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

He’s Back…. Parallelograms are special types of quadrilaterals with unique properties If you know you have a parallelogram, then you can prove that these unique properties exist… With each property we learn, say the following to yourself.. “If a quadrilateral is a parallelogram, then _____________.”

Definition of a Parallelogram ( ) What do you think the definition is based on the diagram? If a quadrilateral is a parallelogram, then opposite sides are parallel. ABCD A B C D Partners: What do we know about the angles of a parallelogram b/c it has parallel sides? What do we know a bout a quadrilateral besides the fact that it has four sides and four angles? What do we know about a parallelogram because it has parallel sides? *consecutive angles = 180

Naming a Parallelogram Use the symbol for parallelogram and name using the 4 vertices in order either clockwise or counter clockwise. ABCD A B C D What do we know a bout a quadrilateral besides the fact that it has four sides and four angles? What do we know about a parallelogram because it has parallel sides?

A B C D The fact that we know opposite sides are parallel, we can deduce addition properties through theorems…

Theorem What did we discuss at the beginning of the lesson about s-s int. angles? Opposite angles of a parallelogram are congruent. A B Do the proof. Note that it is only possible to do the angles one pair at a time. D C

Theorem Opposite sides of a parallelogram are congruent. A B D Do the proof. D What would be our plan for solving this theorem? C

Theorem The diagonals of a parallelogram bisect each other. A B D Do the proof. D What is another name for AC and BD? C

Parallelograms: What we now know… If a quad is a parallelogram, then… From the definition.. opposite sides are parallel From theorems… Consecutive angles = 180 opposite angles are congruent opposite sides are congruent The diagonals of a parallelogram bisect each other

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 ABCD, if m  A = 50, then m  C = 130.  Hint draw a picture

White Board Practice Given ABCD Draw the parallelogram with the diagonals intersecting at E Use different tick marks to show all the segments that are congruent

White Board Practice A B D C

White Board Groups Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. 6 x = 80 y = 100 a = 6 b = 9 R S yº xº 9 b 80º T U a

White Board Groups Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. R S x = 100 y = 45 a = 12 b = 9 yº xº 9 12 a b 45º 35º U T

White Board Groups Given this parallelogram with the diagonals drawn. x = 5 y = 6 22 18 4y - 2 2x + 8

5-2:Ways to Prove a Quad is a Parallelogram Objectives Learn about ways to prove a quadrilateral is a parallelogram Most of today's statements are the converses of yesterdays. The properties of a parallelogram can be used to prove that a quadrilateral is a parallelogram.

What we already know… If a quad is a parallelogram, then… 5 properties What we are going to learn.. What if we don’t know if a quad is a parallelogram, how can we prove that it is one?

Its Friday night, you and your “quad” friends try to get into the parallelogram club…

“If my quad has __(insert any of the statements)_, then it is a parallelogram. both pairs of opposite sides parallel both pairs of opposite sides congruent both pairs of opposite angles congruent diagonals that bisect each other one pair of opposite sides are both congruent and parallel While all of these are useful, and most proofs offer you the choice of method, the third one (Th. 5-5) is the best to use, if possible, because it only involves one pair of sides. Make chart in notes w/ diagram

The diagonals of a quadrilateral _____________ bisect each other A. Sometimes Always Never I don’t know

If the measure of two angles of a quadrilateral are equal, then the quadrilateral is ____________ a parallelogram 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. 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

Whiteboards Open book to page 173 Answer the following… #2 #3 #6 #9

5-3 Theorems Involving Parallel Lines Objectives Apply the theorems about parallel lines and triangles

Theorem (noodle theorem) If three parallel lines cut off congruent segments on one transversal, then they do so on any transversal. A D B A good memory device for this theorem is to call it the “Knife Theorem” because it is like a chefs knife, always cutting uniformly, but cutting different things into different shapes. E C F

White Board Practice BD = 3x – 2; AB = 11 AC = 12x ; BD = 2x +40 T A U S T A U B C D

Theorem (skip) If two lines are parallel, then all points on one line are equidistant from the other line. Demo: 6 volunteers How do we measure the distance from a point to a line? This is another way of defining parallel lines. m n What does equidistant mean?

Theorem (skip) 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 . 1 2 3 A 3 A good device for this theorem is the “1-2-3 Theorem” because all three sides of the triangle are involved. X Y 1 2 B 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. 1 2 3 A If BC is 12 then XY =? 2 This is the converse to the “1-2-3 Theorem”. Its proof is interesting, especially the part about being half the length. X Y 1 3 B C

White Board Practice AB BC AC ST RT RS 12 14 18 15 22 10 9 7.8 Given: R, S, and T are midpoint of the sides of  ABC AB BC AC ST RT RS 12 14 18 15 22 10 9 7.8 B R S C A T

White Board Practice AB BC AC ST RT RS 12 14 18 6 7 9 20 15 22 10 7.5 Given: R, S, and T are midpoint of the sides of  ABC AB BC AC ST RT RS 12 14 18 6 7 9 20 15 22 10 7.5 11 15.6 5 7.8 B R S C A T

ST is parallel to what side? B ST is parallel to what side? BC is parallel to what side? R S C A T

5.4 Special Parallelograms Objectives Apply the definitions and identify the special properties of a rectangle, rhombus and square.

QUADRILATERALS parallelogram Rhombus Rectangle Square

Parallelograms: What we now know… If a quad is a parallelogram, then… From the definition.. opposite sides are parallel From theorems… Consecutive angles are supplementary opposite angles are congruent opposite sides are congruent The diagonals of a parallelogram bisect each other

Rectangle By definition, it is a quadrilateral with four right angles. V While basic, all these definitions also include all of the properties of a parallelogram. Focus not only on what a rectangle has in common with a parallelogram, but what distinguishes them as well. S T

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

Square By definition, it is a quadrilateral with four right angles and four congruent sides. C B What do you notice about the definition compared to the previous two? The square is the most specific type of quadrilateral. The square is the most specific type of quadrilateral. It shares all the properties of a parallelogram, a rectangle and a rhombus. D A

Theorem The diagonals of a rectangle are congruent. WY  XZ What can we conclude about the smaller segments that make up the diagonals? W Z P Prove this. X Y

Finding the special properties of a Rhombus Apply the properties of a parallelogram to find 2 special properties that apply to the Rhombus. Hint: both properties involve angles.  

Theorem The diagonals of a rhombus are perpendicular. K X J L Prove this. Show the difference between a kite and a rhombus. What does the definition of perpendicular lines tell us? M

Theorem Each diagonal of a rhombus bisects the opposite angles. K X J Prove this if there is time. L M

X M Z Y

Theorem The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. X M Be sure to prove this. Do an extensive analysis as it involves some isosceles triangles as well. Z Y Discuss special angle relationships!!

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

White Board Practice Quadrilateral ABCD is a rhombus Find the measure of each angle 1.  ACD = 62 2.  DEC = 90 3.  EDC = 28 4.  ABC = 56 A B E 62º D C

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

White Board Practice Quadrilateral MNOP is a rectangle Find the measure of each angle 1. m  PON = 90 2. m  PMO = 61 3. PL = 12 4. MO = 24 M N 29º 12 L P O

White Board Practice  ABC is a right ; M is the midpoint of AB 1. If AM = 7, then MB = ____, AB = ____, and CM = _____ . mL1 = 40, find the rest. A 4 M 5 3 B C 2 1

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 The diagonals of a rhombus are ___________ congruent.

A. Always B. Sometimes C. Never D. I don’t know The diagonals of a rhombus ___________ bisect each 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. B C Trap. ABCD C There is no symbol for trapezoid. Be sure to note that the definitions of a parallelogram and a trapezoid make them mutually exclusive. How does this definition differ from that of a parallelogram? A D

Anatomy Of a Trapezoid The bases are the parallel sides Base R S V T 1 pair of base angles This slide allows you to define the bases, legs and base angles of a trapezoid. Animate the parts and use the smart pens to write the labels at the appropriate time. 2nd pair of base angles V T Base

Anatomy Of a Trapezoid The legs are the non-parallel sides R S Leg Leg This slide allows you to define the bases, legs and base angles of a trapezoid. Animate the parts and use the smart pens to write the labels at the appropriate time. Leg V T

Isosceles Trapezoid A trapezoid with congruent legs. What do you think the definition is based on the diagram? What do you think would happen if I folded this figure in half? Point out that the bases cannot be congruent and the legs cannot be parallel.

Theorem The base angles of an isosceles trapezoid are congruent. F G E Supplementary Supplementary There are two pairs of base angles, and in addition to being congruent, they are supplementary to the other pair. What is something I can conclude about 2 of the angles (other than congruency) based on the markings of the diagram? E H

The Median of a Trapezoid A segment that joins the midpoints of the legs. B C While the median of a triangle connects the vertex to the midpoint of the opposite side, a trapezoid has no vertex opposite each side, it has an equal number of sides. X Y Note: this applies to any trapezoid A D

The median of a trapezoid is parallel to the bases and its length is the average of the bases. AD + BC 2 = XY B B C C Both of these things appear to be true about a median. Focus on that and try to get them to see that its length is the average. The sketch shows the relationship in a movable diagram. X X Y Y How do we find an average of the bases ? A D D

White Board Practice Complete 1. AD = 25, BC = 13, XY = ______ 19 B C

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

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

One angle of an isosceles trap is 40. Find the other 3 angle measures.