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Geometry 2206 Unit 3: Mrs. Bondi Unit 3: Two-Dimensional Shapes Lesson Topics: Lesson 1: Congruent Figures (PH text 4.1) Lesson 2: Triangle Congruence by SSS and SAS (PH text 4.2) Lesson 3: Triangle Congruence by ASA and AAS (PH text 4.3) Lesson 4: Using Corresponding Parts of Congruent Triangles (PH text 4.4) Lesson 5: Isosceles and Equilateral Triangles (PH text 4.5) Lesson 6: Congruence in Right Triangles (PH text 4.6) Lesson 7: Congruence in Overlapping Triangles (PH text 4.7) Lesson 8: Midsegments of Triangles (PH text 5.1) Lesson 9: Perpendicular and Angle Bisectors (PH text 5.2) Lesson 10: Bisectors in Triangles (PH text 5.3) Lesson 11: Medians and Altitudes (PH text 5.4) Lesson 12: Indirect Proof (PH text 5.5) Lesson 13: Inequalities in One Triangle (PH text 5.6) Lesson 14: Inequalities in Two Triangles (PH text 5.7) Lesson 15: The Polygon-Angle Sum Theorems (PH text 6.1) Lesson 16: Properties of Parallelograms (PH text 6.2) Lesson 17: Proving that a Quadrilateral is a Parallelogram (PH text 6.3) Lesson 18: Properties of Rhombuses, Rectangles, and Squares (PH text 6.4) Lesson 19: Conditions for Rhombuses, Rectangles, and Squares (PH text 6.5) Lesson 20: Trapezoids and Kites (PH text 6.6) Lesson 21: Polygons in the Coordinate Plane (PH text 6.7) Lesson 22: Applying Coordinate Geometry (PH text 6.8) Lesson 23: Proofs Using Coordinate Geometry (PH 6.9) Lesson 24: Proportions in Triangles (PH text 7.5) Lesson 25: Areas of Parallelograms and Triangles (PH text 10.1) Lesson 26: Areas of Trapezoids, Rhombuses and Kites (PH text 10.2) Lesson 27: Areas of Regular Polygons (PH text 10.3) Lesson 28: Perimeters and Areas of Similar Figures (PH text 10.4) Lesson 29: Trigonometry and Area (PH text 10.5) 1 Geometry 2206 Unit 3: Mrs. Bondi Lesson 15: The Polygon-Angle Sum (PH text 6.1) Objective: to find the sum of the both interior and exterior angles of a polygon Theorem 6-1 Polygon Angle-Sum Theorem The sum of the measures of the interior angles of an n-gon is __________________ Example 1: If m∠B = m∠D, find m∠B in ABCD. B A 100° 40° D Equiangular Polygon C Equilateral Polygon Regular Polygon Corollary to Polygon Interior Angle-Sum Theorem The measure each interior angle of a regular n-gon is __________________ Example 2: What is the measure each interior angle of a regular nonagon? 2 Geometry 2206 Unit 3: Theorem 6-2 Mrs. Bondi Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is ________ Example 3: Find the measure of each interior angle. What is the sum of the interior angles? Extend one side at each vertex and find the exterior angle that was formed. What is the sum of the exterior angles? Check out this explanation. http://www.mathsisfun.com/geometry/exterior-angles-polygons.html Example 4: Find the measure of an interior and an exterior angle of a regular pentagon. Practice: HW: p.356 #6-36 even 3 Geometry 2206 Unit 3: Mrs. Bondi Lesson 16: Properties of Parallelograms (PH text 6.2) Objective: to find relationships among angles, sides, and diagonals of parallelograms Parallelogram: Theorem 6-3 Opposite sides of a parallelogram are congruent. Prove the theorem. Given: ABCD Prove: AB CD BC DA B A Statement C D Reason Consecutive Angles: Theorem 6-4 Theorem 6-5 Consecutive angles of a parallelogram are supplementary. Opposite angles of a parallelogram are congruent. A 4 B C A D B C D Geometry 2206 Unit 3: Theorem 6-6 Mrs. Bondi The diagonals of a parallelogram bisect each other. B C A Theorem 6-7 D If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. If AB CD EF and AC CE , then BD DF . A C E (Try dividing an index card into four congruent vertical sections using notebook paper.) Practice: HW: p.364 #9-40, 43 (at least 4 proofs) 5 B D F Geometry 2206 Unit 3: Mrs. Bondi Lesson 17: Proving that a Quadrilateral is a Parallelogram (PH text 6.3) Objective: Theorem 6-8 Prove it: Given: Prove: to determine whether a quadrilateral is a parallelogram If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. B AB CD BC DA ABCD is a C A Statement D Reason Theorem 6-9 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Theorem 6-10 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 6 Geometry 2206 Unit 3: Mrs. Bondi Example 1: Find the value of each variable to make the figure a parallelogram. Theorem 6-11 Prove it: Given: Prove: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. B AC and BD bisect each other at E ABCD is a parallelogram A Statement Theorem 6-12 D Reason If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. 7 C Geometry 2206 Unit 3: Mrs. Bondi Is it possible to prove that the quadrilateral is a parallelogram based on the given information? Explain. 8 Geometry 2206 Unit 3: Mrs. Bondi Practice: HW: p.372 # 6-24 + two additional proofs 9 Geometry 2206 Unit 3: Mrs. Bondi Lesson 18: Properties of Rhombuses, Rectangles, and Squares (PH text 6.4, 6.6) Objectives: to define and classify special types of parallelograms. to use properties of diagonals of rhombuses and rectangles Special Quadrilaterals: Parallelogram: Example: Rhombus: Rectangle: Example: Example: Square: Kite: Example: Example: Trapezoid: Isosceles Trapezoid: Example: Example: Example 1: Judging by appearance, name the quadrilateral in as many ways as possible. 10 Geometry 2206 Unit 3: Mrs. Bondi Example 2: Given C(1, 2), D(5, 2), E(7,-2), F(-1, -2), determine the most precise name for quadrilateral CDEF. Theorem 6-13 Prove it: Given: Prove: The diagonals of a rhombus are perpendicular. Rhombus ABCD AC BD B C A Statement D Reason 11 Geometry 2206 Unit 3: Theorem 6-14 Prove it: Given: Prove: Mrs. Bondi Each diagonal of a rhombus bisects the two opposite angles. Rhombus ABCD 1 2 3 4 B C 1 A 2 Statement 3 4 D Reason Example 3: If mABC = 120, find the measures of the numbered angles. B Rhombus ABCD C 5 1 A 12 2 3 4 D Geometry 2206 Unit 3: Mrs. Bondi Theorem 6-15 The diagonals of a rectangle are congruent. Prove it: Given: Prove: Rectangle ABCD AC DB Statement B C A D Reason Practice: HW: p.379 # 6-44 even, 50-56 even 13 Geometry 2206 Unit 3: Mrs. Bondi Lesson 19: Conditions for Rhombuses, Rectangles, and Squares (PH text 6.5) Objectives: Theorem 6-16 Prove it: Given: Prove: to determine whether a parallelogram is a rhombus or a rectangle. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. ABCD and AC BD ABCD is a rhombus B A Statement C D Reason Theorem 6-17 If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. Theorem 6-18 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. 14 Geometry 2206 Unit 3: Mrs. Bondi Examples: Use the given information to identify each quadrilateral with any names that work for it. Also identify its most specific name. (It is often helpful to draw a sketch.) 1. a diagonal bisects two angles 2. the diagonals are congruent 3. the diagonals bisect each other 4. the angle measures are 35, 145, 35, 145 5. the diagonals are perpendicular Example 6: For what values of x is B C 3x 7x - 2 A ABCD a rhombus? D Example 7: For what values of x is B ABCD a rectangle? C 7x - 2 A 5x - 8 D Practice: HW: p.386 #6-19, 23-24, 32-34 15 (The expressions are labeling the pieces of the diagonals.) Geometry 2206 Unit 3: Mrs. Bondi Lesson 20: Trapezoids and Kites (PH text 6.6) Objective: bases to verify and use properties of trapezoids and kites base angles trapezoid – isosceles trapezoid – Theorem 6-19 Example 1: legs Base angles of an isosceles trapezoid are congruent. 15 Find the height of the trapezoid. 13 13 25 Example 2: A circular dart board has 20 sections meeting in the middle. What are the base angles of the trapezoids formed in the second ring? Theorem 6-20 Prove it: Given: Prove: The diagonals of an isosceles trapezoid are congruent. B Isosceles Trapezoid ABCD with AB DC AC DB C A Statement D Reason 16 Geometry 2206 Unit 3: Mrs. Bondi Midsegment of a trapezoid – Theorem 6-21 Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to its bases AND the length of the midsegment is half the sum of the lengths of the bases Example 2: x+3 For what values of x can we call this a midsegment?. 2x - 4 24 kite – Theorem 6-22 Prove it: Given: Prove: The diagonals of a kite are perpendicular. R Kite RSTW with RS RW and ST WT RT SW W Statement S Reason T 17 Geometry 2206 Unit 3: Mrs. Bondi R Problem: Given Kite RSTW, find the value of RT. RS = 13 cm ST = 15 cm SW = 24 cm RT = ? W S T Example 3: Refer to the kite diagram below. If KT x 4 , KB 2x 4 , JB y 4 , and JT 2 x , find the values of x and y. T J K B Quadrilateral Summary: (include 8 types of figures) Practice: HW: p.394 #8-70 even (do at least two of the proofs) 18 Geometry 2206 Unit 3: Mrs. Bondi 19 Geometry 2206 Unit 3: Mrs. Bondi Mid-Chapter Quiz – p.398 Prepare for a quiz! Algebra Review – Simplifying Radicals – p.399 Be sure you remember these skills! More Radical Practice: 1. A pool is shaped like a rectangle with a length 4 times its width w. What is an expression for the distance between opposite corners of the pool? 2. Evelyn rode her horse along a triangular path. The distance she traveled south was five times the distance she traveled east. Then she rode directly back to her starting point. What is an expression for the total distance she rode? 3. You are making a mosaic design on a square table top. You have already covered half of the table top with 150 1-inch square tile pieces. a. What are the dimensions of the table top? b. What is the measure of the diagonal from one corner to the opposite corner of the table top? 4. The equation r SA gives the radius r of a sphere with surface area SA. What is the radius of a 4 sphere with the given surface area? Use 3.14 for π. a. 1256 in2. b. 200.96 cm 2 c. 379.94 ft2 6. Open-Ended What are three radical expressions that simplify to 2 x 3 ? 20 Geometry 2206 Unit 3: Mrs. Bondi Algebra Review: 21 Geometry 2206 Unit 3: Mrs. Bondi Lesson 21: Polygons in the Coordinate Plane (PH text 6.7) Objective: to classify polygons in the coordinate plane Reminder: In Coordinate Geometry, we combine our knowledge of geometry and algebra to identify shapes, find missing vertices, or prove shape characteristics. This is useful for many real-world applications. Examples: 1) Name the type of triangle is formed by A (2, 0), B (3, 4), C (-1 , 1) Use the distance formula to find length of: AB BC CA 22 Geometry 2206 Unit 3: Mrs. Bondi 2) Prove EFGH is a parallelogram. E (-5, -3), F (-2, 6), G (2 , 7), H (-1 , -2) Use the slope formula to find slope of: EF FG GH HE 3) Give the most precise name for JKLM . J (-2, 4), K (2, 2), L (4 , -2), M (-4 , -4) 23 Geometry 2206 Unit 3: Mrs. Bondi A quadrilateral formed by joining the midpoints of the sides of any quadrilateral will be a parallelogram. 4) Plot the vertices of kite ABCD. What is the shape of the quadrilateral formed by the midpoints? A (-2, 4), B (2, 2), C (4 , -2), D (-4 , -4) Use the midpoint formula to find midpoints of: AB BC CD DA (Is it a special parallelogram? Use slope to check for ║, and distance formula to check side lengths.) Practice: HW: p.404 #18-26 even, 31-33, 37-38, 45-50 24 Geometry 2206 Unit 3: Mrs. Bondi Lesson 22: Applying Coordinate Geometry (PH text 6.8) Objective: to name coordinates of special figures by using their properties We can use Coordinate Geometry to supply missing information. 1) Square ABCD Find the coordinates of the other vertices. A (-3, 3), B (-3, -3), C ( , ), D ( , ) Hints: If this is a square, … What do you know about side lengths? What do you know about the sides’ relationship to one another? What do you know about the slope of the sides? 2) Rhombus EFGH Find the coordinates of the other vertex. E (0, 0), F (3, 4), G (8 , 4 ), H ( , ) Hints: If this is a rhombus, … What do you know about side lengths? What do you know about the diagonals’ relationship to one another? What do you know about the slope of the diagonals? 25 Geometry 2206 Unit 3: 3) Mrs. Bondi Find the coordinates of the missing vertices of isosceles trapezoid EFGH. EF GH E (-1, -2), F (-1, 5), G (4 , 8), H ( , ) In Coordinate Geometry, we can use variables to name coordinates of a figure. This is useful to demonstrate a general case for a given situation. Use variables to name the coordinates of the vertices of each figure. Square Isosceles Triangle Rectangle Rhombus 26 Geometry 2206 Unit 3: Mrs. Bondi What do you know about the diagonals of a parallelogram? How can we find the coordinates of D? We can use coordinates with variables to prove a general geometric relationships. This is called a coordinate proof. Sometimes using a coordinate proof is the easiest way to prove a statement. 27 Geometry 2206 Unit 3: Mrs. Bondi Practice: HW: p.410 #8-30 even, 38-40 28 Geometry 2206 Unit 3: Mrs. Bondi Lesson 23: Proofs Using Coordinate Geometry (PH 6.9) Objective: to prove theorems using figures in the coordinate plane Find the coordinates of the midsegment of trapezoids ABCD and EFGH. Write a coordinate proof – use coordinate geometry to prove that the midpoint of the hypotenuse of right LMN is equidistant from the three vertices. (Hint: use midpoint and distance formulas) Given: LMN is a right triangle L (0, 0), M (2a, 0), N (0, 2b) P is the midpoint of MN LP = MP = NP Prove: Statement Reason 29 Geometry 2206 Unit 3: Mrs. Bondi Statement Given: MN is the midsegment of trapezoid TRAP. Prove: MN||TP, MN||RA, and MN = ½(TP + RA). R (2b, 2c) T (0, 0) A (2d, 2c) P (2a, 0) Practice: HW: p.416 #4-28 even, 34-40 30 Reason Geometry 2206 Unit 3: Mrs. Bondi 31 Geometry 2206 Unit 3: Mrs. Bondi 32 Geometry 2206 Unit 3: Mrs. Bondi 33 Geometry 2206 Unit 3: Mrs. Bondi Chapter 6 Review 34 Geometry 2206 Unit 3: Mrs. Bondi Chapter 6 Review (continued) 35 Geometry 2206 Unit 3: Mrs. Bondi After ch.6 test – get ready for ch.7, p.429, and ch.10, p.611 Additional Review: 1. The lengths of two sides of a polygon are in the ratio 2 : 3. Write expressions for the measures of the two sides in terms of the variable x. 2. ∆HJK ~ ∆RST. Complete each statement. 3. To the nearest inch, a door is 75 in. tall and 35 in. wide. What is the ratio of the width to the height? 4. What is a proportion that has means 9 and 10 and extremes 6 and 15? 5. Find the geometric mean of each pair of numbers. a. 4 and 25 b. 9 and 12 Solve for the value of the variables in each right triangle. 6. 7. 8. 9. 10. 11. 36 Geometry 2206 Unit 3: Mrs. Bondi Lesson 24: Proportions in Triangles (PH text 7.5) Objective: to use the Side-Splitter Theorem and the Triangle-Angle-Bisector Theorem Reminder: Theorem 7-4 Side-Splitter Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Example 1: Find the value of x. Example 2: Find the value of x. Reminder: Corollary to Side-Splitter Theorem If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. Example 3: Find the value of x and y. 37 Geometry 2206 Unit 3: Mrs. Bondi Theorem 7-5 Triangle-Angle-Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. Examples: 4. Find the value of x. 9 5. Find the value of x. 5 6 15 X 3 6. Find the value of x. 7. Find the value of x. 3 cm 15 cm 7 cm 8 x 30 Practice: HW: p.474 #8-40 even, 46-47 38 Geometry 2206 Unit 3: Mrs. Bondi Lesson 25: Areas of Parallelograms and Triangles (PH text 10.1) Objective: to find areas of parallelograms and triangles Theorem 10-1 Area of a Rectangle A = _________ Theorem 10-2 Area of a Parallelogram A = _________ Theorem 10-3 Area of a Triangle A = _________ Base of a parallelogram Altitude Height Examples: 1. Find the area and perimeter of: 15 cm 2. Find the area of the shaded triangular region. 12 cm 3 ft 20 cm 2 ft 2 ft 3. Abby’s Bakery has a plan for a 50’ by 31’ rectangular parking lot. The four parking spaces are congruent parallelograms (10 ft by 16 ft), the driving area is a rectangle, and the two unpaved areas for flowers are congruent triangles. a) Complete the diagram based on the description. 50' 31' b) Explain two different ways to find the area of the region that must be paved. 39 Geometry 2206 Unit 3: Mrs. Bondi 4. In a triangle, a base and the corresponding height are in the ratio 3:2. If the area is 108 in2, determine the length of the base and its corresponding height. D 5. Determine the value of x in parallelogram ABCD. C X 20" 18" A Practice: HW: p.619 #8-34 even, 39-43 40 30" B Geometry 2206 Unit 3: Mrs. Bondi Lesson 26: Areas of Trapezoids, Rhombuses and Kites (PH text 10.2) Objective: to find the area of a trapezoid, rhombus or kite Trapezoid - base b1 base – leg leg h leg – b2 base height – Theorem 10-4 Area of a Trapezoid The area of trapezoid is half the product of the height and the sum of the lengths of the bases. b1 b2 1 A hb1 b2 2 h b2 b1 Examples: 1. Find the area of a trapezoid with a height of 42 in. and bases measuring 121 in. and 145 in. 2. B C Find the area of Trapezoid ABCD. BC = 21 cm AD = 25 cm 60 A D Isosceles Trapezoid - > P T 41 Z > A E R Geometry 2206 Unit 3: Mrs. Bondi > P Examples: 3. Find the area of trapezoid TRAP if ET = 15 ft and AP = 10 ft. > T H A 45 10 2 ft 45 E 6 cm R T 4. Find the area of isosceles trapezoid NRTH. Leave your answer in simplest radical form. 60 N 2 cm O 5. The area of a trapezoid is 160 cm 2 . Its height is 8 cm and the length of its shorter base is 14 cm. Find the length of the longer base. Theorem 10-5 Area of a Kite The area of rhombus or a kite is half the product of the lengths of its diagonals. A Problem: a) b) 1 d1d 2 2 Find the measures of the numbered angles in the rhombus. (justify each) 1 A d1 d 2 Find the area. BE = 5cm; CE = 6 cm B C 2 1 Justify each step. E 4 3 2 A D 42 R Geometry 2206 Unit 3: Mrs. Bondi K Examples: 6. Find the area of kite KITE if EI = 10 ft., KM = 3 ft., and KT = 14 ft. E M T 7. Find the area of the given figure. Practice: HW: p.626 #12-38 even, 43-45 43 I Geometry 2206 Unit 3: Mrs. Bondi Lesson 27: Areas of Regular Polygons (PH text 10.3) Objective: to find the area of a regular polygon Regular Polygon – Center – Radius – Apothem – Postulate 10-1 If two figures are congruent, then their areas are equal. Theorem 10-6 Area of a Regular Polygon The area of a regular polygon is half the product of the apothem and the perimeter. A= 1 ap a= A ap or A 2 2 p= Examples: 1) Find the area of a regular heptagon with a 12 inch apothem, and 10 inch sides. 2) Find the area of a regular decagon with a 9.4 inch apothem, and 9 inch sides. 44 Geometry 2206 Unit 3: Mrs. Bondi 1) The length of an apothem of an equilateral triangle is 7 cm. Find the area. 2) The length of a radius of a regular hexagon is 8 cm. Find the area. Practice: HW: p.632 #8-34 even, 44-47 45 Geometry 2206 Unit 3: Mrs. Bondi Lesson 28: Perimeters and Areas of Similar Figures (PH text 10.4) Objective: to find the relationships between the similarity ratio and the perimeters and areas of similar figures. Theorem 10-6 If the scale factor of two similar figures is a:b, then: a) the ratio of their perimeters is a:b b) the ratio of their areas is a2:b2 Example 1 Find the area of a regular hexagon with side length 3 cm, then find the area of a regular hexagon with side lengths of 6 cm. 3cm 3cm 6cm Example 2 The areas of two similar rectangles are 48 in2 and 75 in2. What is the ratio of their perimeter (scale factor)? Example 3 Madeline used 144 tiles to tile a rectangular kitchen floor. If each dimension of the kitchen were doubled, what is the ratio of the perimeters? X What is the ratio of the areas? Y How many tiles would Madeline need to cover the bigger floor? 2X 2Y 46 Geometry 2206 Unit 3: Mrs. Bondi Example 4 The areas of two similar rectangles are 1875 ft2 and 135 ft2. Find the ratio of their perimeters. Example 5 The similarity ratio of the dimensions of two similar pieces of window glass is 3:5. If the smaller piece cost $2.50, what should be the cost of the larger piece (based on area)? Practice: 47 Geometry 2206 Unit 3: Mrs. Bondi Practice: HW: p.638 #10-46 even, 48-51 48 Geometry 2206 Unit 3: Mrs. Bondi Lesson 29a: Trigonometry and Area (PH text 10.5) Objective: to find areas of regular polygons and triangles using trigonometry The apothem of any regular polygon forms isosceles triangles. You can use trigonometry to help find the area of any regular polygon as long as you know the length of a side, radius or apothem. Reminder: Area of a Regular Polygon A 1 ap 2 or A ap 2 Finding area using trigonometry: Steps: 1. Draw in one apothem and the associated isosceles triangle. 2. Imagine all of these triangles formed within the figure. 3. Divide to find the measure of the angle formed in the right angle at the center, x. 4. Use trigonometry, the given information and the angle measure to find the needed information to compute the area. 5. Compute the area of the polygon. It is best to write out what you will plug into the calculator so you can double check your accuracy. Examples: 1. 2. 3 HW. p.646 #6-12 even, 20, 24, 26 49 Geometry 2206 Unit 3: Mrs. Bondi Lesson 29b: Trigonometry and Area (PH text 10.5) Objective: to find areas of regular polygons and triangles using trigonometry Examples/Practice: 50 Geometry 2206 Unit 3: Mrs. Bondi Additional Practice: The polygons are regular polygons. Find the area of the shaded region. Then find the probability of a pebble, being randomly tossed, hitting the white area. HW: p.646 #14-18 even, 28, 30, 35-36 51 Geometry 2206 Unit 3: Mrs. Bondi Review 3.24-29 (text 7.5 & 10.1-5) 52 Geometry 2206 Unit 3: Mrs. Bondi 13. What is the area of the 12-gon with a radius of 8 cm? 53 Geometry 2206 Unit 3: Mrs. Bondi 54