Quadrilaterals and Polygons Polygon: A plane figure that is formed by three or more segments (no two of which are collinear), and each segment (side) intersects at exactly two other sides – one at each endpoint (Vertex). Which of the following diagrams are polygons?

# of Sides Type of Polygon 3 4 5 6 7 # of Sides Type of Polygon 8 9 10 Polygons are Named & Classified by the Number of Sides They Have # of Sides Type of Polygon 3 4 5 6 7 # of Sides Type of Polygon 8 9 10 12 # Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodagon N-gon What type of polygons are the following?

Convex and Concave Polygons Convex – A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. Concave – A polygon that is not convex Interior Interior Equilateral, Equiangular, and Regular

Diagonals and Interior Angles of a Quadrilateral Diagonal – a segment that connects to non-consecutive vertices. Theorem 6.1 – Interior Angles of a Quadrilateral Theorem The sum of the measures of the interior angles of a quadrilateral is 360O m<1 + m<2 + m<3 + m<4 = 360o 80o 70o xo 2xo

Properties of Parallelograms Theorem 6.2 If a quadrilateral is a parallelogram, then its opposite sides are congruent. PQ = RS and SP = QR Theorem 6.3 If a quadrilateral is a parallelogram, then it opposite angles are congruent. <P = < R and < Q = < S Theorem 6.4 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. m<P + m<Q = 180o, m<Q + m<R = 180o m<R + m<S = 180o, m<S + m<P = 180o Theorem 6.5 If a quadrilateral is a parallelogram, then its diagonals bisect each other. QM = SM and PM = RM _ ~ Q R P S

Using the Properties of Parallelograms FGHJ is a parallelogram. Find the length of: a. JH b. JK F 5 G K 3 J H PQRS is a parallelogram. Find the angle measures: a. m<R b. m<Q Q R 70o P S Find the value of x P Q 3xo 120o S R

Proofs Involving Parallelograms A E B 2 1 D C 3 G F Given: ABCD and AEFG are parallelograms Prove: <1 = < 3 Statements Reasons ~ ABCD & AEFG are Parallelograms 1. Given <1 = < 2 2. Opposite Angles are congruent (6.3) <2 = <3 3. Opposite Angles are Congruent (6.3) <1 = <3 4. Transitive Property of Congruence ~ Plan: Show that both angles are congruent to <2

Given: ABCD is a parallelogram Prove: AB = CD, AD = CB Proving Theorem 6.2 A B D C Given: ABCD is a parallelogram Prove: AB = CD, AD = CB Statements Reasons ~ _ ABCD is a parallelogram 1. Given Draw Diagonal BD 2. Through any two points there exists exactly one line 3. AB || CD, and AD || CB 3. Def. of a parallelogram __ 4. <ABD = < CDB 4. Alternate Interior Angles Theorem 5. <ADB = < CBD 5. Alternate Interior Angles Theorem 6. DB = DB 6. Reflexive Property of Congruence 7. /\ ADB = /\ CBD 7. ASA Congruence Postulate 8. AB = CD, AD = CB 8. CPCTC __ ~ Plan: Insert a diagonal which will allow us to divide the parallelogram into two triangles

Proving Quadrilaterals are Parallelograms Theorem 6.6 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram Q R P S Q R P S Theorem 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram Q R P S (180-x)o xo xo Theorem 6.8 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram Q R P S Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

Concept Summary – Proving Quadrilaterals are Parallelograms Show that both pairs of opposite sides are Show that both pairs of opposite angles are Show that one angle is supplementary to Show that the diagonals Show that one pair of opposite sides are both congruent parallel BOTH consecutive interior <‘s bisect each other congruent and ||

Proving Quadrilaterals are Parallelograms – Coordinate Geometry B(1,3) D (7,1) A(2, -1) How can we prove that the Quad is a parallelogram? 1. Slope - Opposite Sides || 2. Length (Distance Formula) – Opposite sides same length 3. Combination – Show One pair of opposite sides both || and congruent

Rhombuses, Rectangles, and Squares Square – a parallelogram with four congruent sides and four right angles Rhombus – a parallelogram with four congruent sides Rectangle – a parallelogram with four right angles Parallelograms Rhombuses Rectangles Squares

Using Properties of Special Triangles If ABCD is a rectangle, what else do you know about ABCD? A B C D Corollaries about Special Quadrilaterals Rhombus Corollary – A quad is a rhombus if and only if it has four congruent sides Rectangle Corollary – A quad is a rectangle if and only if it has four right angles Square Corollary – A quad is a square if and only if it is a rhombus and a rectangle How can we use these special properties and corollaries of a Rhombus? P Q S R 2y + 3 5y - 6

Theorem 6.11: Theorem 6.12: Theorem 6.13: Using Diagonals of Special Parallelograms Theorem 6.11: A parallelogram is a rhombus if and only if its diagonals are perpendicular Theorem 6.12: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Theorem 6.13: A parallelogram is a rectangle if and only if its diagonals are congruent

Decide if the statement is sometimes, always, or never true. A rhombus is equilateral. 2. The diagonals of a rectangle are _|_. 3. The opposite angles of a rhombus are supplementary. 4. A square is a rectangle. 5. The diagonals of a rectangle bisect each other. 6. The consecutive angles of a square are supplementary. Always Sometimes Quadrilateral ABCD is Rhombus. 7. If m <BAE = 32o, find m<ECD. 8. If m<EDC = 43o, find m<CBA. 9. If m<EAB = 57o, find m<ADC. 10. If m<BEC = (3x -15)o, solve for x. 11. If m<ADE = ((5x – 8)o and m<CBE = (3x +24)o, solve for x 12. If m<BAD = (4x + 14)o and m<ABC = (2x + 10)o, solve for x. A B E D C 32o 86o 66o 35o 16 26

Coordinate Proofs Using the Properties of Rhombuses, Rectangles and Squares Using the distance formula and slope, how can we determine the specific shape of a parallelogram? Rhombus – Rectangle – Square - Based on the following Coordinate values, determine if each parallelogram is a rhombus, a rectangle, or square. P (-2, 3) P(-4, 0) Q(-2, -4) Q(3, 7) R(2, -4) R(6, 4) S(2, 3) S(-1, -3)

Given: HIJK is a parallelogram /\ HOI = /\ JOI Prove: HIJK is a Rhombus Statements Reasons H I O K J ~

Given: RECT is a Rectangle Prove: /\ ART = /\ ACE Statements Reasons ~ T C

Given: PQRT is a Rhombus Prove: PR bisects <TPQ and < QRT, and QT bisects <PTR and <PQP Statements Reasons P Q T R Plan: First prove that Triangle PRQ is congruent to Triangle PRT; and Triangle TPQ is congruent to Triangle TRQ

Trapezoids and Kites A B D C > A Trapezoid is a Quad with exactly one pair of parallel sides. The parallel sides are the BASES. A Trapezoid has exactly two pairs of BASE ANGLES In trapezoid ABCD, Which 2 sides are the bases? The legs? Name the pairs of base angles. A B D C > If the legs of the trapezoid are congruent, then the trapezoid is an Isosceles Trapezoid.

Theorems of Trapezoids If a trapezoid is isosceles, then each pair of base angles is congruent. <A = <B = <C = <D ~ A B D C > Theorem 6.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. ABCD is an isosceles trapezoid A B D C > ~ Theorem 6.16 A trapezoid is isosceles if and only if its diagonals are congruent. ABCD is isosceles if and only if AC = BD _ A B D C >

Kites and Theorems about Kites A kite is a quadrilateral that has two pairs of consecutive congruent sides, But opposite sides are NOT congruent. Theorem 6.18 If a Quad is a Kite, then its diagonals are perpendicular. Theorem 6.19 If a Quad is a kite then exactly one pair of opposite angles are congruent

Using the Properties of a Kite X 12 20 U 12 W Y Z Find the length of WX, XY, YZ, and WZ. J H 132o 60o K G Find the angle measures of <HJK and < HGK

Summarizing the Properties of Quadrilaterals ______________ _________________ ________________ ____________ _____________ ____________ ______________ Kites Parallelograms Trapezoids Rhombus Squares Rectangles Isosceles Trap.

Properties of Quadrilaterals X X X X X X X X X X X X X X X X X X X X X X X X

Area of a Square Postulate Using Area Formulas Area of a Square Postulate The area of a square is the square of the length of its side. Area Congruence Postulate If two polygons are congruent then they have the same area. Area Addition Postulate The area of a region is the sum of the area of its non-overlapping sides. Area of a Rectangle The area of a rectangle is the product of its base and height. A = bh Area of a Parallelogram The area of a parallelogram is the product of a base, and it’s corresponding height Area of a Triangle The area of a triangle is one half the product of a base and its corresponding height A = ½bh h b

Name That Proof

Prove: MRON is a parallelogram Statements Reasons ~ __ Given: /\ RQP = /\ ONP R is the midpoint of MQ Prove: MRON is a parallelogram Statements Reasons ~ __ Q R P O M N 1. /\ RQP = /\ ONP 1. Given R is the midpoint of MQ 2. MR = RQ 2. Definition of a midpoint 3. RQ = NO 3. CPCTC 4. MR = NO 4. Transitive Property of Congruency 5. <QRP = < NOP 5. CPCTC 6. MQ || NO 6. Alternate Interior <‘s Converse 7. MRON is a parallelogram 7. Theorem 6.10 ~ __

Given: UWXZ is a parallelogram, <1 = <8 Prove: UVXY is a parallelogram Statements Reasons U V W Z Y X 2 3 4 1 8 5 6 7 ~ 1. UWXZ is a parallelogram 1. Given 2. UW || ZX 2. Definition of a parallelogram 3. UV || YX 3. Segments of Congruent Segments 4. <Z = <W 4. Opposite <‘s of a parallelogram are = 5. <1 = <8 5. Given 6. <5 = <4 6. Third Angles Theorem 6. <4 = <7 7. Alternate Interior Angles Theorem 6. <5 = <7 8. Transitive Property of Congruence 7. UY || VX 9. Corresponding Angles Converse 8. UVXY is a parallelogram 10. Definition of a Parallelogram ~ __

Given: GIJL is a parallelogram Prove: HIKL is a parallelogram Statements Reasons L K J M G H I GIJL is a parallelogram 1. Given GI || LJ 2. Definition of a parallelogram <GIL = <JLI 3. Alternate Interior Angles Theorem GJ Bisects LI 4. Diagonals of a parallelogram bisect MI = ML 5. Definition of a Segment Bisector <HMI = <KML 6. Vertical Angles Theorem /\ HMI = /\ KML 7. ASA Congruence Postulate MH = MK 8. CPCTC HK and IL Bisect Each other 9. Definition of a Segment Bisector HIKL is a parallelogram 10. Theorem 6.9 __ ~