Unit 4: Polygons

Lesson 1: Angles of Polygons and Quadrilaterals

Diagonal: a diagonal of a polygon is a segment that connects any two nonconsecutive vertices.

Pattern Recognition: Polygon Interior Angles Sum: The sum of the interior angle measures of an n-sided convex polygon is (n – 2 ) *180

A decorative window is designed to have the shape of a regular octagon A decorative window is designed to have the shape of a regular octagon. Find the sum of the measures of the interior angles of the octagon. Answer: 1080

EXTERIOR ANGLE SUM The sum of the exterior angles is The measure of one exterior angle is

The measure of an interior angle of a regular polygon is 135 The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. Answer: The polygon has 8 sides.

The measure of an interior angle of a regular polygon is 144 The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. Answer: The polygon has 10 sides.

Find the measure of each interior angle.

Find the measure of each interior angle. Answer:

Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ.

Find x ----- Meeting Notes (10/3/12 15:23) ----- 31x - 12 = 360

Find x A. 10 B. 12 C. 14 D. 15

pRACTICE….. Sum of the measures of the interior angles of a 11-gon is The measure of an exterior angle of a regular octagon is The number of sides of regular polygon with exterior angle 72 ° is The measure of an interior angle of a regular polygon with 30 sides

Flow Chart Quadrilaterals Parallelogram Rhombus Rectangle Square Trapezoid Parallelogram Rhombus Isosceles Trapezoid Rectangle Square

Lesson 6-1: Parallelogram Definition: A quadrilateral whose opposite sides are parallel. C B A D Symbol: a smaller version of a parallelogram Naming: A parallelogram is named using all four vertices. You can start from any one vertex, but you must continue in a clockwise or counterclockwise direction. For example, the figure above can be either ABCD or ADCB. Lesson 6-1: Parallelogram

Properties of Parallelogram B Properties of Parallelogram P D C 1. Both pairs of opposite sides are congruent. 2. Both pairs of opposite angles are congruent. 3. Consecutive angles are supplementary. 4. Diagonals bisect each other but are not congruent P is the midpoint of .

Examples H K M P L Draw HKLP. HK = _______ and HP = ________ . m<K = m<______ . m<L + m<______ = 180. If m<P = 65, then m<H = ____,m<K = ______ and m<L =____. Draw the diagonals with their point of intersection labeled M. If HM = 5, then ML = ____ . If KM = 7, then KP = ____ . If HL = 15, then ML = ____ . If m<HPK = 36, then m<PKL = _____ . PL KL P P or K 115° 115° 65 5 units 14 units 7.5 units 36; (Alternate interior angles are congruent.)

Proving Quadrilaterals as Parallelograms Theorem 1: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram . H G Theorem 2: E F If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .

Theorem: Theorem 3: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. G H then Quad. EFGH is a parallelogram. M Theorem 4: E F If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram . then Quad. EFGH is a parallelogram. EM = GM and HM = FM

5 ways to prove that a quadrilateral is a parallelogram. 1. Show that both pairs of opposite sides are || . [definition] 2. Show that both pairs of opposite sides are  . 3. Show that one pair of opposite sides are both  and || . 4. Show that both pairs of opposite angles are  . 5. Show that the diagonals bisect each other .

Examples …… Example 1: Find the value of x and y that ensures the quadrilateral is a parallelogram. y+2 6x = 4x+8 2x = 8 x = 4 units 2y = y+2 y = 2 unit 6x 4x+8 2y Find the value of x and y that ensure the quadrilateral is a parallelogram. Example 2: 2x + 8 = 120 2x = 112 x = 56 units 5y + 120 = 180 5y = 60 y = 12 units 120° 5y° (2x + 8)°

Example 3 Use the Distance Formula to determine if A(3, 7), B(9, 10), C(10, 6), D(4, 3) are the vertices of a parallelogram. A. yes B. no

Rectangles Definition: A rectangle is a parallelogram with four right angles. A rectangle is a special type of parallelogram. Thus a rectangle has all the properties of a parallelogram. Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. Lesson 6-3: Rectangles

Properties of Rectangles Theorem: If a parallelogram is a rectangle, then its diagonals are congruent. Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles. E D C B A Converse: If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle.

Examples……. If AE = 3x +2 and BE = 29, find the value of x. If AC = 21, then BE = _______. If m<1 = 4x and m<4 = 2x, find the value of x. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6. x = 9 units 10.5 units x = 18 units 6 5 4 3 2 1 E D C B A m<1=50, m<3=40, m<4=80, m<5=100, m<6=40 Lesson 6-3: Rectangles

Example 5 Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x. A. x = 1 B. x = 3 C. x = 5 D. x = 10

ART Some artists stretch their own canvas over wooden frames ART Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides and the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, DA = 35 inches, BD = 37 inches, and AC = 37 inches, explain how an artist can be sure that the frame is rectangular. Since AB = CD, DA = BC, and AC = BD, AB  CD, DA  BC, and AC  BD. Answer: Because AB  CD and DA  BC, ABCD is a parallelogram. Since AC and BD are congruent diagonals in parallelogram ABCD, it is a rectangle.

Rhombus ≡ ≡ Definition: A rhombus is a parallelogram with four congruent sides. ≡ ≡ Since a rhombus is a parallelogram the following are true: Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other

Properties of a Rhombus Theorem: The diagonals of a rhombus are perpendicular. Theorem: Each diagonal of a rhombus bisects a pair of opposite angles.

Rhombus Examples ..... Given: ABCD is a rhombus. Complete the following. If AB = 9, then AD = ______. If m<1 = 65, the m<2 = _____. m<3 = ______. If m<ADC = 80, the m<DAB = ______. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____. 9 units 65° 90° 100° 10

Square Definition: A square is a parallelogram with four congruent angles and four congruent sides. Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals. Opposite sides are parallel. Four right angles. Four congruent sides. Consecutive angles are supplementary. Diagonals are congruent. Diagonals bisect each other. Diagonals are perpendicular. Each diagonal bisects a pair of opposite angles.

Squares – Examples…... 10 units 10 units 5 units 90° 45° 90° Given: ABCD is a square. Complete the following. If AB = 10, then AD = _____ and DC = _____. If CE = 5, then DE = _____. m<ABC = _____. m<ACD = _____. m<AED = _____. 10 units 10 units 5 units 90° 45° 90°

Conditions of Rhombi and Squares Theorem: If the diagonals of a parallelograms are perpendicular , then the parallelogram is a rhombus. Theorem: If one diagonal bisects a pair of opposite angles then the parallelogram is a rhombus. Theorem: If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. Theorem: If a quadrilateral is both a rectangle and a rhombus, then it is a square.

Lesson 6-5: Trapezoid & Kites Definition: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases and the non-parallel sides are called legs. Trapezoid Base Leg An Isosceles trapezoid is a trapezoid with congruent legs. Isosceles trapezoid Lesson 6-5: Trapezoid & Kites

Properties of Isosceles Trapezoid 1. Both pairs of base angles of an isosceles trapezoid are congruent. 2. The diagonals of an isosceles trapezoid are congruent. B A Base Angles D C Lesson 6-5: Trapezoid & Kites

Lesson 6-5: Trapezoid & Kites Median of a Trapezoid The median of a trapezoid is the segment that joins the midpoints of the legs. The median of a trapezoid is parallel to the bases, and its measure is one-half the sum of the measures of the bases. Median Lesson 6-5: Trapezoid & Kites