2. Each side intersects exactly two other sides, one at each endpoint.
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1 Name Period Quadrilaterals Chapter 6 - GEOMETRY Section 6.1 Polygons GOAL 1: Describing Polygons A polygon is a plane figure that meets the following conditions. 1. It is formed by three or more segments called sides, such that no two sides with a common endpoint are collinear. 2. Each side intersects exactly two other sides, one at each endpoint. Ex. 1 Decide whether the figure is a polygon. If not, explain why A vertex is. The plural of vertex is. You can name a polygon by listing its vertices consecutively. PQRST is one way to name this polygon. What is another way? Polygons are also named by the number of sides they have. # of sides Type of polygon # of sides Type of polygon # of sides Type of polygon 3 Triangle 7 Heptagon 12 Dodecagon 4 Quadrilateral 8 Octagon n n-gon 5 Pentagon 9 Nonagon 6 Hexagon 10 Decagon A polygon is convex if A polygon is concave if Ex. 2 Use the number of sides to tell what kind of polygon the shape is. Then state whether the polygon is convex orconcave A diagonal of a polygon is a. 1
2 Ex. 3 Use the diagram at the right to answer the following. 9. Name the polygon by the number of sides it has. 10. Polygon MNOPQR is one name. State two other names. 11. Name all of the diagonals that have vertex M as an endpoint. 12. Name the consecutive angles to N. A polygon is equilateral if. A polygon is equiangular if. A polygon is regular if. Ex. 4 State whether the polygon is best described as equilateral, equiangular, regular, or none of these GOAL 2: Interior Angles of Quadrilaterals Theorem 6.1 Interior Angles of a Quadrilateral The sum of the measures of the interior angles of a quadrilateral is 360. m 1 m 2 m 3 m Ex. 5 Use the information in the diagram to solve for x Section 6.2 Properties of Parallelograms GOAL 1: Properties of Parallelograms A parallelogram is a. 2
3 Ex. 1 Mark the congruences for the theorems below. Theorem 6.2 If a quadrilateral is a parallelogram, then its opposite sides are congruent. PQ RS and SP QR Theorem 6.3 If a quadrilateral is a parallelogram, then its opposite angles are congruent. P R and Q S Theorem 6.4 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. m P m Q 180, m Q m R 180, m R m S 180, m S m P 180 Theorem 6.5 If a quadrilateral is a parallelogram, then its diagonals bisect each other. QM SM and PM RM Ex. 2 Lets prove Theorem 6.3 in a paragraph proof. Given: ABCD is a parallogram. Prove: A C and B D Opposite sides of a parallelogram are congruent, so and. By the Reflexive Property of Congruence,. ABD CDB because of the Congruence Postulate. Because parts of congruent triangles are congruent, A C. Now draw diagonal AC. By use of the same reasoning, B D. Ex. 3 Decide whether the figure is a parallelogram. If it is not, explain why not
4 Ex. 4 Use the diagram of parallelogram MNOP at the right. Complete the statement and give a reason. 4. MN 5. MN P 6. ON 7. MPO 8. PQ 9. QM 10. MQN 11. NPO Ex. 5 Find the measure in the parallelogram HIJK. Explain your reasoning. 12. HI 13. KH 14. GH 14. HJ 16. m KIH 17. m JIH 18. m KJI 19. m HKI Ex. 6 Find the value of each variable in the parallelogram Section 6.3 Proving Quadrilaterals are Parallelograms GOAL 1: Proving Quadrilaterals are Parallelograms Theorem 6.6 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6.8 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. 4
5 We can also use the definition of a parallelogram to prove that a quadrilateral is a parallelogram. If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram. Ex. 1 Name 6 ways to prove that a quadrilateral is a parallelogram. Ex. 2 Are you given enough information to determine whether the quadrilateral is a parallelogram? Ex. 3 What additional information is needed in order to prove that quadrilateral ABCD is a parallelogram? 7. AB PDC 8. AB DC 9. DCA BAC 10. DE EB 11. m CDA m DAB 180 Ex. 4 What value of x and y will make the polygon a parallelogram?
6 GOAL 2: Using Coordinate Geometry When a figure is in the coordinate plane, you can use the Distance Formula to prove that sides are congruent and you can use the slope formula to prove that sides are parallel. Ex. 5 Prove that the points represent the vertices of a parallelogram. Use two different methods. A( 2, -1), B( 1, 3), C( 6, 5), D( 7, 1) Ex. 6 Draw the quadrilateral ABCD. If the hat rack were expanded outward, would ABCD still be a parallelogram? Explain. Section 6.4 Rhombuses, Rectangles, and Squares GOAL 1: Properties of Special Parallelograms In this lesson you will study three special types of parallelograms: rhombuses, rectangles, and squares. A rhombus is a A rectangle is a A square is a You can use the following corollaries to prove that a quadrilateral is a rhombus, rectangle, or square without proving first that the quadrilateral is a parallelogram. Rhombus Corollary A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary A quadrilateral is a rectangle if and only if it has four right angles. Square Corollary A quadrilateral is a square if and only if it is a rhombus and a rectangle. Ex. 1 Decide whether the statement is sometimes, always, or never true. 1. A square is a rectangle. 2. A parallelogram is a rhombus. 3. A rectangle is a square. 4. A rhombus is a rectangle. 5. A parallelogram is a rectangle. 6. A square is a parallelogram. 6
7 GOAL 2: Using Diagonals of Special Parallelograms The following theorems are about diagonals of rhombuses and rectangles. Theorem 6.11 A parallelogram is a rhombus if and only if its diagonals are perpendicular. ABCD is a rhombus if and only if AC BD Theorem 6.12 A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. ABCD is a rhombus if and only if AC bisects DAB and BCD and BD bisects ADC and CBA Theorem 6.13 A parallelogram is a rectangle if and only if its diagonals are congruent. ABCD is a rectangle if and only if AC BD Remember that is a square is both a rectangle and a rhombus. Ex. 2 List everything you know about squares. (Hint: List everything about parallelograms, rectangles and rhombuses. Ex. 3 Match the properties of a quadrilateral with all of the types of quadrilateral which have that property. 7. The diagonals are congruent. A. Parallelogram 8. Both pairs of opposite sides are congruent. B. Rectangle 9. Both pairs of opposite sides are parallel. C. Rhombus 10. All angles are congruent. D. Square 11. All sides are congruent. 12. Diagonals bisect the angles. Ex. 4 Decide whether the statement is sometimes, always, or never true. 13. A rhombus is equilateral. 14. The diagonals of a rectangle are perpendicular. 15. The opposite angles of a rhombus are supplementary. 16. The diagonals of a rectangle bisect each other. 17. The consecutive angles of a square are supplementary. 7
8 Ex. 5 Find the value of x. 18. MNOP is a square 19. DEFG is a rhombus. 20. WZYZ is a rectangle. Section 6.5 Trapezoids and Kites GOAL 1: Using Properties of Trapezoids A trapezoid is a. The parallel sides are the. A trapezoid has two pairs of. The nonparallel sides are the of the trapezoid. If the legs of a trapezoid are congruent, then the trapezoid is an. Ex. 1 Match the pairs of segments or angles with the term, which describes them in trapezoid PQRS. 1. S and P A. bases 2. QS and PR B. legs 3. QR and PS C. diagonals 4. Q and S D. base angles 5. PQ and RS E. opposite angles Theorem 6.14 If a trapezoid is isosceles, then each pair of base angles is congruent. A B, C D Theorem 6.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. ABCD is an isosceles trapazoid. Theorem 6.16 A trapezoid is isosceles if and only if its diagonals are congruent. ABCD is isosceles if and only if AC BD 8
9 Ex. 2 Complete the statement with always, sometimes or never. 6. A trapezoid is a parallelogram. 7. The bases of a trapezoid are parallel. 8. The base angles of an isosceles trapezoid are congruent. 9. The legs of a trapezoid are congruent. Ex. 3 Find the angle measures of ABCD The midsegment of a trapezoid is the. Theorem 6.17 Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. MN PAD, MN PBC, MN 1 (AD BC) 2 Ex. 4 Find the length of the midsegment RT GOAL 2: Using Properties of Kites A kite is a. Theorem 6.18 If a quadrilateral is a kite, then its diagonals are perpendicular. Theorem 6.19 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. 9
10 Ex. 5 Find the length of the sides to the nearest hundredth or the measure of the angles in kite KITE Section 6.6 Special Quadrilaterals GOAL 1: Summarizing Properties of Quadrilaterals Ex. 1 Summarize the seven special types of quadrilaterals in a diagram. Ex. 2 Put an X in the box if the shape always has the given property. Property gram Rectangle Rhombus Square Kite Trapezoid Isosceles Trapezoid Both pairs of opp. sides are Exactly 1 pair of opp. sides are Diagonals are Diagonals are Diagonals bisect each other Both pairs of opp. Sides are Exactly 1 pair of opp. Sides are All sides are Both pairs of opp. 's are Exactly 1 pair of opp. 's are All 's are Area 10
11 Ex. 3 Identify the special quadrilateral. Use the most specific name Ex. 4 What quadrilateral meet the conditions shown? ABCD is not drawn to scale Section 6.7 Areas of Triangles and Quadrilaterals GOAL 1: Using Area Formulas Area Postulates Postulate 22 Area of a Square Postulate The area of a square is the square of the length of its side, or A s 2 Postulate 23 Area Congruence Postulate If two polygons are congruent, then they have the same area. Postulate 24 Area Addition Postulate The area of a region is the sum of the areas if its nonoverlapping parts. Area Theorems Theorem 6.20 Area of a Rectangle The area of a rectangle is the product of its base and height. A bh Theorem 6.21 Area of a Parallelogram The area of a parallelogram is the product of a base and its corresponding height. A bh Theorem 6.22 Area of a Triangle The area of a triangle is one half the product of a base and its corresponding height. A 1 2 bh 11
12 Ex. 1 Find the area of the polygon GOAL 2: Areas of Trapezoids, Kites, and Rhombuses Theorem 6.23 Area of a Trapazoid The area of a trapezoid is on half the product of the height and the sum of the basses. A 1 2 h(b 1 b 2 ) Theorem 6.24 Area of a Kite The area of a kite is one half the product of the lengths of its diagonals. A 1 2 d 1 d 2 Theorem 6.25 Area of a Rhombus The area of a rhombus is equal to one half the product of the lengths of the diagonals. A 1 2 d 1 d 2 Ex. 2 Find the area of the polygon
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