Chapter 6: Quadrilaterals Fall 2008 Geometry

6.1 Polygons A polygon is a closed plane figure that is formed by three or more segments called sides, such that no two sides with a common endpoint are collinear. Each endpoint of a side is a vertex of the polygon. Name a polygon by listing the vertices in clockwise or counterclockwise order.

Polygons State whether the figure is a polygon.

Identifying Polygons 3 sides Triangle 4 sidesQuadrilateral 5 sidesPentagon 6 sidesHexagon 7 sidesHeptagon 8 sidesOctagon 9 sidesNonagon 10 sidesDecagon 12 sidesDodecagon n sidesn-gon

Polygons A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon

Polygons A polygon is concave if it is not convex.

Polygons A polygon is equilateral if all of its sides are congruent. A polygon is equiangular if all of its angles are congruent. A polygon is regular if it is equilateral and equiangular.

Polygons Determine if the polygon is regular.

Polygons A diagonal of a polygon is a segment that joins two nonconsecutive vertices. E R M L B

Polygons The sum of the measures of the interior angles of a quadrilateral is 360. – A + B + C + D = 360 A D C B

Homework 6.1 Pg. 325 # 12 – 34, 37 – 39, 41 – 46

6.2 Properties of Parallelograms A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Theorems about Parallelograms If a quadrilateral is a parallelogram, then its opposite sides are congruent. –AB = CD and AD = BC A B C D

Theorems about Parallelograms If a quadrilateral is a parallelogram, then its opposite angles are congruent. – A = C and D = B A B C D

Theorems about Parallelograms If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. D + C = 180 ; D + A = 180 A + B = 180 ; B + C = 180 A B C D

Theorems about Parallelograms If a quadrilateral is a parallelogram, then its diagonals bisect each other. AM = MC and DM = MB A B C D M

Examples FGHJ is a parallelogram. Find the unknown lengths. –JH = _____ –JK = _____ F JH G 3K 5 F JH G 3K

Examples PQRS is a parallelogram. Find the angle measures. –m R = –m Q = 70 P SR Q

Examples PQRS is a parallelogram. Find the value of x. 3x P SR Q 120

6.3 Proving Quadrilaterals are Parallelograms For the 4 theorems about parallelograms, their converses are also true.

Theorems If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorems If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorems If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

Theorems If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

One more… If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

Examples Is there enough given information to determine that the quadrilateral is a parallelogram?

Examples Is there enough given information to determine that the quadrilateral is a parallelogram?

Examples Is there enough given information to determine that the quadrilateral is a parallelogram?

How would you prove ABCD? A DC B

A DC B

AB C D

Homework 6.2 – 6.3 Pg. 334 # 20 – 37 Pg. 342 # 9 – 19, 32 – 33

6.4 Rhombuses, Rectangles, and Squares A rhombus is a parallelogram with four congruent sides.

Rhombuses, Rectangles, and Squares A rectangle is a parallelogram with four right angles.

Rhombuses, Rectangles, and Squares A square is a parallelogram with four congruent sides and four right angles.

Special Parallelograms Parallelograms squares rhombuses rectangles

Corollaries about Special Parallelograms Rhombus Corollary –A quadrilateral is a rhombus iff it has 4 congruent sides. Rectangle Corollary –A quadrilateral is a rectangle iff it has 4 right angles. Square Corollary –A quadrilateral is a square iff it is a rhombus and a rectangle.

Theorems A parallelogram is a rhombus iff its diagonals are perpendicular.

Theorems A parallelogram is a rhombus iff each diagonal bisects a pair of opposite angles.

Theorems A parallelogram is a rectangle iff its diagonals are congruent.

Examples Always, Sometimes, or Never –A rectangle is a parallelogram. –A parallelogram is a rhombus. –A rectangle is a rhombus. –A square is a rectangle.

Examples (A) Parallelogram; (B) Rectangle; (C) Rhombus; (D) Square –All sides are congruent. –All angles are congruent. –Opposite angles are congruent. –The diagonals are congruent.

Examples MNPQ is a rectangle. What is the value of x? 2x Q P NM

6.4 Homework Pg. 351 # 12-21, 25-43

6.5 Trapezoids and Kites A trapezoid is a quadrilateral with exactly one pair of parallel sides. –The parallel sides are the bases. –A trapezoid has two pairs of base angles. –The nonparallel sides are called the legs. If the legs are congruent then it is an isosceles trapezoid.

Theorems If a trapezoid is isosceles, then each pair of base angles is congruent. A = B, C = D A D C B

Theorems If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. A D C B

Theorems A trapezoid is isosceles iff its diagonals are congruent. –ABCD is isosceles iff AC = BD A D C B

Midsegments of Trapezoids The midsegment of a trapezoid is the segment that connects the midpoints of its legs. midsegment

Midsegment Theorem The midsegment of a trapezoid is parallel to each base and its length is ½ the sum of the lengths of the bases. –MN ll AD, MN ll BC, MN = ½ (AD + BC) B M A D N C

Kites A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Theorems about kites If a quadrilateral is a kite, then its diagonals are perpendicular. A C D B

Theorems about kites If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. - A = C, B = D A C D B

Name the bases of trap. ABCD A D CB

Trapezoid, Isosceles Trap., Kite, or None

Find the length of the midsegment 7 11

Find the length of the midsegment 12 6

Find the angle measures of JKLM J LK M 44

Find the angle measures of JKLM J LK M 82

Find the angle measures of JKLM J LK M

6.6 Special Quadrilaterals quadrilateral Kite parallelogram trapezoid rhombus rectangle squareisosceles trapezoid

Example 1 Quadrilateral ABCD has at least one pair of opposite sides congruent. What kinds of quadrilaterals meet this condition?

Check which shapes always have the given property. PropertyPara.Rect.RhombusSquareKiteTrap. Both pairs of opp. Sides = Exactly 1 pair of opp. Sides = All sides are = Both pairs of opp. = Exactly 1 pair of opp. = All = ~ ~ ~ ~ ~ ~

Check which shapes always have the given property. PropertyPara.Rect.RhombusSquareKiteTrap. Diagonals are = Diagonals. Diag. bisect each other ~

Homework Trapezoid worksheet and 6.6 B Worksheet out of workbook.

6.7 Areas of Triangles and Quadrilaterals Area of a Rectangle = bh Area of a Parallelogram = bh Area of a Triangle = ½ bh b b b h h h

Areas Area of a Trapezoid = ½ h (b 1 + b 2 ) Area of a Kite = ½ d 1 d 2 Area of a Rhombus = ½ d 1 d 2 b1b1 b2b2 h

Homework Trapezoid worksheet, Practice 6.6 B