Geometry 81 Angles of Polygons


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1 . Sum of Measures of Interior ngles Geometry 81 ngles of Polygons 1. Interior angles  The sum of the measures of the angles of each polygon can be found by adding the measures of the angles of a triangle. 2. Theorem 81 (Interior angle sum Theorem) If a convex polygon has n sides and S is the sum of the measures of the interior angles, then S = 180(n  2). onvex Polygon # of sides # of Vs Sum of ngle Measures triangle 3 1 (1 180) or 180 quadrilateral 4 2 (2 180) or 360 pentagon 5 3 (3 180) or 540 hexagon 6 4 (4 180) or 720 heptagon 7 5 (5 180) or 900 octagon 8 6 (6 180) or 1080 Ex 1: Find the measure of each interior angle of a regular pentagon. Use the interior angle sum theorem to find the sum of the angle measures. S = 180(n  2) S = 180(  2) S = so each angle is /5 or. Ex 2: Find the measure of each interior angle. R (11x + 4) o S 5x o U 5x o (11x + 4) o T. Sum of Measures of Exterior ngles x x x x x
2 1. Theorem 82 Exterior ngle Sum Theorem  If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex is 360. (exterior angle)(# of sides) = 360 Ex 3: Find the measures of an exterior and interior angle for a regular hexagon. HW: Geometry 81 p odd, odd 4748, odd, odd Hon: 57
3 Geometry 82 Parallelograms. Sides and angles of parallelograms 1. quadrilateral with opposite sides is called a parallelogram. Y. Theorems 1. Theorem 83 Opposite sides of a parallelogram are congruent abbreviation: Opp. sides of Y are. 2. Theorem 84 Opposite angles of a parallelogram are congruent abbreviation: Opp. s of Y are. 3. Theorem 85 onsecutive angles of a parallelogram are supplementary. abbreviation: ons. s of Y are suppl. 4. Theorem 86 If a parallelogram has one right angle, it has 4 right angles. abbreviation: If Y has 1 rt., it has 4 rt. s. Ex. 1: Write a 2 column proof for Theorem 84 Given: Y Prove:, Statements Reasons 1. Y 1. Given 2., and are supplementary 3. and are supplementary and are supplementary 4., 4.
4 Ex. 2: RSTU is a parallelogram, find m URT, m RST, and y. R U 3y S 18 o 40o 18 T. iagonals of Parallelograms R S 1. Theorem 87 The diagonals of a parallelogram bisect each other. So: RQ QT and U Q T Ex: 3: What are the coordinates of the intersection of the diagonals of parallelogram MNPR with vertices M(3, 0), N(1, 3), P(5, 4), R(3, 1). 2. Theorem 88 Each diagonal of a parallelogram separates the parallelogram into two congruent triangles. abbreviation: iag. separates Y into 2 V s. HW: Geometry 82 p all, 3739, 5051, odd, 5657, odd Hon: 36, 46 Geometry 83 Tests for Parallelograms. onditions for a parallelogram
5 1. Theorem If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. abbreviation: If both pairs of opp. sides are, then quad. is Y. 2. Theorem If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. abbreviation: If both pairs of opp. 's are, then quad. is Y. 3. Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. abbreviation: If diag. bisect each other, then quad. is Y. 4. Theorem If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. abbreviation: If one pair of opp sides is and, then quad. is Y. Ex 1: etermine whether the quadrilateral is a parallelogram Ex 2: Find x and y so that each quadrilateral is a parallelogram. a.) E 8y 15 F 6x 12 2x + 16 b.) H o (5y + 283) y + 10 (6y + 14 o ) G
6 . Parallelograms and the oordinate Plane You can use the istance Formula (or Pythagorean Theorem) to determine if a quadrilateral is a parallelogram in the coordinate plane. Ex 3: The coordinates of the vertices of a quadrilateral PQRS are P(5,3), Q(1,5), R(6,1) and S(2,1). etermine if quadrilateral PQRS is a parallelogram. quadrilateral is a parallelogram if and only if any one of the following is true: 1. oth pairs of opposite sides are parallel. (efinition) 2. oth pairs of opposite sides are congruent. (Theorem 89) 3. iagonals bisect each other. (Theorem 810) 4. oth pairs of opposite angles are congruent (Theorem 811) 5. pair of opposite sides is both parallel and congruent (Theorem 812) HW: Geometry 83 p , odd, 37, 4550, odd Hon: 2324, 33, 35, 39 Geometry 84 Rectangles. Properties of Rectangles
7 1. efinition  rectangle is a quadrilateral with four right angles. if both pairs of opposite angles are congruent, then it is a parallelogram.  Thus a rectangle is a parallelogram. M 2. Theorem 813 If a parallelogram is a rectangle, then its diagonals are congruent. abbreviation: if Y is a rectangle, diag. are. Ex 1: Quadrilateral RSTU is a rectangle. If RT = 6x + 4, and SU = 7x 4, find x. R U S T Ex 2: Find x and y if MNPL is a rectangle. M 6 y + 2 o N L 5x + 8 o 3x + 2 o P 3. Theorem 814 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Ex. 3: etermine whether parallelogram is a rectangle, given (2, 1), (4, 3), (5, 0), and (1, 2).
8 HW: Geometry 84 p odd, Hon: 42 Geometry 85 Rhombi and Squares M. Properties of Rhombi 1. rhombus is a quadrilateral with all 4
9 sides congruent. 2. Theorem The diagonals of a rhombus are perpendicular. and 3. Theorem If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. 4. Theorem Each diagonal of a rhombus bisects a pair of opposite angles. so and M Ex 1: a.) Find y if 2 m 1 y 10. = L 1 Q N b.) Find m PNL if m MPL = 64. P Ex 2: etermine whether parallelogram is a rhombus, a rectangle, or a square for (4, 2), (2, 6) (6, 4), (4, 4). List all that apply. (2, 6) (6, 4) (4, 2) (4, 4) Ex 3: square table has four legs that are 2 feet apart. The table is placed over an umbrella stand so that the hole in the center of the table lines up with the hole in the stand. How far away from a leg is the center of the hole?
10 Properties of Rhombi and Squares Rhombi Squares 1. rhombus has all the properties of a 1. square has all the properties of a parallelogram. parallelogram 2. ll sides are congruent. 2. square has all the properties of a rectangle. 3. iagonals are perpendicular. 3. square has all the properties of a rhombus. 4. iagonals bisect the angles of the rhombus. HW: Geometry 85 p , 2123, 2631, 32, 4647, odd Hon: 37, 38, 40, 42, 44 Geometry 86 Trapezoids. Properties of Trapezoid 1. trapezoid is a quadrilateral with exactly one pair of parallel sides. 2. The parallel sides are called bases. 3. The base angles are formed by the base and one of the legs. 4. The nonparallel sides are called legs. base
11 5. If the legs are congruent, then the trapezoid is an isosceles trapezoid. 6. Theorem oth pairs of base angles of an isosceles trapezoid are congruent. 7. Theorem The diagonals of an isosceles trapezoid are congruent. Ex 1: Finish the flow proof of Theorem Given: MNOP is an isosceles trapezoid Prove: MO NP Proof: M N P O MNOP is an isosceles V MP NO MPO NOP PO PO Ex 2: is a quadrilateral with vertices (5, 1), (3, 1), (2, 3), and (2, 4). a.) Verify that is a trapezoid.
12 b.) etermine whether is an isosceles trapezoid. Explain.. Medians of Trapezoids 1. The segment that joins midpoints of the legs of a trapezoid is the median. (sometimes called the midsegment) 2. Theorem The median of a trapezoid is parallel to the bases and its measure is onehalf the sum of the bases. 1 Ex: MN = ( + ) 2 Ex 3: EFG is an isosceles trapezoid with median MN. a.) Find G if EF = 20 and MN = 30. M N E 3 4 M N 1 2 G b.) Find m 1, m 2, m 3, and m 4 if m 1 = 3x + 5 and m 3 = 6x 5. HW: Geometry 86 p odd, 1318, 2228, 4248, odd Hon: 19, 20, 32, 35 Geometry 87 oordinate Proof With Quadrilaterals. Position Figures Ex 1: Position and label a square with sides a units long on the coordinate plane. 1.) Let,,, and be the vertices of the square. (0, ) (, ) (0, 0) (, 0)
13 2.) Place the square with vertex at the origin, along the positive xaxis, and along the yaxis. Label the vertices,, and. 3.) The ycoordinate of is 0 because the vertex is on the xaxis. Since the side length is a, the xcoordinate is. 4.) is on the yaxis so the xcoordinate is 0. The ycoordinate is 0 + a or. 5.) The xcoordinate of is also. The ycoordinate is 0 + a, or because side is a units long. Ex 2: Name the missing coordinate for the isosceles trapezoid. (?,?) (a2b, c) (0, 0) (a, 0). Prove Theorems 1. Once we have place a figure on the coordinate plane, we can use slope formula, distance formula and midpoint formula to prove theorems. Ex 3: Place a square on the coordinate plane. Label the midpoints of the sides, M, N, P, and Q. Write a coordinate proof to show MNPQ is a square. a.) y midpoint formula, the coordinates of M, N, O, and P are as follows: M (, ) N (, ) P (, ) Q (, ) (0, 2a) (2a, 2a) b.) Find the slopes of QP, MN, QM, and PN. (0, 0) (2a, 0)
14 Each pair of opposite sides is, therefore MNPQ is a, and a c.) Use the distance formula to find QP and QM. MNPQ is a square because each pair of opposite sides is parallel, and consecutive sides form and are. HW: Geometry 87 p , 2425, 2839
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