Chapter 8. Properties of Quadrilaterals
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1 Chapter 8 Properties of Quadrilaterals
2 8.1 Properties of Parallelograms Objective: To use the properties of parallelograms Parallelogram Theorem Description Picture Theorem 8.1 The opposite sides of a parallelogram are congruent. Theorem 8.2 The opposite angles of a parallelogram are congruent. Theorem 8.3 The consecutive angles of a parallelogram are supplementary. Theorem 8.4 The diagonals of a parallelogram bisect each other. Example One: Find the m DAB Find the m ADC Example Two: Find KL If m LKN = 58 0, and m KNL = 32 0, then m MNL =, and m LMN = Example Three: Find all the congruent parts of UVWX.
3 8.2 Properties of Rhombuses Objective: To use the properties of rhombuses. Theorem Description Picture Theorem 8.5 Theorem 8.6 Each diagonal of a rhombus bisects a pair of opposite angles. Each diagonal of a rhombus is the perpendicular bisector of the other. Example One: BEC = If AC = 12 ft, AE = ABD = BAE = Example Two: ABCD is a Rhombus If AE = 12, then CE = If DE =9, then DB = If AD =13, then AB =
4 Is a square a type of rhombus? Is a rhombus a type of square? Is a rhombus a type of parallelogram? Use this figure to create a rhombus. True/false. The diagonals of any rhombus are congruent. True/false. Each diagonal of a rhombus bisects a pair of angles. True/false. Every rhombus has point symmetry. True/false. Every rhombus has two lines of symmetry.
5 8.3 Properties of Rectangles and Squares Objective: To use the properties of rectangles and squares. Theorem Description Picture Theorem 8.7 The diagonal of a rectangle are congruent. Square Rectangle Examples: ABCD is a rectangle. AC = 5 cm. PHMT is a rectangle. Find each angle measure. m HTM = 21 0 DC = AD = DB = CX = m 1 = m 2 = m 3 = m 4 = m 5 =
6 MNOP is a square. MQ = 14.5mm How many lines of symmetry does a the figure have? Which lines of symmetry result in the square being a rhombus? m 1 = m 2 = Which lines of symmetry result in the square being a rectangle? m 3 = m 4 = QO = QN = PN = MN= MO= What are all the properties of a rectangle? What are all the properties of a square? Draw a parallelogram with diagonals that satisfy each of the given conditions: a. The diagonals are congruent but not perpendicular b. The diagonals are perpendicular but not congruent. c. The diagonals are congruent and perpendicular.
7 8.4 Properties of Trapezoids Objective: To Use the properties of trapezoids and isosceles trapezoids Theorem Description Picture Theorem 8.8. The median of any trapezoid is parallel to the bases and has a length equal to half the sum of the base lengths. Theorem 8.9 Base angles of an isosceles trapezoid are congruent. Examples: LMNP is a trapezoid. Name: a. Bases b. Legs c. Base angles d. The median Given NP = 24, LM = 12, & LQ = 6 find: e. QS = f. QP = g. m P = h. m 1 = i. m 2 = Does an isosceles trapezoid have point symmetry? Do all trapezoids have congruent diagonals? Do the diagonals of a trapezoid bisect a pair of angles?
8 Types of Quadrilaterals 8.5 Proving Properties of Quadrilaterals Objective: To use given information to prove a statement true. Give a reason for each statement: 1. Given: AB bisects CD at point X. Conclusion: CX DX 2. Given: Lines l and m meet to form a right angle Conclusion: l is perpendicular to m 3. Given: ABCD is a rectangle. Conclusion: AC and BD bisect each other.
9 8.6 Finding Quadrilaterals that are Parallelograms Objective: To determine whether a quadrilateral must be a parallelogram, rectangle, rhombus or square. What information would you need for a quadrilateral to be a : a) Parallelogram b) Rhombus c) Rectangle d) Square If.then statements :
10 Theorem If. Then Picture Theorem 8.12 Theorem 8.13(a) Theorem 8.13(b) Theorem 8.13(c) Theorem 8.14 Theorem 8.15 Theorem 8.16 If the diagonals of a quadrilateral bisect each other, If the diagonals of a quadrilateral bisect each other, and are congruent If the diagonals of a quadrilateral bisect each other, and are perpendicular If the diagonals of a quadrilateral bisect each other, and are congruent and perpendicular If both pairs of opposite sides of a quadrilateral are congruent, If both opposite angels of a quadrilateral are congruent, If two sides of a quadrilateral are both parallel and congruent Must each quadrilateral be a parallelogram? If yes, why?
11 8.7 Problem Solving Strategy: Finding a Pattern Objective: To solve problems by finding a pattern. STRATEGY WHEN TO USE IT Finding a Pattern The problem gives information or data that can be observed to have a pattern. **To use this strategy successfully, you need to be sure the pattern will really continue. 1. ABCH, HCDG, and GDEF are trapezoids. The dashed lines are medians. a. Above each median, write its length. b. Seven numbers appear in the figure. Describe the pattern in the numbers. c. Describe the pattern in the lengths of the medians. d. Suppose there are nine more trapezoids that were drawn using the same pattern. Find the length of the median of the last trapezoid. 2. ABCD is a rhombus. a. For each rhombus, find the m A. b. Describe the pattern in the m A. c. Suppose that there are 10 more rhombuses drawn in the same pattern Find m A in the final rhombus.
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