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Transcript

Geometry 2205 Unit 1: Mrs. Bondi Unit 1 Coordinate Geometry Topics Lesson 1: Points, Lines, and Planes (PH text 1.2) Lesson 2: Measuring Segments (PH text 1.3) Lesson 3: Measuring Angles (PH text 1.4) Lesson 4: Exploring Angle Pairs (PH text 1.5, 2.6) Lesson 5: Midpoint and Distance in the Coordinate Plane (PH text 1.7) Lesson 6: Perimeter, Circumference and Area (PH text 1.8) Lesson 7: Patterns and Inductive Reasoning (PH text 2.1) Lesson 8: Conditional Statements (PH text 2.2) Lesson 9: Biconditionals and Definitions (PH text 2.3) Lesson 10: Deductive Reasoning (PH text 2.4) Lesson 11: Reasoning in Algebra and Geometry (PH text 2.5) Lesson 12: Lines and Angles (PH text 3.1) Lesson 13: Properties of Parallel Lines (PH text 3.2) Lesson 14: Proving Lines Parallel (PH text 3.3) Lesson 15: Parallel and Perpendicular Lines (PH text 3.4) Lesson 16: Parallel Lines and Triangles (PH text 3.5) Lesson 17: Equations of Lines in the Coordinate Plane (PH text 3.7) Lesson 18: Slopes of Parallel and Perpendicular Lines (PH text 3.8) Big Ideas and Essential Questions: Some geometric relationships can be described as functional relationships. Relations and functions are mathematical relationships that can be represented and analyzed by using words, tables, graphs, and equations. Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms. Mathematical statements can be justified through deductive and inductive reasoning and proof. How can you use coordinates and algebraic techniques to represent, interpret, and verify geometric relationships? How do you use the ideas of direct and indirect proof, and counter-examples to verify valid conjectures and refute invalid conjectures? 1 Geometry 2205 Unit 1: Mrs. Bondi Lesson 1: Points, Lines, and Planes (PH text 1.2) Objective: Student will be able to Why??? 1) Understand basic terms of geometry 2) Understand basic postulates of geometry To lay the foundation for your study of geometry Vocabulary: Definition: Diagram: Point: Line: Plane: Space: Segment: Ray: Opposite Rays: (Yes) B A (No) E C D 2 F Geometry 2205 Unit 1: Vocabulary: Mrs. Bondi Definition: Diagram: Collinear Points: Non-collinear Points: Coplanar: Non-coplanar: Intersection: Postulate: Axiom: 3 Geometry 2205 Mrs. Bondi Unit 1: Postulate 1-1: Through any two points A and B there is Draw an example: “exactly” _____________________________ Postulate 1-2: If two distinct lines intersect, then they intersect Draw an example: in “exactly” ____________________________ Line AB intersects line CD at point P. Postulate 1-3: If two planes intersect, then they intersect R T in _______________________________________ W S The intersection of plane RST and plane WST is _______. Postulate 1-4: Through any three non-collinear points E H F G there is “exactly” ________________________ D Points A, B, and C determine plane _______. A B There are three possible ways to determine a specific plane (draw an example of each) a) 3 non-collinear points b) a line and a point not on the line 4 c) 2 different intersecting lines C Geometry 2205 Mrs. Bondi Unit 1: Use figure 1 to answer #1 – 3. P N T H O E 1) Point ______ and point _______ are collinear with point R and point H. R S 2) The intersection of line PR and line OT is ______. W 3) Line ______ and line ______ intersect at point O. Figure 1 Use Figure 2 to answer # 4 – 7. 4) Name a point that is coplanar with points F, B, and C. ______ 5) What is the intersection of plane HFDG and plane BCDF? ______ 6) What is the intersection of plane AEG and plane BAC? ______ H G E A F B D Figure 2 7) What is the intersection of plane HACD and line GD? ____________ 8) Which of the following is not an acceptable name for the plane shown below? A) Plane RSZ D) Plane RSTW B) Plane RSWZ C) Plane WSZ E) Plane STZ R Do together: p.16 #1-6, 62-64 HW day 1: p.16 #8-14, 67-80 HW day 2: p.16 #15-26,28-46 even, 50, 54-58 even 5 T W S Z C Geometry 2205 Unit 1: Mrs. Bondi Lesson 2: Measuring Segments (PH text 1.3) Objective: Student will be able to determine the length of a segment Vocabulary: coordinate – distance – congruent segments – midpoint – segment bisector – Postulate 1-5: (Ruler Postulate) Every point on a line can be paired with a real number, called the coordinate of the point. The distance between any two points on the number line is the ___________________ ___________________ of the ____________________ of the coordinates. A Postulate 1-6: (Segment Addition Postulate) C B If three points A, B, and C are ____________________ and B is ___________________ A and C, then AC = _______ + ________ Example: 1) Point M is on LN . LM = 23 and MN = 18. What is the measure of LN ? 2) 2) If AC 50 , AB 2x 8 and BC 3x 12 , then solve for x to determine the length of AB , and BC . x = _____ A B C AB = _____ BC = _____ 6 Geometry 2205 Mrs. Bondi Unit 1: Congruent Segments: segments with the ______________ _______________________ If AB CD then AB CD. Midpoint of a segment – divides a segment into _______ ____________ ____________________ Label as CD and use tick marks to show that M is the midpoint of CD . Segment bisector – a point, line, or ray that intersects a segment at its __________________ Label CD and midpoint M to show that MN is the segment bisector of CD . Example: 2) In the first diagram B is the midpoint of AC . Solve for x and find the length of AB , BC , and AC if AB 6x 4 and BC 3x 8 . x = _____ A B C AB = _____ BC = _____ AC = _____ Do together: p.23 #1-7, 19 HW: p.24 #8-28 even, 29, 35-36 (Copy diagrams into your homework. DO NOT WRITE IN THE TEXTBOOK!) 7 Geometry 2205 Unit 1: Mrs. Bondi Lesson 3: Measuring Angles (PH text 1.4) Objective: A Student will be able to determine the measure of an angle N Vocabulary: G Angle: formed by two _____________ with the same ______________________ Sides: Vertex: a point where two or more _________________ meet or where two sides of a polygon meet. The plural is ________________________ Angle Postulate 1-7: (Protractor Postulate) Every ray on an angle can be paired with a real number from 0 to 180. Find the measure of each of the following angles using the protractor. 4) m CAT = _______ 5) m MAF = _______ 6) m MAT = _______ 7) m FAC = _______ 8) m CAM = _______ 9) m TAF = _______ Types of angles classified by angle measure: Zero Angle: an angle with a measure of _______. Acute Angle: an angle with a measure between _______ and _______. Right Angle: an angle with a measure of _______. Obtuse Angle: an angle with a measure between _______ and _______. Straight Angle: an angle with a measure of _______. Congruent Angles: angles with _____________________________________. If mA mB , then A B . 8 Vertex Geometry 2205 Mrs. Bondi Unit 1: Postulate 1-8: (Angle Addition Postulate) A O If point B is in the interior of AOC , then mAOC = __________ + __________. C B B A O C Examples: 1) M is in the interior of LON . LOM = 34o. MON = 18o. What is the measure of LON ? B 2) In the diagram to the right, mAOB =52 mBOC = _______ A O C Practice: 1) Use letters to name the numbered angles in the given diagrams. A a) 1 = ____________ or _____________ B 1 2 b) 2 = ____________ or _____________ D c) 3 = ____________ or _____________ or _____________ C F 3 E 2) If the mGDE 2x 3 , mFDE 6x 1 , and mGDF 3x 78 , solve for x to determine the mGDE , mFDE , and mGDF . x = _____ G mGDE = _____ F E mFDE = _____ D mGDF = _____ Do together: p.31 #1-5 HW: p.31 #6-23 even, 40 (Copy diagrams!) 9 G Geometry 2205 Unit 1: Mrs. Bondi Lesson 4: Exploring Angle Pairs (PH text 1.5, 2.6) Objective: Students will be able to identify special angle pairs and use their relationships to find angle measures. Review: Draw a diagram where B is in the interior of AOC . If mAOB 40 , mBOC 3x 15 , and mAOC 4x 15 solve for x and justify each step. Angle bisector - ____________________________________________ Label LOV , and draw point E in the interior of the angle. Mark the diagram to show that OE bisects LOV . Practice: 2) In the second diagram DF is the angle bisector of GDE . Solve for x 1 and find the measure of GDF and EDG if mGDF x 9 and mEDG 3x 42 . 2 x = _____ E mGDF = _____ F mEDG = _____ D G Important Angle Pairs: Vertical Angles: Adjacent Angles: Example: Example: Complementary Angles: Supplementary Angles: Example: Example: 10 Geometry 2205 Unit 1: Mrs. Bondi Important Postulate and Theorems: Postulate 1-9 Linear Pair Postulate If two angles for a linear pair, then they are supplementary. Theorem 2-1 Vertical Angles Theorem: Vertical angles are congruent Theorem 2-2 Congruent Supplements Theorem: If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent Theorem 2-3 Congruent Complements Theorem: If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent Theorem 2-4 All right angles are congruent. Theorem 2-5 If two angles are both congruent and supplementary, then each is a right angle. NEVER ASSUME! A linear pair can be assumed to be on a straight line. NO other assumptions should be made! You must see congruent markings, or be told of segment/angle relationships. DO NOT ASSUME!!! HW day 1: p.38 #7-41 (Copy diagrams!) HW day 2: p.124 #6-11, 14-17, 19, 20-22 (Copy diagrams!) Good Idea: p.41 Mid-Chapter Quiz FUN STUFF! Basic constructions, constructing shapes with a straight edge and compass is quite interesting. North Penn is not including it as a part of our curriculum, but I will gladly work through this with you outside of class time. If we have time at the end of the year, we will return to it. There are several lessons on constructions in your textbook, beginning with lesson 1.6. 11 Geometry 2205 Unit 1: Mrs. Bondi Lesson 5a: Midpoint Formula in the Coordinate Plane (PH text 1.7) Objectives: to find the midpoint of a segment. The midpoint of a segment is the halfway point between two endpoints. The coordinates of a midpoint are the averages of the coordinates of the endpoints. The midpoint of two coordinates on a number line can be found by ____________________________ -3 10 The midpoint formula: For endpoints P x1 , y1 and Q x2 , y2 on the coordinate plane the midpoint m can be expressed by: x x y y M = 1 2 , 1 2 2 2 Examples: 1. Find the midpoint of A(2, -1) and B(4, -3) 2. Find the midpoint of P(−1, 6) and Q(5, 0) We can also use the midpoint formula to find an endpoint if we know the midpoint and one of the endpoints. Example: 3. Given the midpoint of AB is M(2, 6), and endpoint A(-8, 9), find the coordinates of endpoint B. 4. Given the midpoint of AB is M(4.5, 0.5), and endpoint C(7, 5), find the coordinates of endpoint D. 12 Geometry 2205 Unit 1: Mrs. Bondi HW: p.54 #6-21 Lesson 5b: Distance Formula in the Coordinate Plane (PH text 1.7) Objectives: to find the distance between two points in a coordinate plane. Coordinate Plane Review: 1. Label the following on the coordinate plane below: Quadrants, x-axis, y-axis, origin 2. Graph the points ( –3, 4 ), ( 1, 1 ), ( –3, 1) and connect them to form a triangle. 3. Mark the lengths of the legs of the triangle by counting units. Use the Pythagorean Theorem to find the length of the hypotenuse. a 2 b2 c2 Now use the distance formula to find the length between (1,1) and ( –3, 4 ). d= x2 x1 13 2 y 2 y1 2 Geometry 2205 Unit 1: Mrs. Bondi The distance formula: For points P x1, y1 and Q x2 , y2 in the coordinate plane, the distance d between the points is given by: d= x2 x1 2 y 2 y1 2 Round answers to the nearest tenth. 4. Find the distance between (1, 4) and (−2, −5). 5. Find the distance between (−3, 2) and (3, −2). Real life situations can make use of this formula by transforming locations into coordinates. 6. One hiker is 4 miles west and 3 miles north of the campground. Another is 6 miles east and 3 miles south of the campground. How far apart are the hikers? (the camp ground is at (0, 0) ) 7. Mickey travels 15 miles west, then 20 miles north. Jamie travels 5 miles east, then 10 miles south. How far apart are they? 14 Geometry 2205 Unit 1: Mrs. Bondi 8. Quadrilateral KLMN has vertices with coordinates K(-3, -2), L(-5, 6), M(2, 6) and N(4, -2). Show that LK MN . HW: p.54 #22-34 15 Geometry 2205 Unit 1: Mrs. Bondi Midpoint and Distance Practice x2 x1 y2 y1 The distance formula: d The midpoint formula: x x y y2 M 1 2, 1 2 2 2 16 2 Geometry 2205 Unit 1: Mrs. Bondi Lesson 6: Perimeter, Circumference and Area (PH text 1.8) Objective: Student will be able to: 1) find the area and perimeter of basic polygons 2) find the circumference and area of circles Vocabulary: Perimeter (P) – the distance ______________________________________ or the sum of the lengths of the sides of the polygon Circumference (C) – the distance ___________________________________ Area (A) – the number of ______________________ enclosed by the polygon Formulas: Square: P = ________ A = ________ Rectangle: P = ________ A = ________ Triangle: P = ________ A = ________ Circle: P = ________ A = ________ Circle Notes: Name using … π= Examples: 1. Find the perimeter/circumference and area of each of the following: a. b. 10 in 12 inches 2 ft 2. Find the perimeter and area of PET . P (5,6), E (5,-2) and T (-1,-2) y x 17 c. d = 8 inches Geometry 2205 Mrs. Bondi Unit 1: 3. You are designing a rectangular banner for the Winter Ball. The banner will be 2 yards long and 4 feet wide. a) How much material will you need? b) If the material costs $8.95 per square yard, what will be the total cost? 4. You have 32 yd of fencing, and you want to make a rectangular pen for a calf that you are raising for a 4-H project. a) What are the dimensions of the rectangle that will give the maximum area? b) What is the maximum area? Area Addition Postulate - Postulate 1-10: The area of a region is ___________ of the areas of the non-overlapping parts. 5. Find the area of the figure using two different methods. 4yd 4yd 2yd 1 yd 2yd 1 yd 3 yd 4 yd 3 yd 4 yd 3 yd 3 yd 6. Amy is planning to install a new, 4 ft wide sidewalk around her 20 ft x 28 ft rectangular pool. Find the amount of cement that she will need. If she lines the outer edge of the sidewalk with brick, how many linear feet of brick will she need? (Draw a diagram before you begin.) 18 Geometry 2205 Unit 1: Mrs. Bondi HW: p.64 #9, 10-26 even, 32-35, 42 19 Geometry 2205 Unit 1: Mrs. Bondi Lesson 7: Patterns and Inductive Reasoning (PH text 2.1) Objective: Students will be able to use inductive reasoning to make conjectures. Inductive Reasoning – reasoning based on patterns you ___________ conjecture – a ________________ you reach using inductive reasoning (predict and test) counterexample – an example that shows that a conjecture is ____________ Examples: Look for a pattern. What are the next three terms in each sequence? 1) 1, 1, 2, 3, 5, … 2) M, T, W, … Practice: 20 Geometry 2205 Unit 1: Mrs. Bondi HW: p.85 #6-48 even (skip 18?) 21 Geometry 2205 Unit 1: Mrs. Bondi Lesson 8: Conditional Statements (PH text 2.2) Objective: Students will be able to: 1) recognize conditional statements and their parts 2) write converses, inverses, and contrapositives of conditionals (TONS of) Vocabulary: Conditional: another name for an ______-_______ statement. It has ____ parts. Example – “If two angles are congruent, then they have equal measures.” If p, then q. or “p implies q” q p Hypothesis: the part of the conditional statement that ____________ the _____. (Do not include the word “if” as part of the hypothesis.) Conclusion: the part of the conditional statement that ____________ the ______. (Do not include the word “then” as part of the conclusion.) Practice 1) Underline the hypothesis once and the conclusion twice. Try a Venn Diagram. a) If a triangle is scalene, then none of the sides are congruent. b) Jess can go out to play if it is not raining. Truth Value – a conditional can be either _________ or __________ To be a true conditional statement, the conclusion must be true EVERY time the hypothesis is true. One counterexample is enough to prove a conditional false. Counterexample: an example that shows the statement is __________________. Practice 2) Provide a counterexample to show each statement is false. a) If a number is divisible by 4, then it is divisible by 6. ________________ b) If AB BC , then B is the midpoint of AC . _______________________ 22 Geometry 2205 Mrs. Bondi Unit 1: To determine the truth value of a conditional, evaluate both the hypothesis and conclusion. It may be helpful to use a Venn Diagram. Create a Venn Diagram for the conditional: If it is your birthday, then you will have cake. Look at the following cases to determine the truth value of the conditional. Conditional: If it is your birthday, then you will have cake. What is the statement’s truth value? Evaluate the truth value for each case. Case 1: It is your birthday, you have cake. Case 2: It is your birthday, you do not have cake. Case 3: It is not your birthday, you have cake. Case 4: It is not your birthday, you do not have cake. (NOTE: If the hypothesis is false, then the truth value of the conditional is always _________.) The counterexample must have a true hypothesis. Practice 3) Decide whether the statement is true or false. If false, provide a counterexample. b) If x 0 , then x 2 x . (True or False) a) If x 2 25 , then x 5 . (True or False) Counterexample: _____________ Counterexample: _____________ Negation: means the ________________ (verb – “to negate”) “~p” means ________ Practice 4) Negate each statement. a) It is Saturday. b) You slept late. 23 Geometry 2205 Mrs. Bondi Unit 1: Every conditional has three related conditional statements. We use negation to form the related statements. Conditional Example: Converse: If it is Saturday, then you slept late. _____________________________ the hypothesis and conclusion of the conditional. Example - _________________________________________________ Inverse: ________________ BOTH the hypothesis and conclusion of the conditional. Example - _________________________________________________ Contrapositive: ______________ BOTH the hypothesis and conclusion of the _________________. or BOTH ________________ and ________________ the hypothesis and conclusion. Example - _________________________________________________ Equivalent Statements – two statements with the same truth value A conditional and its contrapositive are equivalent statements. A converse and inverse are equivalent statements. They must either both be true, or both be false. Practice 5) Write the Conditional, Converse, Inverse, and Contrapositive Statements. a) “I will take my umbrella if it is raining out.” (If-Then): ___________________________________________________ (Converse): ___________________________________________________ (Inverse)____________________________________________________ (Contrapositive)_________________________________________________ b) Choosey Mothers choose Jiff. (If-Then): ___________________________________________________ (Converse): ___________________________________________________ (Inverse)____________________________________________________ (Contrapositive)_________________________________________________ HW: p.93 #6-18 even, 51-58 24 Geometry 2205 Unit 1: Mrs. Bondi Lesson 9: Biconditionals and Definitions (PH text 2.3) Objective: Students will be able to write biconditionals and recognize good definitions. Biconditional statement: when a ________________ and its ________________ are BOTH true, they can be combined into one statement joined by “if and only if”. Example - _________________________________________________ Practice 1) Given: “Two segments are congruent if and only if they have the same measure.” a) Write the biconditional statement as a conditional. _________________________________________________________ b) Write the converse of the conditional. _________________________________________________________ Practice 2) If two angles are supplementary, then the sum of their measure is 180. a) Write the converse of the statement. _________________________________________________________ b) Is the original statement true? ______ Is the converse true? ______ c) If both statements are true, then write a biconditional statement. If neither is true, then write a counterexample. _________________________________________________________ Good Definitions are statements that can help you identify or classify an object. A good definition has three important components. Uses clearly understood terms – commonly used or already defined Is precise – avoid words like large, sort of, almost, etc. Is reversible – can be written as a true biconditional 25 Geometry 2205 Mrs. Bondi Unit 1: Examples/Vocabulary: Parallel lines are _________________ lines that do not intersect Draw and label line AB parallel to line CD. (Use a geometer for neatness/accuracy.) The symbol for parallel is Write the definition as a biconditional: _____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________ Perpendicular lines are two lines that intersect to form ________________________ Draw and label line EF perpendicular to line GH at point P. Use appropriate marks to indicate the lines are perpendicular. (Use a geometer for neatness/accuracy.) The symbol for perpendicular is Write the definition as a Biconditional: _____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________ Skew lines are noncoplanar lines that are not _______________ and do not _______________ Draw and label line JK skew to line LM. (Use a geometer for neatness/accuracy.) Write the definition as a Biconditional: _____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________ Perpendicular bisector of a segment is a line, segment, or ray that is __________________ to the segment at its _________________. Label CD with midpoint M and draw MN so that it is the perpendicular bisector CD . (Use a geometer for neatness/accuracy.) Write the definition as a Biconditional: _____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________ HW: p.101 #10-15, 22-26, 31, 38-41 GOOD IDEA: mid-chapter quiz p.105 26 Geometry 2205 Unit 1: Mrs. Bondi Lesson 10: Deductive Reasoning (PH text 2.4) Objective: Students will be able to use deductive reasoning to form a conjecture. Deductive Reasoning (also called Logical Reasoning) – the process of reasoning logically from given statements or facts to a conclusion Property: Law of Detachment If the hypothesis of a true conditional is true, then the conclusion is true. Example 1: If Rebekah maintains distinguished honor roll for seventh grade, she may get a cell phone. Rebekah ended the year with distinguished honor roll. You conclude: _____________________________________________________________ Property: Law of Syllogism Allows you to draw a conclusion based on two true conditional statements, when the conclusion of one statement is the hypothesis of the other statement. Example 2: If it is Monday, Bridget has piano lessons at 4:00. If Bridget has a piano lesson at 4:00, I pick her up from school. Today is Monday. You conclude: _____________________________________________________________ We can use the Law of Detachment and the Law of Syllogism, along with the properties, postulates and theorems to justify our steps in finding a solution to a problem or proving a statement. Example 3: Refer to the diagram below. Name two pairs of vertical angles. 1 4 3 2 Example 4: Using the diagram from example 3, if m1 150 , find the measures of 2 and 3 . Justify each step. 27 Geometry 2205 Unit 1: Mrs. Bondi HW: p.110 #6-16, 36-39 28 Geometry 2205 Unit 1: Mrs. Bondi Lesson 11: Reasoning in Algebra and Geometry (PH text 2.5) Objective: Students will be able to connect reasoning in algebra and geometry. You must remember your basic properties: Properties of Equality/Congruence and Real Numbers: Property Definition Addition Property Subtraction Property Multiplication Property Division Property Substitution Property Distributive Property Reflexive Property Symmetric Property Transitive Property Proof – a convincing argument that uses deductive reasoning – logically shows why a conjecture is true A proof can be written as a paragraph, two-columns (statements and reasons), or a flow chart. Each statement must be justified, and follow logically from the statements before it. Usually, the first statement is the given statement, and the last statement is what you want to prove. The statements that come between are the statements needed to logically progress from the given statement to the statement to be proven. 29 Geometry 2205 Unit 1: Examples: 5) Given: Prove: Mrs. Bondi 3x + 5 = x – 7 x = -6 Statement Reason 3x + 5 = x – 7 6) Given: Prove: 6 4 1 3 Statement 1 6 Reason Practice: HW: p.116 #4-19, 24, 35-41 30 2 5 3 4 Geometry 2205 Unit 1: Mrs. Bondi Extra Practice: 31 Geometry 2205 Unit 1: Mrs. Bondi 32 Geometry 2205 Unit 1: Mrs. Bondi HW: p.124 #12-13, 18, 20-24, 30, 40-42 Before beginning ch.3, do p.137. 33 Geometry 2205 Unit 1: Mrs. Bondi Lesson 12: Lines and Angles (PH text 3.1) Objective: Students will be able to 1) identify relationships between figures in space. 2) identify angles formed by two lines and a transversal. Types of Lines: 1) Parallel Lines: a) are _______________________ b) do NOT _____________________ c) The symbol for parallel is *Algebraic Connection: The slopes of 2 parallel lines are __________________. 2) Perpendicular Lines: a) are _______________________ b) do ______________________ c) The symbol for perpendicular is *Algebraic Connection: The slopes of perpendicular lines are ______________ ______________ 3) Skew Lines: a) are not __________________ b) are not __________________ c) do not _________________ d) There is no symbol for skew lines. Parallel Planes: planes that ______ ______ __________________ H Examples: Use Figure 1 to answer # 1 – 6. G E A 1) List all lines parallel to line HG ( HG ). F B 2) List all lines perpendicular to HG . Figure 1 D C 3) List all lines skew to line HG . 6) What is the intersection of HF and line BF ? 4) Which planes contain HG ? 7) Name two ⊥ segments that are both || to plane HGD? 5) Name 3 pair of parallel planes in the figure. 8) What is the intersection of plane HFA and plane BCD? 34 Geometry 2205 Unit 1: Mrs. Bondi 1 2 Use colored pencils to identify the described angle pairs in the diagrams. a 3 4 Transversal – 5 b 7 6 8 Interior – c Exterior – Alternate Interior Angles – Alternate Interior Angles – Same-Side Interior Angles – Same-Side Interior Angles – Corresponding Angles – Corresponding Angles – Alternate Exterior Angles – Alternate Exterior Angles – 35 Geometry 2205 Unit 1: Mrs. Bondi Parallel lines indicator 1 a 3 5 b 7 2 4 6 8 c Notice that the angle relationships exist regardless of the relationship (parallel or not) of the lines being intersected by the transversal. The angle pairs are named the same here, where the lines are parallel, and in the previous diagram, where they are not. 4. Name the relationship of each pair of angles in #2. HW: p.143 #10-16, 21-35, 37-42 (draw a diagram for #25-28) 36 Geometry 2205 Unit 1: Mrs. Bondi Lesson 13: Properties of Parallel Lines (PH text 3.2) Objective: Students will be able to prove theorems about parallel lines. Parallel Lines and Transversals: The angle relationships we explored in the last lesson hold special meaning when they are formed by parallel lines and a transversal. When the transversal intersects parallel lines, the special angle pairs formed are either congruent, supplementary, or both. Postulate 3-1 Same-Side Interior Angles Postulate Theorem 3-1 Alternate Interior Angles Theorem Theorem 3-2 Corresponding Angles Theorem Theorem 3-3 Alternate Exterior Angles Theorem 37 Geometry 2205 Mrs. Bondi Unit 1: Let’s prove the theorems! Theorem 3-1 Given: Alternate Interior Angles Theorem a b Statement Corresponding Angles Theorem a b Statement Statement Prove: 1 5 Reason Theorem 3-3 Given: 3 6 Reason Theorem 3-2 Given: Prove: Alternate Exterior Angles Theorem a b Prove: 1 8 Reason 38 Geometry 2205 Unit 1: Mrs. Bondi Parallel Line Proof Practice 5 1 7 8 4 10 r 6 l 1) Given: Prove: Statement 3) r s m1 m2 l m Prove: 1 and 4 are supplementary Statement s m 2) Given: Prove: Reason Given: 9 2 3 Statement 4) Given: Prove: Reason Statement 39 r s 1 and 3 are supplementary Reason l m, 1 and 4 are supplementary Reason Geometry 2205 Unit 1: Mrs. Bondi A B 1 2 3 C 5) Given: AB CD Prove: m3 m2 mABD Statement D 6) Given: BC bisects ABD AB CD Reason HW: p.153 #7-30 even & 23 40 Prove: m3 m2 Statement Reason Geometry 2205 Unit 1: Mrs. Bondi Lesson 14: Proving Lines Parallel (PH text 3.3) Objective: Students will be able to determine whether two lines are parallel. The angle relationships we studied in the last lesson are actually biconditional. They can also allow us to prove lines are parallel. We will treat each of these four theorems as biconditional theorems. Theorem 3-4 Converse of Corresponding Angles Theorem Theorem 3-5 Converse of Alternate Interior Angles Theorem Theorem 3-6 Converse of Same-Side Interior Angles Postulate Theorem 3-7 Converse of Alternate Exterior Angles Theorem 41 Geometry 2205 Mrs. Bondi Unit 1: Prove Theorem 3-4: Converse of Alternate Interior Angles Theorem with a flow chart proof. Given: m3 m6 Prove: a b m3 m6 Given m3 m4 180 m4 m6 180 _______________ a b _______________ _______________ Prove the Converse of Alternate Exterior Angles Theorem with a flow chart proof. Given: m1 m8 Prove: a b m1 m8 Given m1 m4 _______________ a _______________ m5 m8 _______________ What types of quadrilaterals can have angles with the following measures? Why? 1) 120, 60, 150, 30 2) 42 72, 108, 45, 135 b _______________ Geometry 2205 Unit 1: Mrs. Bondi 3) What is the value for x that will make a b ? a 4x + 6 2x – 12 b 4) What is the measure of each angle? c Practice: 5) Given: l m , m4 m3 Prove: r s Statement HW: p.160 #6-10, 12-28, 31-32 43 Reason Geometry 2205 Unit 1: Mrs. Bondi 44 Geometry 2205 Unit 1: Mrs. Bondi HW: p.160 #11, 29, 40, 41, 47-57 45 Geometry 2205 Unit 1: Mrs. Bondi Lesson 15: Parallel and Perpendicular Lines (PH text 3.4) Objective: Theorem 3-8 Students will be able to relate parallel and perpendicular lines. If two lines are parallel to the same line, then they are parallel to each other. If a b and b c , then a c . Theorem 3-9 In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. If m t and n t , then m n . Theorem 3-10 In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. If n l and l m , then m n . Example 1) A quilter has cut pieces of fabric that are isosceles trapezoids 2 in wide and 3 inches long on the longest side. The angles measure 135 and 45. When four pieces are fit together to form a square with short sides facing inward, will the sides be parallel? Explain. 46 Geometry 2205 Mrs. Bondi Unit 1: Prove Theorem 3-9 Given: e g and f g Prove: e f Statement Reason e f g 2) Prove Cherry St intersects Apple St perpendicularly. You know that Berry St and Apple St are both perpendicular to Date St. You also know that Berry St is perpendicular to Cherry St. (Hint: you may benefit from drawing a diagram first.) Practice: HW: p.167 #5-6, pick two proofs 7-10, 16, 19-22, 31-32 47 Geometry 2205 Unit 1: Mrs. Bondi Lesson 16: Parallel Lines and Triangles (PH text 3.5) Objective: Students will be able to use parallel lines to prove a theorem about triangles. to find measures of angles of triangles. Classifying Triangles: Triangle: Equiangular: Scalene: Acute Triangle: Isosceles: Right Triangle: Equilateral Triangle: Obtuse Triangle: Postulate 3-2 Parallel Postulate Through a point not on a line, there is one and only one line parallel to the given line. Theorem 3-11 Triangle Angle-Sum Theorem The sum of the measures of a triangle is 180. Example 1: Find the values of x, y and z. P Q z 90° x S y 110° R 40° 48 Geometry 2205 Mrs. Bondi Unit 1: Exterior angle of a polygon – an angle formed by a _____________ and an extension of an __________________ _____________ Remote interior angles – in a triangle, the two non-adjacent angles Theorem 3-12 Triangle Exterior Angle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. m1 m2 m3 Example 2: If m∠NGL = 30 and m∠GNL = 110, find the m∠1. N 1 L G Practice: Find m1. 1. 2. 3. 49 Geometry 2205 Unit 1: Mrs. Bondi Algebra Find the value of each variable. 4. 5. 6. 7. a. Which of the numbered angles are exterior angles? b. Name the remote interior angles for each exterior angle. c. Which two exterior angles share the same remote interior angles? Explain. HW 1: p.175 #8-21 50 Geometry 2205 Mrs. Bondi Unit 1: More Practice with angles in Triangles: Algebra Use the given information to find the unknown angle measures in the triangle. 13. The ratio of the angle measures of the acute angles in a right triangle is 1 : 3. 14. The measure of one angle of a triangle is 61. The other two angles are in a ratio of 2 : 5. 15. The measure of the exterior angle of a triangle is 110. The measures of its remote interior angles are in a ratio of 2 : 3. 16. Think About a Plan The measure of an exterior angle of DEF is 4x. The measure of one of this angle’s remote interior angles is x + 23. The measure of the other remote interior angle is 2x + 12. Find the value of x, the measure of each angle of the triangle, and the measure of the exterior angle. • How can drawing a picture help you find the answers? • How are the exterior angle and the third angle of the triangle related? 51 Geometry 2205 Unit 1: Mrs. Bondi Find the values of the variables and the measures of the angles. 17. 18. 19. 20. HW: p.176 #22-33 (Mid-Chapter Quiz p.181) 52 Geometry 2205 Unit 1: Mrs. Bondi Lesson 17: Equations of Lines in the Coordinate Plane (PH text 3.7) Objective: Students will be able to graph and write linear equations. Reminder: Slope: A slope will have a predictable “look”. Positive Slope – Negative Slope – Zero Slope – Undefined Slope – Class Practice: Find the slope of each line. Rise Run = ______ Rise Run = ______ Find the slope of a line through ( –2, 1) and (3, 5). Then graph it to check. Graphing Lines using slope intercept form: m = ________________ Rise Run = ______ Draw a line through the point (1, 2) with the slope 5/4. y = mx + b b = _________________________ 53 Geometry 2205 Unit 1: Mrs. Bondi To write an equation for a line from a graph, get the slope and y-intercept from the graph and substitute into the form y = mx +b. 1. Find the y-intercept. b = _____ 2. Find the slope using two points. m = _____ 3. Substitute into y = mx + b to write the equation for the line. Equations of Lines Reminder: slope-intercept form point slope form standard form Write an equation in slope-intercept form of the line that passes through points ( 1, 4 ) and ( 3, 10 ). Write an equation of the horizontal line that passes through ( -2, 5 ). Write an equation of the vertical line that passes through ( -2, 5 ). 54 Geometry 2205 Unit 1: Mrs. Bondi Practice: HW: p.193 #7, 8-12 even, 20-30 even, 36-39 55 Geometry 2205 Unit 1: Mrs. Bondi Lesson 18: Slopes of Parallel and Perpendicular Lines (PH text 3.8) Objective: Students will be able to relate slope to parallel and perpendicular lines. Parallel Lines (||) have slopes that are ____________________________________. (y-intercepts must be different) Perpendicular Lines (⊥) have slopes that are ____________________________________. (lines must intersect) (the product of the slopes of the lines is -1) Example: Find the slope of the line, and lines perpendicular and parallel to the line, for each equation. 1 y x 5 y = 3x - 2 y=x 3x + 5y = 7 2 m = _______ m = _______ m = _______ m = _______ m⊥= _______ m⊥= _______ m⊥= _______ m⊥= _______ m|| = _______ m|| = _______ m|| = _______ m|| = _______ 1) Write an equation for a line in slope intercept form that contains the point ( 5, 4 ) and is parallel to y = 2x – 2 . 2) Write an equation for a line in slope intercept form that contains the point ( 1, 6 ) and is perpendicular to y = 3x – 4 . 56 Geometry 2205 Unit 1: Mrs. Bondi 3) LK is the perpendicular bisector of HJ . Solve for x if LKJ 3 x 3. 4 L K H J x = _____ 4. Use slopes to determine whether the opposite sides of quadrilateral WXYZ are parallel. W(1, 1), X(3, 1), Y(2, 4), Z(2, 3) 5. Are 1 and 2 perpendicular? Explain. 57 Geometry 2205 Unit 1: Mrs. Bondi Practice: HW: p.201 #5, 8-26 even, 33 58 Geometry 2205 Unit 1: Mrs. Bondi ---------------------------------------------------------------------------------------------------------------------Unit 1 Exam Review: (textbook practice – all answers are in back of book – be prepared to use multiple skills in the same question) Ch.1 - p.70-74 #1-4, 7-25, 30-41 Ch.2 - p.129-132 #1-38 Ch.3 – p.206-210 #1-29, 34-46 59