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Quadrilaterals and Polygons

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?.

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Quadrilaterals and Polygons

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  1. 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?

  2. Polygons are Named & Classified by the Number of Sides They Have Octagon Nonagon Decagon Dodagon N-gon Triangle Quadrilateral Pentagon Hexagon Heptagon What type of polygons are the following?

  3. Interior Interior 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 Equilateral, Equiangular, and Regular

  4. 80o 70o xo 2xo 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

  5. Q R P S Q R P S Q R P S Q R P S 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 _ _ _ _ ~ ~ ~ ~ _ _ _ _ ~ ~ Properties of Parallelograms

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

  7. 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 ~ ~ ~ Proofs Involving Parallelograms Plan: Show that both angles are congruent to <2

  8. 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 ~ __ __ ~ ~ __ __ __ __ ~ ~ Proving Theorem 6.2 Plan: Insert a diagonal which will allow us to divide the parallelogram into two triangles

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

  10. congruent parallel congruent BOTH consecutive interior <‘s bisect each other congruent and || Concept Summary – Proving Quadrilaterals are Parallelograms • Show that both pairs of opposite sides are • 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

  11. C(6,5) B(1,3) D (7,1) A(2, -1) Proving Quadrilaterals are Parallelograms – Coordinate Geometry 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

  12. Parallelograms Rhombuses Rectangles Squares 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

  13. A B C D If ABCD is a rectangle, what else do you know about ABCD? 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 Using Properties of Special Triangles

  14. 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 Using Diagonals of Special Parallelograms

  15. A B E D C 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. • 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 Sometimes Always Always Always 32o 86o 66o 35o 16 26

  16. 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)

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

  18. R E A T C Given: RECT is a Rectangle Prove: /\ ART = /\ ACE Statements Reasons ~

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

  20. 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 A B D C > > > > If the legs of the trapezoid are congruent, then the trapezoid is an Isosceles Trapezoid. Trapezoids and Kites

  21. A B D C A B D C A B D C > > > Theorem 6.14 If a trapezoid is isosceles, then each pair of base angles is congruent. <A = <B = <C = <D ~ ~ ~ > > > Theorem 6.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. ABCD is an isosceles trapezoid Theorem 6.16 A trapezoid is isosceles if and only if its diagonals are congruent. ABCD is isosceles if and only if AC = BD _ _ ~ Theorems of Trapezoids

  22. 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

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

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

  25. 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 X X X

  26. 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 A = bh 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 h b h b 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.

  27. Name That Proof

  28. Q R P O M N ~ Given: /\ RQP = /\ ONP R is the midpoint of MQ Prove: MRON is a parallelogram Statements Reasons __ ~ 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 __ __ ~ __ __ ~ __ __ ~ ~ __ __

  29. U V W Z Y X 2 3 4 1 8 5 6 7 ~ Given: UWXZ is a parallelogram, <1 = <8 Prove: UVXY is a parallelogram Statements Reasons 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 __ __ __ __ ~ ~ ~ ~ ~ __ __

  30. L K J M G H I Given: GIJL is a parallelogram Prove: HIKL is a parallelogram Statements Reasons • 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 __ __ ~ __ __ ~ ~ ~ __ __ ~

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