Angles of Polygons. Sum of Measures' of Exterior Angles There is a simple relationship among the exterior angles of a convex polygon.

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1 Angles of Polgons Sum of Measures of Interior Angles The segments that connect the nonconsecutive sides of a polgon are called diagonals. Drawing all of the diagonals from one vertex of an n-gon separates the polgon into n - 2 triangles. The sum of the measures of the interior angles of the polgon can be found b adding the measures of the interior angles of those n - 2 triangles. Interior Angle If a convex polgon has n sides, and S is the sum of the measures of its interior angles, Sum Theorem then S = 180(n - 2). Example 1 A convex polgon has 13 sides. Find the sum of the measures of the interior angles. S = 180(n - 2) = 180(13-2) = 180(11) = 1980 Example 2 The measure of an interior angle of a regular polgon is 120. Find the number of sides. The number of sides is n, so the sum of the measures of the interior angles is 120n. S = 180(n - 2) 120n = 180(n - 2) 120n = 180n n = -360 n=6 Sum of Measures' of Exterior Angles There is a simple relationship among the exterior angles of a convex polgon. Exterior Angle If a polgon is convex, then the sum of the measures of the exterior angles, Sum Theorem one at each vertex, is 360. Example 1 Find the sum of the measures of the exterior angles, one at each vertex, of a convex 27-gon. For an. convex polgon, the sum of the measures of its exterior angles, one at each vertex, IS 360. Example 2 Find the measure of each exterior angle of regular hexagon ABCDEF. The sum of the measures of the exterior angles is 360 and a hexagon has 6 angles. If n is the measure of each exterior angle, then 6n = 360 n = 60

2 Parallelograms l UfQ Sides and Angles of Parallelograms A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are four important properties of parallelograms. s, R If PQRS is a parallelogram, then The opposite sides of a parallelogram are congruent. The opposite angles of a parallelogram are congruent. The consecutive angles of a parallelogram are supplementar. If a parallelogram has one right angle, then it has four right angles. PO =:0 SR and PS =:oor LP =:0 LR and LS =:0 LO LP and LS are supplementar; LS and LR are supplementar; L Rand L 0 are supplementar; L 0 and L P are supplementar. If mlp = 90, then mlo = 90, mlr = 90, and mls = 90. Example IfABCD is a parallelogram, find a and b. - - AB and CD are opposite sides, so AB == CD. 2a = 34 a = 17 LA and LC are opposite angles, so LA == LC. 8b = 112 b = 14 Diagonals of Parallelograms Two important properties of parallelograms deal with their diagonals. l:2?1b D C If ABeD is a parallelogram, then: The diagonals of a parallelogram bisect each other. Each diagonal separates a parallelogram into two congruent triangles. AP = PC and DP = PB,6.ACD =:0,6.CAB and,6.adb =:0,6.CBD Example Find x and in parallelogram ABCD. The diagonals bisect each other, so AE = CE and DE = BE. 6x = 24 4 = 18 x = 4 = 4.5 Tests for Parallelograms Conditions for a Parallelogram There are man was to establish that a quadrilateral is a parallelogram.,..---/fb D=---~ t><lb D C If: both pairs of opposite sides are parallel, both pairs of opposite sides are congruent, both pairs of opposite angles are congruent, the diagonals bisect each other, one pair of opposite sides is congruent and parallel, then: the figure is a parallelogram. If: AB II DC and AD II BC, - AB =:0 DC and AD =:0 BC, LABC =:0 LADC and LDAB =:0 LBCD, AE =:0 CE and DE =:0 BE, AB II CD and AB =:0 CD, or AD II BC and AD =:obc, then: ABCD is a parallelogram.

3 Properties of Rectangles A rectangle is a quadrilateral with four right angles. Here are the properties of rectangles. A rectangle has all the properties of a parallelogram. Opposite sides are parallel. Opposite angles are congruent. Opposite sides are congruent. Consecutive angles are supplementar. The diagonals bisect each other. Also: All four angles are right angles. The diagonals are congruent. Example 1 In rectangle RSTU above, US = 6x + 3 andrt = 7x - 2. Findx. The diagonals of a rectangle bisect each other, so US = RT. 6x + 3 = 7x - 2 3=x-2 5=x - - TI'~7r U~R LUTS, LTSR, LSRU, and LRUTare right angles. TR=US Example 2 In rectangle RSTU above, mlstr = 8x + 3 and mlutr = 16x - 9. Find mlstr. LUTS is a right angle, so mlstr + mlutr = 90. 8x x - 9 = x - 6 = x = 96 x=4 mlstr = 8x + 3 = 8(4) + 3 or 35.Properties of Rhombi A rhombus is a quadrilateral with four congruent sides. Opposite sides are congruent, so a rhombus is also a parallelogram and has all of the properties of a parallelogram. Rhombi also have the following properties. The diagonals are perpendicular.. - MH.l RO ~H MK)d Each diagonal bisects a pair of opposite angles. If the diagonals of a parallelogram are perpendicular, then the figure is a rhombus. MH bisects LRMO and LRHo. RO bisects L MRH and L MOH. If RHOM is a parallelogram and RO.l MH, then RHOM is a rhombus. Example In rhombus ABCD, mlbac = 32. Find the measure of each numbered angle. B2 ABCD is a rhombus, so the diagonals are perpendicular and MBE is a right triangle. Thus ml4 = 90 and mll = or 58. The ~ E 3 diagonals in a rhombus bisect the vertex angles, so ml1 = ml2. Thus, ml2 = 58. o A 32 4 C A rhombus is a parallelogram, so the opposite sides are parallel. LBAC and L3 are alternate interior angles for parallel lines, so ml3 = 32. Properties of Squares A square ha~-ali the pr~p-~~tie~ of a rho~bus -~nd all the properties of a rectangle. Example Find the measure of each numbered angle of square ABCD. Using properties of rhombi and rectangles, the diagonals are perpendicular and congruent. MBE is a right triangle, so ml1 = ml2 = 90. Each vertex angle is a right angle and the diagonals bisect the vertex angles, so ml3 = ml4 = ml5 = 45.

4 Trapezoids Properties of Trapezoids A trapezoid is a quadrilateral with exactl one pair of parallel sides. The parallel sides are called bases and the nonparallel sides are called legs. If the legs are congruent, the trapezoid is an isosceles trapezoid. In an isosceles trapezoid both pairs of base angles are congruent. S b"'~ E'eg~ R base U STUR is an isosceles trapezoid. SR s: TU; LR s: LU, LS s: LT Example The vertices ofabcd are A(-3, -1), B(-1, 3), C(2, 3), and D(4, -1). Verif that ABCD is a trapezoid. ~ 3 - (-1) 4 slope ofab = -1 _ (-3) = 2" = 2 AB = Y(-3 - (-1))2 + (-1-3) (-1) 0 slope ofad = 4 _ (-3) = '7 = 0 = V = V20 = 2\ slope ofbc = 2 _ (-1) = 3 = 0 CD = Y(2-4)2 + (3 - (-1))2 B C J ~, II J \ II 0, x A D = V = V20 = 2\15 slope of CD = = - = Exactl two sides are parallel, AD and BC, so ABCD is a trapezoid. AB = CD, so ABCD is an isosceles trapezoid. Medians of Trapezoids The median of a trap~zoid is the H J segment that joins the midpoints of the legs. It is parallel to the ~ bases, and its length is one-half the sum of the lengths of the bases. In trapezoid HJKL, MN = ~ (HJ + LK). L~'\K Example MN = ~(RS + at) 1 30 = 2"(3x x - 5) 1 30 = -(12x) 2 30 = 6x 5=x MN is the median of trapezoid RSTU. Find x.

5 Parallelograms on the Coordinate Plane On the coordinate plane, the Distance Formula and the Slope Formula can be used to test if a quadrilateral is a parallelogram. Example Determine whether ABCD is a parallelogram. The vertices are A(-2,3), B(3, 2), C(2, -1), and D(-3,0). Y2 - Yl Method 1: Use the Slope Formula, m = X slope ofad = -2 _ (-3) = T = (-1) 3 slope ofbc = 3 _ 2 = T = slope ofab = 3 _ (-2) = slope of CD = 2 - (-3) = -5 A J l" I-i- B,I II 0 r 0 -l" I C X Opposite sides have the same slope, so AB II CD and AD II BC. Both pairs of opposite sides are parallel, so ABCD is a parallelogram. Method 2: Use the Distance Formula, d = Y(x 2 - AB = Y(-2-3)2 + (3-2)2 = Y or V26 CD = Y(2 - (-3))2 + (-1-0)2 = Y or V26 AD = Y(-2 - (-3))2 + (3-0)2 = V1+9 or V10 BC = Y(3-2)2 + (2 - (-1))2 = V1+9 or V10 x 1 )2 + (Y2 - Y1)2. Both pairs of opposite sides have the same length, so ABCD is a parallelogram. In the coordinate plane ou can use the Distance Formula, the Slope Formula, and properties of diagonals to show that a figure is a rectangle. Example Determine whether A(-3,0), B(-2,3), C(4, 1), and D(3, -2) are the vertices of a rectangle. Method 1: Use the Slope Formula slope ofab = -2 _ (-3) = Tor 3 slope ofad = 3 _ (-3) = (3 or slope of CD = 3 _ 4 = -1 or 3 slope ofbc = 4 _ (-2) = (3 or -3 Opposite sides are parallel, so the figure is a parallelogram. Consecutive sides are perpendicular, so ABCD is a rectangle. Method 2: Use the Midpoint and Distance Formulas. The midpoint ofac is ( , 0 ; 1) = (~, ~) and the midpoint of BD is B I " r--.. c II A r--. o'\. I x ~ i'~~ 0 (-22+ 3, 3 ; 2) = (t, ~). The diagonals have the same midpoint so the bisect each other. Thus, ABCD is a parallelogram. AC = Y(-3-4)2 + (0-1)2 = Y = V50 or 5\12 BD = Y(-2-3)2 + (3 - (-2))2 = Y = V50 or 5\12 The diagonals are congruent. ABCD is a parallelogram with diagonals that bisect each other, so it is a rectangle.

6 Coordinate Proof and Quadrilaterals Position Figures Coordinate proofs use properties oflines and segments to prove geometric properties. The first step in writing a coordinate proof is to place the figure on the coordinate plane in a convenient wa. Use the following guidelines for placing a figure on the coordinate plane. 1. Use the origin as a vertex, so one set of coordinates is (0, 0), or use the origin as the center of the figure. 2. Place at least one side of the quadrilateral on an axis so ou will have some zero coordinates. 3. Tr to keep the quadrilateral in the first quadrant so ou will have positive coordinates. 4. Use coordinates that make the computations as eas as possible. For example, use even numbers if ou are going to be finding midpoints. Example Position and label a rectangle with sides a and b units long on the coordinate plane. Place one vertex at the origin for R, so one vertex is R(O, 0). - - Place side RU along the x-axis and side RS along the -axis, with the rectangle in the first quadrant. The sides are a and b units, so label two vertices S(O, a) and U(b, 0). Vertex Tis b units right and a units up, so the fourth vertex is T(b, a). 8(0. a)i-i--j ,t(b, a) h R(O, 0) U(b,O) x Prove Theorems After a figure has been placed on the coordinate plane and labeled, a coordinate proof can be used to prove a theorem or verif a propert. The Distance Formula, the Slope Formula, and the Midpoint Theorem are often used in a coordinate proof. Example Write a coordinate proof to show that the diagonals of a square are perpendicular. The first step is to position and label a square on the coordinate plane. Place it in the first quadrant, with one side on each axis. Label the vertices and draw the diagonals. Given: square RSTU Prove: SU 1. RT Pr f 0 - a The s ope 0 f SU ' 00 : I IS a _ 0 = -1, and the slope ofrt IS a _ 0 = 1. The product of the two slopes is -1, so SU 1. RT. 8(0, a)i.---'" U(a,O) x