6.1: Date: Geometry. Polygon Number of Triangles Sum of Interior Angles

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1 6.1: Date: Geometry Polygon Number of Triangles Sum of Interior Angles Triangle: # of sides: # of triangles: Quadrilateral: # of sides: # of triangles: Pentagon: # of sides: # of triangles: Hexagon: # of sides: # of triangles: Heptagon: # of sides: # of triangles: Octagon: # of sides: # of triangles:

2 Theorem 6-1: Polygon Angle-Sum Theorem The sum of the measures of the angles of an n-gon is: Ex 1). What is the sum of the angle measures of a 10-gon? Ex 2). What is the sum of the angle measures of a 13-gon? An polygon An polygon A polygon is a polygon with all sides is a polygon with all angles is a polygon that is both congruent. congruent. equilateral & equiangular. Corollary to the Polygon Angle-Sum Theorem The measure of angle of a polygon is Ex 3). Marcy creates a floor tile pattern using squares, regular hexagons, and regular dodecagons (12-sided polygon). What is the measure of each angle in one regular dodecagon?

3 Ex 4). What is m D in quadrilateral ABCD? Theorem 6-2: Polygon Exterior Angle-Sum Theorem The sum of the measures of the angles of a polygon, one at each vertex, is. Ex 5). What is the measure of an exterior angle of a regular hexagon? Ex 6). Find the value of each variable. Ex 7). Find the value of each variable. Homework: pg. 374 #1 3, 5, 7 13, 15 23, 26 28

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5 6.2 Day 1: Date: Geometry Properties of Parallelograms

6 Ex 1). KLMN is a parallelogram. Find MN and KN. Ex 2). STVW is a parallelogram. Find the missing angle measures. Ex 3). JKLM is a parallelogram. Find JN. Ex 4). Find the values of x and y in the parallelogram. Homework: pg. 381 #1 4, 7 10, 12 16, 21 23

7 6.2 Day 2: Date: Geometry Ex 1). Solve a system of linear equations to find the values of a and b in parallelogram HIJK. What are HJ and IK? Ex 2). Solve a system of linear equations to find the values of x and y in the parallelogram. Ex 3). Solve a system of linear equations to find the values of x and y in the parallelogram.

8 Theorem 6-7: If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Ex 4). In the figure below, RW SV and SV TU. If RS = ST = 5 and WV = 7, what is WU? Ex 5). Find each length. a) EB: d) BD: b) AF: e) AK: c) CD: f) GJ: Homework: pg. 385 #1, 2 15, 21 28

9 6.3: Date: Geometry The following theorems are converses of the theorems from lesson 6-2. If a quadrilateral has any of the following properties, then it must be a. Theorem 6-8: If both pairs of opposite sides of a quadrilateral are, then the quadrilateral is a parallelogram. Theorem 6-9: If an angle of a quadrilateral is supplementary to both of its angles, then the quadrilateral is a parallelogram. Theorem 6-10: If both pairs of opposite angles of a quadrilateral are, then the quadrilateral is a parallelogram. Theorem 6-11: If the diagonals of a quadrilateral, then the quadrilateral is a parallelogram. Theorem 6-12: If one pair of opposite sides is both congruent and, then the quadrilateral is a parallelogram. Ex 1). For what value of x must RSTU be a parallelogram? Ex 2). For what values of x and y must EFGH be a parallelogram? Ex 3). Can you prove that the quadrilateral is a parallelogram based on the given information? Explain. a) b) c)

10 Ex 4). Can you prove the quadrilateral is a parallelogram based on the given information? Explain. a) Given: AE = CE = 14, DB = 2DE b) Given: m Y = 49, m Z = 131, m W = 49 Ex 5). A table tray can be adjusted up or down by raising or lowering the table on its hinged legs as shown. Will the table always remain parallel to the surface it sits on? Explain. Ex 6). For what value of x must the quadrilateral be a parallelogram? a) b) c) d) Homework: pg. 392 #1 3, 7 14, 19 21, 27, 28

11 6.4: Date: Geometry Special Parallelograms: A is a parallelogram with sides. A is a parallelogram with angles. A is a parallelogram with sides and angles. Ex 1). Is parallelogram FGHJ a rhombus, a rectangle, or a square? Explain. Ex 2). List the quadrilaterals that have the given property. Choose among parallelogram, rhombus, rectangle, and square. a) Opposite angles are supplementary b) Consecutive sides are congruent c) Consecutive sides are perpendicular d) Consecutive angles are congruent Ex 3). Decide whether the parallelogram is a rhombus, rectangle, or square. Explain. a) b) c)

12 More on Rhombuses: Theorem 6-13: If a parallelogram is a rhombus, then its diagonals are. Theorem 6-14: If a parallelogram is a rhombus, then each diagonal a pair of opposite angles. Ex 4). What are the measures of the numbered angles in the rhombus? a) b) More on Rectangles: If a parallelogram is a rectangle, then its diagonals are. Ex 5). In rectangle MNOP, PN = 7x 8 and MO = 4x What is the length of PN? FYI: Everything that is true for a rhombus and rectangle is true for a square! Homework: pg. 400 #1 4, 9 23, 24 32(e), 34, 35

13 6.5: Date: Geometry The following theorems are converses of lesson 6.4. Each theorem allows you to prove that a parallelogram is either a rhombus or a rectangle. Theorem 6-16: If the diagonals of a parallelogram are, then the parallelogram is a. Theorem 6-17: If one diagonal of a parallelogram a pair of opposite angles, then the parallelogram is a. Theorem 6-18: If the diagonals of a parallelogram are, then the parallelogram is a. *Notice that if a parallelogram is a rectangle and a rhombus, then it is a. Ex 1). Can you conclude that the parallelogram is a rhombus, rectangle, or square? Explain. a) b) Ex 2). For what value of x is ABCD a square?

14 Ex 3). For what value of x is DEFG a rhombus? Ex 4). LN = 54. For what values of x is LMNO a rectangle? Ex 5). For what value of x is RSTU a rhombus? What is m SRT? What is m URS? Homework: pg. 407 #1 5, 8 13, 16 18, 23 29

15 6.6: Date: Geometry TRAPEZOID Theorem 6-19: If a quadrilateral is an isosceles trapezoid, then each pair of base angles is. Ex 1). RSTU is an isosceles trapezoid and m S = 75. What are m R, m T, and m U? Ex 2). LMNO is an isosceles trapezoid and m L = 75. What are m M, m N, and m O? Theorem 6-20: If a quadrilateral is an isosceles trapezoid, then its diagonals are. Ex 3). WXYZ is an isosceles trapezoid, Ex 4). AC = x + 5 and DB = 2x 2. and WY = 12, what is XZ? Find the value of x and each diagonal.

16 Theorem 6-21: If a quadrilateral is a trapezoid then, 1. The midsegment is to the bases 2. The length of the midsegment is of the sum of the lengths of the bases. Ex 5). TU is the midsegment of trapezoid WXYZ. What is x? Ex 6). Find GH. Ex 7). Find the lengths of the segments with variable expressions. KITE Theorem 6-22: If a quadrilateral is a kite, then its diagonals are.

17 Ex 8). ABCD is a kite. What are m 1 and m 2? The following are kites, find the missing angles: Ex 9). Ex 10). Find the value of the variable in each kite. Ex 11). Ex 12). Ex 13). Determine whether each statement is true or false. Explain. a) All kites are quadrilaterals b) A kite is a parallelogram c) A kite can have congruent diagonals d) Both diagonals of a kite bisect angles at the vertices. Homework: 6.6 Practice Worksheet

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19 6.7: Date: Geometry You will need Distance Formula: Slope Formula: Classifying Triangles: Scalene Isosceles Equilateral Ex 1). Is Triangle RST scalene, isosceles, or equilateral? Ex 2). Is parallelogram ABCD a rhombus? Explain.

20 Ex 3). Parallelogram MNPQ has vertices M(0, 1), N(-1, 4), P(2, 5), and Q(3, 2). Is MNPQ a rectangle? Is it a square? Ex 4). Determine whether the parallelogram is a rhombus, rectangle, square, or none. Explain. P(-1, 2), O(0, 0), S(4, 0), T(3, 2) Homework: pg. 428 #1, 2, 5 7, 10, 12, 23

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