallelograms: What we now know If a quad is a, then From the definition.. 1. opposite sides are parallel From theorems 1. onsecutive angles = 180. opposite angles are congruent. opposite sides are congruent 4. he diagonals of a bisect each other rue or False Every is a quadrilateral rue or False Every quadrilateral is a rue or False ll angles of a are congruent rue or False ll sides of a are congruent rue or False White oard Practice White oard Practice In, if m = 50, then m = 10. Given Hint draw a picture raw the with the diagonals intersecting at E Use different tick marks to show all the segments that are congruent

3 White oard Groups Quadrilateral U is a. Find the values of x, y, a, and b. x = 80 y = 100 a = 6 b = 9 U 9 yº 80º a 6 xº b White oard Groups Quadrilateral U is a. Find the values of x, y, a, and b. U 1 45º 5º xº 9 b yº a x = 100 y = 45 a = 1 b = 9 Given this with the diagonals drawn. x = 5 y = 6 White oard Groups 5-:Ways to Prove a Quad is a Parallelogram Learn about ways to prove a quadrilateral is a What we already know If a quad is a, then 5 properties What we are going to learn.. What if we don t know if a quad is a, how can we prove that it is one? Its Friday night, you and your quad friends try to get into the club If my quad has (insert any of the statements)_, then it is a. 1. both pairs of opposite sides parallel. both pairs of opposite sides congruent. both pairs of opposite angles congruent 4. diagonals that bisect each other 5. one pair of opposite sides are both congruent and parallel he diagonals of a quadrilateral bisect each other. ometimes. lways. Never. I don t know If the measure of two angles of a quadrilateral are equal, then the quadrilateral is a ) ometimes ) lways ) Never ) I don t know ake chart in notes w/ diagram

4 If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a. ometimes. lways. Never. I don t know If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a.) ometimes.) lways.) Never.) I don t know Whiteboards Open book to page 17 nswer the following # # #6 #9 5- heorems Involving Parallel Lines pply the theorems about parallel lines and triangles heorem (noodle theorem) If three parallel lines cut off congruent segments on one transversal, then they do so on any transversal. E F 1. = x ; = 11. = 1x ; = x +40 White oard Practice U heorem (skip) If two lines are parallel, then all points on one line are equidistant from the other line. emo: 6 volunteers How do we measure the distance from a point to a line? m heorem (skip) line that contains the midpoint of one side 1 of a triangle and is parallel to a another side passes through the midpoint of the third side. heorem segment that joins the midpoints of two sides 1 of a triangle is parallel to the third side and its length is half the length of the third side. If is 1 then =? 1 1 What does equidistant mean? n 4

5 White oard Practice Given:,, and are midpoint of the sides of White oard Practice Given:,, and are midpoint of the sides of is parallel to what side? is parallel to what side? pecial Parallelograms pply the definitions and identify the special properties of a rectangle, rhombus and square. QUILEL ectangle hombus quare Parallelograms: What we now know If a quad is a, then From the definition.. 1. opposite sides are parallel From theorems 1. onsecutive angles are supplementary. opposite angles are congruent. opposite sides are congruent 4. he diagonals of a bisect each other ectangle y definition, it is a quadrilateral with four right angles. V hombus y definition, it is a quadrilateral with four congruent sides. quare y definition, it is a quadrilateral with four right angles and four congruent sides. he square is the most specific type of quadrilateral. What do you notice about the definition compared to the previous two? 5

6 heorem he diagonals of a rectangle are congruent. W Z What can we conclude about the smaller segments that make up the diagonals? W Z Finding the special properties of a hombus pply the properties of a to find special properties that apply to the hombus. Hint: both properties involve angles. heorem he diagonals of a rhombus are perpendicular. K P J L What does the definition of perpendicular lines tell us? heorem Each diagonal of a rhombus bisects the opposite angles. K heorem he midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. J L Z Z iscuss special angle relationships!! White oard Practice White oard Practice White oard Practice Quadrilateral is a rhombus Find the measure of each angle 1.. E. E E 4. 6º Quadrilateral is a rhombus Find the measure of each angle 1. = 6. E = 90. E = 8 E 4. = 56 6º Quadrilateral NOP is a rectangle Find the measure of each angle 1. m PON = 9º. m PO =. PL = L 4. O = P 1 N O 6

7 White oard Practice Quadrilateral NOP is a rectangle Find the measure of each angle 1. m PON = 90 9º. m PO = 61. PL = 1 L 4. O = 4 1 N White oard Practice is a right ; is the midpoint of 1. If = 7, then =, =, and =.. ml1 = 40, find the rest lways. ometimes. Never. I don t know square is a rhombus P O 1. lways. ometimes. Never. I don t know he diagonals of a bisect the angles of the.. lways. ometimes. Never. I don t know he diagonals of a rhombus are congruent.. lways. ometimes. Never. I don t know he diagonals of a rhombus bisect each other. 5.5 rapezoids pply the definitions and learn the properties of a trapezoid and an isosceles trapezoid. rapezoid quadrilateral with exactly one pair of parallel sides. rap. How does this definition differ from that of a? natomy Of a rapezoid he bases are the parallel sides ase 1 pair of base angles nd pair of base angles V ase 7

8 natomy Of a rapezoid he legs are the non-parallel sides Isosceles rapezoid trapezoid with congruent legs. heorem he base angles of an isosceles trapezoid are congruent. F G Leg V Leg What do do you you think would the happen definition if I is folded based this on figure the in diagram? half? upplementary E What is something I can conclude about of the angles (other than congruency) based on the markings of the diagram? upplementary H he edian of a rapezoid segment that joins the midpoints of the legs. he median of a trapezoid is parallel to the bases and its length is the average of the bases. + = White oard Practice omplete 1. = 5, = 1, = 19 Note: this applies to any trapezoid How do we find an average of the bases? White oard Practice omplete. = 9, = 4, = 19 White oard Practice omplete 4. = 7y + 6, = 5y -, = y 5, y =.5 One angle of an isosceles trap is 40. Find the other angle measures. 8