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Chapter 5 Quadrilaterals • Apply the definition of a parallelogram • Prove that certain quadrilaterals are parallelograms • Apply the theorems and definitions about the special quadrilaterals 5-1 Properties of Parallelograms Objectives • Apply the definition of a parallelogram • List the other properties of a parallelogram through new theorems Quadrilaterals • Any 4 sided figure Definition of a Parallelogram ( ) If the opposite sides of a quadrilateral are parallel, then it is a parallelogram. ABCD A D B C Naming a Parallelogram Use the symbol for parallelogram and name using the 4 vertices in order either clockwise or counter clockwise. ABCD A D B C Theorem Opposite sides of a parallelogram are congruent. A D B C Theorem Opposite angles of a parallelogram are congruent. A D B C Theorem The diagonals of a parallelogram bisect each other. A D B C Remote Time • True or False True or False • Every parallelogram is a quadrilateral True or False • Every quadrilateral is a parallelogram True or False • All angles of a parallelogram are congruent True or False • All sides of a parallelogram are congruent True or False • In RSTU, RS | |TU. Hint draw a picture True or False • In ABCD, if m A = 50, then m C = 130. Hint draw a picture True or False • In XWYZ, XY WZ Hint draw a picture True or False • In ABCD, AC and BD bisect each other Hint draw a picture White Board Practice Given ABCD Name all pairs of parallel sides White Board Practice Given AB || DC BC || AD ABCD White Board Practice Given ABCD Name all pairs of congruent angles White Board Practice Given ABCD BAD DCB ABC CDA BEA DEC BEC DEA CBD ADB ABD CDB BCA DAC BAC DCA White Board Practice Given ABCD Name all pairs of congruent segments White Board Practice Given AB CD BC DA BE ED AE EC ABCD White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. 6 R xº S yº 9 b 80º U a T White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. x = 80 y = 45 a=6 b=9 White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. R S xº yº 9 12 a b U 45º 35º T White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. x = 100 y = 45 a = 12 b=9 White Board Groups • Given this parallelogram with the diagonals drawn. White Board Groups • Given this parallelogram with the diagonals drawn. x=5 y=6 5-2:Ways to Prove that Quadrilaterals are Parallelograms Objectives • Learn about ways to prove a quadrilateral is a parallelogram Use the Definition of a Parallelogram • Show that both pairs of opposite sides of a quadrilateral are parallel • Then the quadrilateral is a parallelogram A D B C Theorem • Show that both pairs of opposite sides are congruent. • If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram. A D B C Theorem • Show that one pair of opposite sides are both congruent and parallel. • If one pair of opposite sides of a quadrilateral are both congruent and parallel, then it is a parallelogram. A D B C Theorem • Show that both pairs of opposite angles are congruent. • If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. A D B C Theorem • Show that the diagonals bisect each other. • If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. A B X D C Five ways to prove a Quadrilateral is a Parallelogram • Show that both pairs of opposite sides parallel • Show that both pairs of opposite sides congruent • Show that one pair of opposite sides are both congruent and parallel • Show that both pairs of opposite angles congruent • Show that diagonals that bisect each other The diagonals of a quadrilateral _____________ bisect each other A. B. C. D. Sometimes Always Never I don’t know If the measure of two angles of a quadrilateral are equal, then the quadrilateral is ____________ a parallelogram A) B) C) D) Sometimes Always Never I don’t know If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is ___________ a parallelogram A. B. C. D. Sometimes Always Never I don’t know If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is __________ a parallelogram A.) B.) C.) D.) Sometimes Always Never I don’t know To prove a quadrilateral is a parallelogram, it is ________ enough to show that one pair of opposite sides is parallel. A.) B.) C.) D.) Sometimes Always Never I don’t know 5-3 Theorems Involving Parallel Lines Objectives • Apply the theorems about parallel lines and triangles Theorem If two lines are parallel, then all points on one line are equidistant from the other. m n Theorem If three parallel lines cut off congruent segments on one transversal, then they do so on any transversal. A D B E C F Theorem A line that contains the midpoint of one side of a triangle and is parallel to a another side passes through the midpoint of the third side. A X B Y C Theorem A segment that joins the midpoints of two sides of a triangle is parallel to the third side and its length is half the length of the third side. A X B Y C White Board Practice • Given: R, S, and T are midpoint of the sides of ABC AB BC AC ST RT RS 12 14 B 18 R 15 10 22 S 10 9 7.8 A T C White Board Practice • Given: R, S, and T are midpoint of the sides of ABC AB BC AC ST RT RS 12 14 18 6 7 B 9 R 20 15 22 10 10 18 15.6 5 S 7.5 11 9 7.8 A T C White Board Practice • Given that AR | | BS | | CT; RS ST R S A T B C White Board Practice • Given that AR | | BS | | CT; RS ST If RS = 12, then ST = ____ R A S T B C White Board Practice • Given that AR | | BS | | CT; RS ST If RS = 12, then ST = 12 R A S T B C White Board Practice • Given that AR | | BS | | CT; RS ST If AB = 8, then BC = ___ R A S T B C White Board Practice • Given that AR | | BS | | CT; RS ST If AB = 8, then BC = 8 R A S T B C White Board Practice • Given that AR | | BS | | CT; RS ST If AC = 20, then AB = ___ R A S T B C White Board Practice • Given that AR | | BS | | CT; RS ST If AC = 20, then AB = 10 R A S T B C White Board Practice • Given that AR | | BS | | CT; RS ST If AC = 10x, then BC =____ R A S T B C White Board Practice • Given that AR | | BS | | CT; RS ST If AC = 10x, then BC = 5x R A S T B C 5.4 Special Parallelograms Objectives • Apply the definitions and identify the special properties of a rectangle, rhombus and square. Rectangle By definition, it is a quadrilateral with four right angles. R S V T Rhombus By definition, it is a quadrilateral with four congruent sides. B C A D Square By definition, it is a quadrilateral with four right angles and four congruent sides. B A C D Theorem The diagonals of a rectangle are congruent. WY XZ W Z P X Y Theorem The diagonals of a rhombus are perpendicular. K X J L M Theorem Each diagonal of a rhombus bisects the opposite angles. K X J L M Theorem The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. A X B C Theorem If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. R S V T Theorem If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. B C A D White Board Practice • Quadrilateral ABCD is a rhombus Find the measure of each angle A 1. ACD 2. DEC E 3. EDC 4. ABC B 62º D C White Board Practice • Quadrilateral ABCD is a rhombus Find the measure of each angle A 1. ACD = 62 2. DEC = 90 E 3. EDC = 28 4. ABC = 56 B 62º D C White Board Practice • Quadrilateral MNOP is a rectangle Find the measure of each angle 1. m PON = M 29º 2. m PMO = L 3. PL = 4. MO = P N 12 O White Board Practice • Quadrilateral MNOP is a rectangle Find the measure of each angle 1. m PON = 90 M 29º 2. m PMO = 61 L 3. PL = 12 4. MO = 24 P N 12 O White Board Practice • ABC is a right ; M is the midpoint of AB 1. If AM = 7, then MB = ____, AB = ____, and CM = _____ . A M C B White Board Practice • ABC is a right ; M is the midpoint of AB 1. If AM = 7, then MB = 7, AB = 14, and CM = 7 . A M C B White Board Practice • ABC is a right ; M is the midpoint of AB 1. If AB = x, then AM = ____, MB = _____, and MC = _____ . A M C B White Board Practice • ABC is a right ; M is the midpoint of AB 1. If AB = x, then AM = ½ x, MB = ½ x, and MC = ½ x . A M C B Remote Time A. B. C. D. Always Sometimes Never I don’t know A. B. C. D. Always Sometimes Never I don’t know • A square is ____________ a rhombus A. B. C. D. Always Sometimes Never I don’t know • The diagonals of a parallelogram ____________ bisect the angles of the parallelogram. A. B. C. D. Always Sometimes Never I don’t know • A quadrilateral with one pairs of sides congruent and one pair parallel is _________________ a parallelogram. A. B. C. D. Always Sometimes Never I don’t know • The diagonals of a rhombus are ___________ congruent. A. B. C. D. Always Sometimes Never I don’t know • A rectangle ______________ has consecutive sides congruent. A. B. C. D. Always Sometimes Never I don’t know • A rectangle ____________ has perpendicular diagonals. A. B. C. D. Always Sometimes Never I don’t know • The diagonals of a rhombus ___________ bisect each other. A. B. C. D. Always Sometimes Never I don’t know • The diagonals of a parallelogram are ____________ perpendicular bisectors of eah other. 5.5 Trapezoids Objectives • Apply the definitions and learn the properties of a trapezoid and an isosceles trapezoid. Trapezoid A quadrilateral with exactly one pair of parallel sides. Trap. ABCD A B C D Anatomy Of a Trapezoid • The bases are the parallel sides Base R S V T Base Anatomy Of a Trapezoid • The legs are the non-parallel sides R S Leg Leg V T Isosceles Trapezoid A trapezoid with congruent legs. J M K L Theorem 5-18 The base angles of an isosceles trapezoid are congruent. F E G H The Median of a Trapezoid A segment that joins the midpoints of the legs. B X A C Y D Theorem The median of a trapezoid is parallel to the bases and its length is the average of the bases. B X A A C Y D White Board Practice • Complete 1. AD = 25, BC = 13, XY = ______ B X A C Y D White Board Practice • Complete 1. AD = 25, BC = 13, XY = 19 B X A C Y D White Board Practice • Complete 2. AX = 11, YD = 8, AB = _____, DC = ____ B X A C Y D White Board Practice • Complete 2. AX = 11, YD = 8, AB = 22, DC = 16 B X A C Y D White Board Practice • Complete 3. AD = 29, XY = 24, BC =______ B X A C Y D White Board Practice • Complete 3. AD = 29, XY = 24, BC =19 B X A C Y D White Board Practice • Complete 4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y =____ B X A C Y D White Board Practice • Complete 4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y = 3.5 B X A C Y D Homework Set 5.5 • WS PM 28 • 5-5 #1-27 odd • Quiz next class day

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