Date Name of Lesson 6.1 Angles of Polygons 1.5 days 6.2 Parallelograms 1 day 6.3 Tests for Parallelograms 1.5 days Quiz 6.1-6.3 0.5 days 6.4 Rectangles 1 day 6.5 Rhombi and Squares 1 day 6.6 Trapezoids and Kites 1 day Review and test 2 days 1
6.1 Quadrilaterals Notes Find the Interior Angles Sum of a Polygon Example 1: A. Find the sum of the measures of the interior angles of a convex nonagon. Example 1: B. Find the measure of each interior angle of parallelogram RSTU. Guided Practice 1: A. Find the sum of the measures of the interior angles of a convex octagon. Guided Practice 1: B. Find the value of x. 2
Example 2: ARCHITECTURE 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. Guided Practice 2: A pottery mold makes bowls that are in the shape of a regular heptagon. Find the measure of one of the interior angles of the bowl. Find Number of Sides Given Interior Angle Measure Example 3: The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon. Guided practice 3: The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. 3
Find Exterior Angle Measures of a Polygon Example 4: A. Find the value of x in the diagram. Example 4: B. Find the measure of each exterior angle of a regular decagon. 4
Guided Practice 4: A. Find the value of x in the diagram. Guided Practice 4: B. Find the measure of each exterior angle of a regular pentagon. 5
6.2 Parallelograms Notes 6
Use Properties of Parallelograms Examples1: A. CONSTRUCTION In Find AD. ABCD suppose m B = 32, CD = 80 inches, BC = 15 inches. Examples 1: B. CONSTRUCTION In ABCD suppose m B = 32, CD = 80 inches, BC = 15 inches. Find m C. Example 1: C. CONSTRUCTION In Find m D. ABCD suppose m B = 32, CD = 80 inches, BC = 15 inches. Guided Practice 1: A. ABCD is a parallelogram. Find AB. Guided Practice 1: B. ABCD is a parallelogram. Find m C. Guided Practice 1: C. ABCD is a parallelogram. Find m D. 7
More practice: Find the value of each variable in the parallelogram. 1. 2. 3. 4. 8
Use Properties of Parallelograms and Algebra Example 2: A. If WXYZ is a parallelogram, find the value of r. Example 2: B. If WXYZ is a parallelogram, find the value of s. Example 2: C. If WXYZ is a parallelogram, find the value of t. Guided Practice 2: A. If ABCD is a parallelogram, find the value of x. Guided Practice 2: B. If ABCD is a parallelogram, find the value of p. Guided Practice 2: C. If ABCD is a parallelogram, find the value of k. 9
More practice: Find the missing variables. 5. Parallelograms and Coordinate Geometry 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)? Guided Practice 3: What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with vertices L(0, 3), M( 2, 1), N(1, 5), O(3, 1)? More examples: 6. Determine the coordinates of the intersection of the diagonals of parallelogram FGHJ with vertices F( 2, 4), G(3, 5), H(2, 3), and J( 3, 4). Steps: 1. Name the diagonals. 2. Find the midpoint of the diagonals. 10
11
6.3 Tests for Parallelograms Notes 12
Identify Parallelograms Example 1: Determine whether the quadrilateral is a parallelogram. Justify your answer. Guided Practice 1: Which method would prove the quadrilateral is a parallelogram? A. Both pairs of opp. sides. B. Both pairs of opp. sides.c. C. Both pairs of opp. s. D. One pair of opp. sides both and. Mores Examples: Determine whether the quadrilateral is a parallelogram. Justify your answer. 1. 2. 3. 13
Use Parallelograms to Prove Relationships Example 2: MECHANICS Scissor lifts, like the platform lift shown, are commonly applied to tools intended to lift heavy items. In the diagram, A C and B D. Explain why the consecutive angles will always be supplementary regardless of the height of the platform. Guided Practice 2: 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. Use Parallelograms and Algebra to Find Values Example 3: Find x and y so that the quadrilateral is a parallelogram. Guided Practice 3: Find m so that the quadrilateral is a parallelogram. 14
More Examples: 4. If FK = 3x 1, KG = 4y + 3, JK = 6y 2, and KH = 2x + 3, find x and y so that the quadrilateral is a parallelogram. Example 4: COORDINATE GEOMETRY Graph 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. Guided Practice 4: Graph quadrilateral EFGH with vertices E( 2, 2), F(2, 0), G(1, 5), and H( 3, 2). Determine whether the quadrilateral is a parallelogram. 15
More Examples: Graph each quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify your answer. 7. K(2, 3), L(8, 4), M(7, 2), and N(1, 3) Steps: 1. Find the distance of each side. 2. Find the slopes of sides. Example 5: A student is given the following information and then asked to write a paragraph proof. Determine which statement would correctly complete the student s proof. Given: Parallelogram PRST and Parallelogram PQVU Prove: V S Proof: We are given Parallelogram PRST and Parallelogram PQVU. Since opposite angles of a parallelogram are congruent, P V and P S.. A. Therefore, V S by the Transitive Property of Congruence. B. Therefore, V S by the Transformative Property of Congruence. C. Therefore, V S by the Reflective Property of Congruence. D. Therefore, V S by the Reflexive Property of Congruence. 16
6.4 Rectangles Notes Example 1: CONSTRUCTION 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. Guided Practice 1: Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ. Use Properties of Rectangles and Algebra Example 2: Quadrilateral RSTU is a rectangle. If m RTU = 8x + 4 and m SUR = 3x 2, find x. 17
Guided Practice 2: Quadrilateral EFGH is a rectangle. If m FGE = 6x 5 and m HFE = 4x 5, find x. Proving Rectangle Relationships Example 3: 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. 18
Guided Practice 3: Max is building a swimming pool in his backyard. He measures the length and width of the pool so that opposite sides are parallel. He also measures the diagonals of the pool to make sure that they are congruent. How does he know that the measure of each corner is 90? More examples: A rectangular park has two walking paths as shown. 1. If PS = 180 meters and PR = 200 meters, find QT. 2. If m PRS = 64, find m SQR. 3. Using the rectangular park, if TS = 120 meters find PR. 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 slopes. 19
Guided Practice 4: Quadrilateral WXYZ has vertices W( 2, 1), X( 1, 3), Y(3, 1), and Z(2, 1). Determine whether WXYZ is a rectangle by using the Distance Formula. More examples: 4. Quadrilateral PQRS has verticies P( 5, 3), Q(1, 1), R( 1, 4), and S( 7, 0). Determine whether PQRS is a rectangle. Find the area. 1. Find the distance of each side. 2. Find the distance of the diagonals. 3. Find the area. 20
5. What is the value of x in the rectangle? 6. What is the value of x in the rectangle? 21
6.5 Rhombi and Squares Notes Use Properties of a Rhombus Example 1: A. The diagonals of rhombus WXYZ intersect at V. If m WZX = 39.5, find m ZYX. Example 1: B. ALGEBRA The diagonals of rhombus WXYZ intersect at V. If WX = 8x 5 and WZ = 6x + 3, find x. 22
Guided Practice 1: A. ABCD is a rhombus. Find m CDB if m ABC = 126. Guided Practice1: B. ABCD is a rhombus. If BC = 4x 5 and CD = 2x + 7, find x. More Examples: The diagonals of rhombus FGHJ intersect at K. Use the given information to find the each measure or value. 1. If m FJH = 82, find m KHJ. 2. If GH = x + 9 and JH = 5x 2, find x. The diagonals of rhombus WXYZ intersect at V. Use the given information to find the each measure or value. 3. If m WZX = 39.5, find m ZYX. 4. If WX = 8x 5 and WZ = 6x + 3, find x. 23
5. JKLM is a rhombus. If m JML = 84 and m JKM = (5x + 16), find the value of x. 24
Example 2: Write a paragraph proof. Given: LMNP is a parallelogram. 1 2 and 2 6 Prove: LMNP is a rhombus. Guided Practice 2: Is there enough information given to prove that ABCD is a rhombus? Given: ABCD is a parallelogram. AD DC Prove: ADCD is a rhombus Example 3: GARDENING 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? 25
Guided Practice 3: Sachin has a shape he knows to be a parallelogram and all four sides are congruent. Which information does he need to know to determine whether it is also a square? A. The diagonal bisects a pair of opposite angles. B. The diagonals bisect each other. C. The diagonals are perpendicular. D. The diagonals are congruent. Classify Quadrilaterals Using Coordinate Geometry 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. Guided Practice 4: Determine whether parallelogram EFGH is a rhombus, a rectangle, or a square for E(0, 2), F( 3, 0), G( 1, 3), and H(2, 1). List all that apply 26
More examples: Guided Practice 5. Determine whether parallelogram JKLM with vertices J( 7, 2), K(0, 4), L(9, 2), and M(2, 4) is a rhombus, a rectangle, or a square. List all that apply. Explain (Remember to find distances, slopes, midpoints to help explain.) 27
6.6 Trapezoids and Kites Notes Use Properties of Isosceles Trapezoids Example 1: A. BASKET Each side of the basket shown is an isosceles trapezoid. If m JML = 130, KN = 6.7 feet, and MN = 3.6 feet, find m MJK. Example 1: B. BASKET Each side of the basket shown is an isosceles trapezoid. If m JML = 130, KN = 6.7 feet, and MN = 3.6 feet, find JL. 28
Guided Practice 1: A. Each side of the basket shown is an isosceles trapezoid. If m FGH = 124, FI = 9.8 feet, and IG = 4.3 feet, find m EFG. Guided Practice 1: B. Each side of the basket shown is an isosceles trapezoid. If m FGH = 124, FI = 9.8 feet, and EG = 14.1 feet, find IH. 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. Guided Practice 2: Quadrilateral QRST has vertices Q( 1, 0), R(2, 2), S(5, 0), and T( 1, 4). Determine whether QRST is a trapezoid and if so, determine whether it is an isosceles trapezoid. 29
Example 3: In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x? Guided Practice 3: WXYZ is an isosceles trapezoid with median Find XY if JK = 18 and WZ = 25. 30
More examples: The speaker shown is an isosceles trapezoid. If m FJH = 85, FK = 8 inches, and JG = 19 inches, find each measure. 1. m FGH 2. KH Each side of the basket shown is an isosceles trapezoid. If m JML = 130, KN = 6.7 feet, and MN = 3.6 feet, find each measure. 3. m MJK 4. JL 5. In the figure LH is the midsegment of trapezoid FGJK. What is the value of x? 31
6. Quadrilateral ABCD has vertices A( 3, 4), B(2, 5), C(3, 3), and D( 1, 0). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. 7. 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. 32
Example 4: A. If WXYZ is a kite, find m XYZ. Example 4: B. If MNPQ is a kite, find NP. Guided Practice 4: A. If BCDE is a kite, find m CDE. Guided Practice 4: B. If JKLM is a kite, find KL. More examples: 9. If FGHJ is a kite, find m GFJ. 10. If WXYZ is a kite, find ZY. 33