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Chapter 8: Quadrilaterals Section: 8.1 – Find Angle Measures in Polygons Aim: To find angle measures in polygons. Date: 2/17/12 What is a polygon? It is a closed plane figure that is formed by three or more sides. Convex Polygons: If you continue a line from each side of the polygon and it does not contain a point inside the polygon . Concave Polygons: If you continue a line from each side of the polygon and it does contain a point inside the polygon . Polygon Interior Angles Theorem The sum of the measures of the interior angles of a convex n-gon is (n - 2) • 180°. m1 m2 ... mn n 2 180 1 2 6 3 5 4 Find the sum of the measures of the interior angles of a convex octagon. SOLUTION An octagon has 8 sides. Use the Polygon Interior Angles Theorem. Substitute 8 for n. (n – 2) 180° = (8 – 2) 180° Subtract. = 6 180° = 1080° Multiply. EXAMPLE 2 Example 2 The sum of the measures of the interior angles of a convex polygon is 900°. Classify the polygon by the number of sides. SOLUTION (n –2) 180° = 900° n –2 = 5 n =7 Polygon Interior Angles Theorem Divide each side by 180°. Add 2 to each side. What is a Quadrilateral? A four sided figure With four angles that all add up to 360° There are six most well known types of quadrilaterals: Parallelogram Rhombus Rectangle Square Trapezoid Kites EXAMPLE 3 Find the value of x in the diagram shown. The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve. x° + 108° + 121° + 59° = 360° x + 288 = 360 x = 72 Corollary to Theorem 8.1 Combine like terms. Subtract 288 from each side. Use the diagram at the right. Find m S and m T. 103°, 103° The measures of three of the interior angles of a quadrilateral are 89°, 110°, and 46°. Find the measure of the fourth interior angle. 115° Polygon Exterior Angles Theorem The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360° m1 m2 ... _ mn 360 SOLUTION Use the Polygon Exterior Angles Theorem to write and solve an equation. x° + 2x° + 89° + 67° = 360° Polygon Exterior Angles Theorem 3x + 156 = 360 Combine like terms. x = 68 Solve for x. Chapter 8: Quadrilaterals Section: 8.2 – Use Properties of Parallelograms Aim: To find angle and side measures in a parallelogram. Date: 2/27 What is a Parallelogram? A quadrilateral Opposite sides are both parallel and equal Theorem 8.3: if a quadrilateral is a parallelogram, then its opposite sides are congruent. Opposite angles are equal Theorem 8.4: if a quadrilateral is a parallelogram, then its opposite angles are congruent. Find the values of x and y. y-8 72° x° ABCD is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x. AB = CD y - 8 = 36 x = 44 By Theorem 8.3, Opposite sides of a are . Substitute y - 8 for AB and 36 for CD. Add 8 from each side. A In 36 C, or m A=m C. So, x ° = 72°. ABCD, x = 72 and y = 44. GUIDED PRACTICE 1. Find FG and m Solve on your own: G. 8, 60° 2. Find the values of x and y. 25, 15 Theorem 8.5 If quadrilateral is a parallelogram, then its consecutive angles are supplementary. y° x° x° y° x y 180 GUIDED PRACTICE Find the indicated measure in 3. NM ANSWER 5. m 2 4. KM ANSWER JML ANSWER 6. m 4 JKLM. 70° KML ANSWER 40° Chapter 8: Quadrilaterals Section: 8.3 – Show that a Quadrilateral is a Parallelogram Aim: To use properties to identify a parallelogram. Date: 3/8/11 Theorems If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorems If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Examples For what value of x is quadrilateral CDEF a parallelogram? FN = DN 5x – 8 = 3x 2x – 8 = 0 2x = 8 Set the segment lengths equal. Substitute 5x –8 for FN and 3x for DN. Subtract 3x from each side. Add 8 to each side. Divide each side by 2. x=4 When x = 4, FN = 5(4) –8 = 12 and DN = 3(4) = 12. Quadrilateral CDEF is a parallelogram when x = 4. GUIDED PRACTICE For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning. 2; The diagonals of a parallelogram bisect each other so solve 2x = 10 – 3x for x. Chapter 8: Quadrilaterals Section: 8.4 – Properties of Rhombuses, Rectangles, and Squares. Aim: To use properties of rhombuses, rectangles, and squares. Date: 3/9/11 What is a Rhombus? Has four congruent sides Sometimes called a diamond Diagonals are perpendicular Diagonals bisect a pair of opposite angles. What is a Rectangle? Rectangles are special parallelograms Two sets of parallel sides Opposite sides are parallel and equal All angles are equal = 90°. Diagonals are congruent. What is a Square? A square is a special rectangle All angles are equal Opposite sides are parallel Each angle is 90° All sides are equal EXAMPLE For1any rhombus QRST, decide whether the statement is always or sometimes true. a. Q S a. By definition, a rhombus is a parallelogram with four congruent sides. By Theorem 8.4, opposite angles of a parallelogram are congruent. Q So, S .The statement is always true. EXAMPLE 1 For any rhombus QRST, decide whether the statement is always or sometimes true. b. Q R b. If rhombus QRST is a square, then all four angles are congruent right R if QRST is a angles. So, Q square. Because not all rhombuses are also squares, the statement is sometimes true. EXAMPLE 2 Classify the special quadrilateral. The quadrilateral has four congruent sides. One of the angles is not a right angle, so the rhombus is not also a square. The quadrilateral is a rhombus. 1. For any rectangle EFGH, is it always or sometimes true that FG GH ? Explain your reasoning. Sometimes; this is only true if EFGH is a square. 2. A quadrilateral has four congruent sides and four congruent angles. Sketch the quadrilateral and classify it. Square Chapter 8: Quadrilaterals Section: 8.5 – Properties of Trapezoids and Kites. Aim: To use properties of trapezoids. Date: 3/11/11 What is a Trapezoid? One set of parallel sides (which are the bases). Special Trapezoid: Isosceles Trapezoid •Each pair of base angles are congruent. •Congruent Diagonals. •Can also tell that it’s an Iso. Trap. If there is only one pair of congruent base angles. Midsegments Segment that connects the midpoints of its legs. Theorem: parallel to each base and its length is one half the sum of the lengths of the bases. In the diagram, MN is the midsegment of trapezoid PQRS. Find MN. Use Theorem to find MN. MN = 1 (PQ + SR) 2 1 = 2 (12 + 28) = 20 Apply Theorem Substitute 12 for PQ and 28 for XU. Simplify. The length MN is 20 inches. In 1 and 2, use the diagram of trapezoid EFGH. 1. If EG = FH, is trapezoid EFGH isosceles? Explain. Yes; if the diagonals are congruent then the trapezoid is isosceles. 2. If m HEF = 70o and m FGH = 110o, is trapezoid EFGH isosceles? Explain. Yes; m EFG = 70° by Consecutive Interior Angles Theorem making EFGH an isosceles trapezoid by having one pair of base angles congruent. GUIDED 3. InPRACTICE trapezoid JKLM, J and M are right angles, and JK = 9 cm. The length of the midsegment NP of trapezoid JKLM is 12 cm. Sketch trapezoid JKLM and its midsegment. Find ML. Explain your reasoning. J 9 cm N 12 cm M K P L 1 15 cm; Solve 2 ( 9 + x ) = 12 for x to find ML. Chapter 8: Quadrilaterals Section: 8.5 – Properties of Trapezoids and Kites. Aim: To use properties of kites. Date: 3/14/11 PIE DAY! What is a Kite? Has two pairs of consecutive congruent sides, but opposite sides are not congruent. Exactly one pair of opposite angles are congruent. The congruent opposite angles are always between the noncongruent sides. Diagonals are perpendicular. EXAMPLE 4 Find m D in the kite shown at the right. By Theorem, DEFG has exactly one pair of congruent opposite angles. Because E G, D and F must be congruent. So, m D = m F.Write and solve an equation to find m D. EXAMPLE 4 m D+m F +124o + 80o = 360o Corollary to Theorem 8.1 m D+m D +124o + 80o = 360o Substitute m 2(m D) +204o = 360o m D = 78o D for m Combine like terms. Solve for m D. F. GUIDED PRACTICE 1. In a kite, the measures of the angles are 3xo, 75o, 90o, and 120o. Find the value of x. What are the measures of the angles that are congruent? 25; 75o