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 AB C D

Naming a Parallelogram Use the symbol for parallelogram and name using the 4 vertices in order either clockwise or counter clockwise. ABCD AB C D

Opposite sides of a parallelogram are congruent. Theorem AB C D

Opposite angles of a parallelogram are congruent. Theorem AB C D

The diagonals of a parallelogram bisect each other. Theorem AB C D

Remote Time 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 ABCD AB || DC BC || AD

White Board Practice Given ABCD Name all pairs of congruent angles

White Board Practice Given ABCD  BAD   DCB  CBD   ADB  ABC   CDA  ABD   CDB  BEA   DEC  BCA   DAC  BEC   DEA  BAC   DCA

White Board Practice Given ABCD Name all pairs of congruent segments

White Board Practice Given ABCD AB  CD BC  DA BE  ED AE  EC

Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. White Board Groups a R U T S 9 b 6 yºyº 80º xºxº

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. White Board Groups a R U T S 9 b 12 yºyº 35º xºxº 45º

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 18 2x + 8 4y

Given this parallelogram with the diagonals drawn. x = 5 y = 6 White Board Groups

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 AB C D

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 B C D

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 B C D

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. AB C D

Theorem Show that the diagonals bisect each other. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. AB C D X

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. Sometimes B.Always C.Never D.I don’t know

If the measure of two angles of a quadrilateral are equal, then the quadrilateral is ____________ a parallelogram A)Sometimes B)Always C)Never D)I don’t know

If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is ___________ a parallelogram A. Sometimes B. Always C. Never D. I don’t know

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is __________ a parallelogram A.) Sometimes B.) Always C.) Never D.) 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.) Sometimes B.) Always C.) Never D.) 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 B C D E 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 BC XY

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 BC XY

White Board Practice Given: R, S, and T are midpoint of the sides of  ABC C A B T R S ABBCACSTRTRS

White Board Practice Given: R, S, and T are midpoint of the sides of  ABC C A B T R S ABBCACSTRTRS

White Board Practice Given that AR | | BS | | CT; RS  ST A B C T S R

White Board Practice Given that AR | | BS | | CT; RS  ST If RS = 12, then ST = ____ A B C T S R

White Board Practice Given that AR | | BS | | CT; RS  ST If RS = 12, then ST = 12 A B C T S R

White Board Practice Given that AR | | BS | | CT; RS  ST If AB = 8, then BC = ___ A B C T S R

White Board Practice Given that AR | | BS | | CT; RS  ST If AB = 8, then BC = 8 A B C T S R

White Board Practice Given that AR | | BS | | CT; RS  ST If AC = 20, then AB = ___ A B C T S R

White Board Practice Given that AR | | BS | | CT; RS  ST If AC = 20, then AB = 10 A B C T S R

White Board Practice Given that AR | | BS | | CT; RS  ST If AC = 10x, then BC =____ A B C T S R

White Board Practice Given that AR | | BS | | CT; RS  ST If AC = 10x, then BC = 5x A B C T S R

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 T V

Rhombus By definition, it is a quadrilateral with four congruent sides. A B C D

Square By definition, it is a quadrilateral with four right angles and four congruent sides. A B C D

Theorem The diagonals of a rectangle are congruent. WY  XZ W XY Z P

Theorem The diagonals of a rhombus are perpendicular. J K L M X

Theorem Each diagonal of a rhombus bisects the opposite angles. J K L M X

Theorem The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. A BC X

Theorem If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. R S T V

Theorem If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. A B C D

White Board Practice Quadrilateral ABCD is a rhombus Find the measure of each angle 1.  ACD 2.  DEC 3.  EDC 4.  ABC D A B C E 62º

White Board Practice Quadrilateral ABCD is a rhombus Find the measure of each angle 1.  ACD =  DEC =  EDC =  ABC = 56 D A B C E 62º

White Board Practice Quadrilateral MNOP is a rectangle Find the measure of each angle 1. m  PON = 2. m  PMO = 3. PL = 4. MO = P MN O L 29º 12

White Board Practice Quadrilateral MNOP is a rectangle Find the measure of each angle 1. m  PON = m  PMO = PL = MO = 24 P MN O L 29º 12

White Board Practice  ABC is a right  ; M is the midpoint of AB 1. If AM = 7, then MB = ____, AB = ____, and CM = _____. C A B M

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. C A B M

White Board Practice  ABC is a right  ; M is the midpoint of AB 1. If AB = x, then AM = ____, MB = _____, and MC = _____. C A B M

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. C A B M

Remote Time A.Always B.Sometimes C.Never D.I don’t know

A. Always B. Sometimes C. Never D. I don’t know A square is ____________ a rhombus

A. Always B. Sometimes C. Never D. I don’t know The diagonals of a parallelogram ____________ bisect the angles of the parallelogram.

A. Always B. Sometimes C. Never D. I don’t know A quadrilateral with one pairs of sides congruent and one pair parallel is _________________ a parallelogram.

A. Always B. Sometimes C. Never D. I don’t know The diagonals of a rhombus are ___________ congruent.

A. Always B. Sometimes C. Never D. I don’t know A rectangle ______________ has consecutive sides congruent.

A. Always B. Sometimes C. Never D. I don’t know A rectangle ____________ has perpendicular diagonals.

A. Always B. Sometimes C. Never D. I don’t know The diagonals of a rhombus ___________ bisect each other.

A. Always B. Sometimes C. Never D. 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. A B C D Trap. ABCD

Anatomy Of a Trapezoid R S TV Base The bases are the parallel sides

Anatomy Of a Trapezoid R S TV Leg The legs are the non-parallel sides

Isosceles Trapezoid A trapezoid with congruent legs. J KL M

Theorem 5-18 The base angles of an isosceles trapezoid are congruent. E F G H

The Median of a Trapezoid A segment that joins the midpoints of the legs. A B C D X Y

Theorem The median of a trapezoid is parallel to the bases and its length is the average of the bases. B C D X Y A A B C D X Y

White Board Practice B C D X Y A Complete 1. AD = 25, BC = 13, XY = ______

White Board Practice B C D X Y A Complete 1. AD = 25, BC = 13, XY = 19

White Board Practice B C D X Y A Complete 2. AX = 11, YD = 8, AB = _____, DC = ____

White Board Practice B C D X Y A Complete 2. AX = 11, YD = 8, AB = 22, DC = 16

White Board Practice B C D X Y A Complete 3. AD = 29, XY = 24, BC =______

White Board Practice B C D X Y A Complete 3. AD = 29, XY = 24, BC =19

White Board Practice B C D X Y A Complete 4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y =____

White Board Practice B C D X Y A Complete 4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y = 3.5

Homework Set 5.5 WS PM #1-27 odd Quiz next class day