Chapter 6: Quadrilaterals. of a polygon is a segment that connects any two nonconsecutive. Triangle Quadrilateral Pentagon Hexagon

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1 Lesson 6-1: Angles of Polygons A vertices. Example: Date: of a polygon is a segment that connects any two nonconsecutive Triangle Quadrilateral Pentagon Hexagon Since the sum of the angle measures of a triangle is, we can make a table and look for a pattern to find the sum of the angle measures for any convex polygon. Polygon Number of Sides Number of Triangles Sum of Interior Angle Measures Triangle Quadrilateral Pentagon Hexagon n-gon This leads us to: Page 1

2 Number of Sides n Polygon Use a protractor to measure each angle to the nearest degree. Then find the sum of these angles. Example 1: A. Find the sum of the measures of the interior angles of a convex nonagon. B. Find the measure of each interior angle of parallelogram RSTU. Page 2

3 Example 2: A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the measure of one of the interior angles of the pentagon. Example 3: The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon. Example 4: A. Find the value of x in the diagram. B. Find the measure of each exterior angle of a regular decagon. Page 3

4 Lesson 6-2: Parallelograms A Example Date: is a quadrilateral with both pairs of opposite sides parallel. Example 1: In parallelogram ABCD, suppose m B = 32, CD = 80, and BC = 15. A. Find AD B. Find m C C. Find m D Page 4

5 Example 2: WXYZ is a parallelogram. A. Find the value of r. B. Find the value of s. C. Find the value t. Example 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), and R(3, 1)? y x Page 5

6 Lesson 6-3: Tests for Parallelograms If a quadrilateral has each pair of opposite sides Date:, then by definition, it is a parallelogram. Other ways to determine if a quadrilateral is a parallelogram include: Example 1: Determine whether each quadrilateral is a parallelogram. Justify your answer. a. b. c. d. e. 8 cm 2 cm 2 cm 8 cm Page 6

7 Example 2: MULTIPLE CHOICE The diagram shows a car jack used to raise a car from the ground. In the diagram, AD BC and AB DC. Based on this information, which statement will be true, regardless of the height of the car jack. A. A B B. A C C. AB BC D. m A + m C = 180 Example 3: Find x and y so that the quadrilateral is a parallelogram. You may need to use the distance, slope, and midpoint formulas to determine whether a quadrilateral in the coordinate plane is a parallelogram. Use the distance formula to show that the pairs of opposite sides are congruent. Use the slope formula to show that the pairs of opposite sides have the same slope. Use the midpoint formula to show that each diagonal has the same midpoint (they intersect at the same point). Example 4: Quadrilateral QRST has vertices Q( 1, 3), R(3, 1), S(2, 3), and T( 2, 1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula. y x Page 7

8 Lesson 6-4: Rectangles A rectangle is a parallelogram with four Date: angles. By definition, a rectangle has the following properties: All four angles are angles. Opposite sides are and. Opposite angles are. Consecutive angles are. Diagonals each other. A rectangle is a parallelogram, but a parallelogram isn t necessarily a rectangle! Example 1: A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet, and LN = 6.5 feet, find KM. Example 2: Quadrilateral RSTU is a rectangle. If m RTU = 8x + 4 and m SUR = 3x 2, find x. Page 8

9 Example 3: 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. Example 4: Quadrilateral JKLM has vertices J( 2, 3), K(1, 4), L(3, 2), and M(0, 3). Determine whether JKLM is a rectangle using the Distance Formula. y x Page 9

10 Lesson 6-5: Rhombi & Squares Date: A is a parallelogram with all four sides congruent. It has all the properties of a parallelogram in addition to: Example 1: A. The diagonals of rhombus WXYZ intersect at V. If m WZX = 39.5, find m ZYX. B. The diagonals of rhombus WXYZ intersect at V. If WX = 8x 5 and WZ = 6x + 3, find x. Page 10

11 A is a parallelogram with four congruent sides and four right angles. A parallelogram that is both a rectangle and a rhombus is also a square. A square is a rhombus, but a rhombus isn t necessarily a square. A square is a rectangle, but a rectangle isn t necessarily a square. Example 2: A. If LMNP is a parallelogram, 1 2, and 2 6, is LMNP is a rhombus? B. If ABCD is a parallelogram and AD DC, is ADCD is a rhombus? Page 11

12 Example 3: Hector is measuring the boundary of a new garden. He wants the garden to be square. He has set each of the corner stakes 6 feet apart. What does Hector need to know to make sure that the garden is square? Since a rhombus has four congruent sides, one diagonal separates the rhombus into two congruent isosceles triangles. Both diagonals would separate the rhombus into four congruent right triangles. If the diagonals of a parallelogram are congruent then it is a. If the diagonals of a parallelogram are perpendicular, then it is a If the diagonals of a parallelogram are both congruent and perpendicular, then it is a,, and. Example 4: Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A( 2, 1), B( 1, 3), C(3, 2), and D(2, 2). List all that apply. Explain. y x Page 12

13 Lesson 6-6: Trapezoids and Kites Date: A is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called the. The nonparallel sides are called the. The are formed by the base and one of the legs. If the legs are congruent, then it is called an. Example: Example 1: Each side of the basket shown is an isosceles trapezoid. A. If m JML = 130, KN = 6.7 feet, and LN = 3.6 feet, find m MJK. B. If m JML = 130, KN = 6.7 feet, and JL is 10.3 feet, find MN. Page 13

14 Example 2: Quadrilateral ABCD has vertices A(5, 1), B( 3, 1), C( 2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. y x The legs of the trapezoid. Example: of a trapezoid is the segment that connects the midpoints of the Example 3: In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x? Page 14

15 A is a quadrilateral with exactly two pairs of consecutive congruent sides. (unlike a parallelogram, the opposite sides are not parallel or congruent) Example: Example 4: A. If WXYZ is a kite, find m XYZ. B. If MNPQ is a kite, find NP. Page 15