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Geometry 2205 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) Lesson 30: Geometric Probability 1 Geometry 2205 Unit 3: Mrs. Bondi Get Ready for Ch.6 Define: Polygon: Complete: Polygon Chart: (Create a chart to help you remember the names and number of sides of polygons with up to 12 sides.) 1 2 3 4 5 6 7 8 9 10 11 12 Do p.349 #1-15. 2 Geometry 2205 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? 3 Geometry 2205 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 #6-24 even, 32 4 Geometry 2205 Unit 3: Mrs. Bondi 5 Geometry 2205 Unit 3: Mrs. Bondi 6 Geometry 2205 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 7 B C A D B C D Geometry 2205 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 #10-30 even, 45-48 8 B D F Geometry 2205 Unit 3: Mrs. Bondi 9 Geometry 2205 Unit 3: Mrs. Bondi 10 Geometry 2205 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. 11 Geometry 2205 Unit 3: Mrs. Bondi Example 1: Find the value of each variable to make the figure a parallelogram. Theorem 6-11 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6-12 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. Is it possible to prove that the quadrilateral is a parallelogram based on the given information? Explain. 12 Geometry 2205 Unit 3: Mrs. Bondi Practice: HW: p.372 # 6-16 even, 22, 24, 29, 32-34 13 Geometry 2205 Unit 3: Mrs. Bondi 14 Geometry 2205 Unit 3: Mrs. Bondi 15 Geometry 2205 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. 16 Geometry 2205 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 Theorem 6-14 The diagonals of a rhombus are perpendicular. Each diagonal of a rhombus bisects the two opposite angles. Example 3: If mABC = 120, find the measures of the numbered angles. B Rhombus ABCD C 5 1 A 17 2 3 4 D Geometry 2205 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 # 6-38 even, 42-46 even, 59-61 18 Geometry 2205 Unit 3: Mrs. Bondi 19 Geometry 2205 Unit 3: Mrs. Bondi 20 Geometry 2205 Unit 3: Mrs. Bondi Lesson 19: Conditions for Rhombuses, Rectangles, and Squares (PH text 6.5) Objectives: to determine whether a parallelogram is a rhombus or a rectangle. Theorem 6-16 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. 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. 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 21 Geometry 2205 Unit 3: Mrs. Bondi 6. Example 6: For what values of x is B 3x 2x + 12 A ABCD a rhombus? C D Example 7: For what values of x is B ABCD a rectangle? C 7x + 2 A 5x + 8 D Practice: HW: p.386 #6-18 even, 32-34, 36-38 22 (The expressions are labeling the pieces of the diagonals.) Geometry 2205 Unit 3: Mrs. Bondi 23 Geometry 2205 Unit 3: Mrs. Bondi 24 Geometry 2205 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 The diagonals of an isosceles trapezoid are congruent. Midsegment of a trapezoid – 25 Geometry 2205 Unit 3: Theorem 6-21 Mrs. Bondi 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 The diagonals of a kite are perpendicular. R Problem: Given Kite RSTW, find the value of RT. RS = 13 cm ST = 15 cm SW = 24 cm RT = ? W S T 26 Geometry 2205 Unit 3: Mrs. Bondi 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 #8-36 even, 57-62, 67-70 27 Geometry 2205 Unit 3: Mrs. Bondi 28 Geometry 2205 Unit 3: Mrs. Bondi 29 Geometry 2205 Unit 3: Mrs. Bondi 30 Geometry 2205 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 5. Open-Ended What are three radical expressions that simplify to 2 x 3 ? 31 Geometry 2205 Unit 3: Mrs. Bondi Algebra Review: 32 Geometry 2205 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 33 Geometry 2205 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) 34 Geometry 2205 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.403 #6-24 even, 33, 45-50 (need graph paper) 35 Geometry 2205 Unit 3: Mrs. Bondi 36 Geometry 2205 Unit 3: Mrs. Bondi 37 Geometry 2205 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? 38 Geometry 2205 Unit 3: 3) Mrs. Bondi Find the coordinates of the missing vertices of isosceles trapezoid EFGH. FG EH 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 39 Geometry 2205 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. 40 Geometry 2205 Unit 3: Mrs. Bondi Practice: HW: p.410 #8-13, 17-19, 38-40 41 Geometry 2205 Unit 3: Mrs. Bondi 42 Geometry 2205 Unit 3: Mrs. Bondi 43 Geometry 2205 Unit 3: Mrs. Bondi Lesson 23: Proofs Using Coordinate Geometry (PH 6.9) Objective: to prove theorems using figures in the coordinate plane 44 Geometry 2205 Unit 3: Mrs. Bondi 45 Geometry 2205 Unit 3: Mrs. Bondi 46 Geometry 2205 Unit 3: Mrs. Bondi Chapter 6 Review 47 Geometry 2205 Unit 3: Mrs. Bondi Chapter 6 Review (continued) 48