Topic 6 Goals and Common Core Standards Ms. Helgeson Identify, name, and describe polygons. Use the sum of the measures of the interior angles of a quad. Use some properties of parallelogram. Prove that a quad is a parallelogram. Use coordinate geometry with parallelograms

Use properties of sides and angles of rhombuses, rectangles, and squares Use properties of diagonals of rhombuses, rectangles, and squares. Use properties of trapezoids Use properties of kites Identify special quadrilaterals based on limited info Prove that a quad is a special type of quad.

CC.9-12.G.CO.13 CC.9-12.G.CO.11 CC.9-12.G.SRT.5 CC.9-12.G.GPE.4 CC.9-12.G.CO.9

Quadrilaterals and Other Polygons Topic 6 Quadrilaterals and Other Polygons

6.1 Polygons A polygon is a plane figure that meets the following conditions. 1. It is formed by three or more segments called sides, such that no two sides with a common endpoint are collinear. 2. Each side intersects exactly two other sides, one at each endpoint. Q R vertex P side S T vertex

Identifying Polygons POLYGON NOT A POLYGON

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

POLYGONS Number of Sides Type of Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon Number of Sides Type of Polygon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon

interior convex polygon concave polygon or non-convex A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that is not convex is called nonconvex or concave

A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Polygon PQRST has 2 diagonals from point Q, QT and QS. diagonals

# of sides 3 4 5 6 n diagonals 1 2 n - 3 triangles n - 2 Sum of <‘s 180 360 540 720 180(n – 2) There are n – 2 triangles in every n-sided polygon. Each triangle has an angle sum of 180˚. Polygon Interior Angle-Sum Theorem: Interior angle sum of an n-sided polygon = 180(n – 2) Measure of an interior angle of a regular n-gon is 180(n – 2) n The sum of the exterior angle measures of any convex polygon is 360˚. Page 248 examples 4 and 5.

Theorem Interior Angles of a Quadrilateral The sum of the measures of the interior angles of a quadrilateral is 360º. m<1 + m<2 + m<3 + m<4 = 360º 4 1 3 2

Find m<U and m<V. 5xº 118º (3x + 10)º 72º

Find m<D. 4xº 4xº 100º 2xº 2xº

Quadrilaterals Isos. T.

6.2 Trapezoids and Kites What is the difference between a trapezoid and a kite? A trapezoid is a quadrilateral with exactly one pair of parallel sides. A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Trapezoid (Not a Parallelogram) 4 sided figure. Exactly one pair of opposite sides are parallel. The median (midsegment) of a trapezoid is parallel to the bases, and its measure is ½ the sum of the measures of the bases.

Example: XY is the median of trap. MNOP XY = 5k-7 MN = k+7 PO = 3k+3 K=___ MN=___ PO=___ XY=___ M N X Y P O

Example: Trap GHIJ with median KL. GH= 3x-1 KL=10 JI= 7x+1 x=_____ G H K L J I

3x + 2 2x + 4 2x + 1

Isosceles Trapezoid 4 sided figure, 1 pair of opposite sides are parallel. Other pair not parallel. Legs are congruent. Diagonals are congruent. Both pairs of base angles are congruent. base base angles base angles leg leg base

Example: Isosceles Trap ABCD AC = 7x BD = 5x+4 x=_____ D C A B

Trapezoids B A E F G H C D EF = ½ (AB – DC) GE = FH

ABCD is a trapezoid with median MN. If DC = 6 and AB = 16, find ME, FN, AND EF. If DC = 3x, AB = 2x², and EF = 7, find the value of x. D C N M E F B A

Examples: One angle of an isosceles trapezoid has measure 57. Find the measures of the other angles. Two congruent angles of an isosceles trapezoid have measures 3x + 10 and 5x – 10. Find the value of x and and then give the measures of all angles of the trapezoid.

If AD = x and BE = x + 6, then x = ? And CF = ?. TA = AB = BC AND TD = DE = EF If AD = x and BE = x + 6, then x = ? And CF = ?. If AD = x + y, BE = 20, and CF = 4x – y, then CF = ? And y = ?. If AD = x + 3, BE = x + y, and CF = 36, then x = ? And y = ? T A D E B F C

Kite A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. The diagonals are perpendicular. Exactly one pair of opposite angles are congruent.

Kite C D B Seg. AC perp. to Seg. BD m<BAD = m<BCD; A m<ABC ≠ m<ADC A

Example 1: GHJK is kite. Find HP. √29 P 5 J G K

X + 30 X 125 Rstu IS A KITE. Find m< R, m< S, and m< T.

6.3 Properties of Parallelograms Both pairs of opposite sides are parallel. (Def) Both pairs of opposite sides are congruent. (Thm) Both pairs of opposite angles are congruent. (Thm) Consecutive angles are supplementary. (Thm) Diagonals bisect each other. (Thm) One pair of sides are congruent and parallel (Thm) Each diagonal divides quad. Into 2 congruent triangles. 4 sides figure - Quadrilateral

Example: Parallelogram TAXI AX = 3y TI = 2y+10 TA = 3y+10 Find: AX=____ TI=____ TA=____ XI=____ m<R = x+20 m<T = 2x-30 Find: m<R=____ m<S=____ m<T=____ m<U=____ Example: Parallelogram RSTU

Example: Parallelogram RSTU U T X R S RX = 3x-4 XT = 2x+4 Find: RX=___ XT=___ RT=___

6.4 Proving Quad is a Parallelogram A quadrilateral is a parallelogram if any one of the following is true. Def: Both pairs of opposites are parallel. Thm: Both pairs of opposites sides are congruent. Thm: Both pairs of opposite angles are congruent. Thm: Diagonals bisects each other.

5. Thm: One pair of opposite sides are parallel and congruent 5. Thm: One pair of opposite sides are parallel and congruent. 6: Thm: Show that one angle is supplementary to both consecutive angles.

Slope Formula: m= y2-y1 x2-x1 Parallel lines: same slopes m1=m2 Perpendicular lines: neg. recip. m1m2=-1 Distance formula √(x2-x1)² + (y2-y1)²

6.5 & 6.6 Rhombuses, Rectangles, and Squares

Rectangles (Parallelogram) Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. 4 sided figure. Contains 4 right angles. Diagonals are congruent.

Example: Quadrilateral QUAD is a rectangle Example: Quadrilateral QUAD is a rectangle. m<2 = 52, m<3 = 16x-12 m<2=____ m<3=____ m<1=____ Q U 2 m<2=_____________ m<3=_____________ m<1=_____________ 1 3 D A

Rhombus (Parallelogram) Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. 4 sided figure. All sides are congruent. Diagonals bisect the opposite angles. Diagonals are perpendicular.

Example: Rhombus BCDE m<1 = 2x+20 m<2 = 5x-4 x=____ m<1=____ Example: Use above figure. m<3 = y²+26 y=____ x=____ m<1=____ m<2=____ m<3=____ B C 1 2 3 F E D

Example: Rhombus WXYZ m<XYZ=68 m<WXZ=____ Z Y W X

Square (Parallelogram) Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. 4 sided figure. Contains 4 right angles. All sides are congruent. Diagonals bisect the angles. Diagonals are congruent. Diagonals are perpendicular.

6.6 Special quadrilaterals (page 364) Quadrilateral Kite Parallelogram Trapezoid Rhombus Rectangle Isos. Trap. Square