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Geometry Section 2.1 __________________________________________________________ Start Thinking: Aaron looks back in his agenda and notices that two Tuesdays ago he had a pop quiz and last Tuesday they also had a pop quiz. It is Monday. What might Aaron want to do? Inductive Reasoning: ____________________________________________________________________________________ ____________________________________________________________________________________. Conjecture: ____________________________________________________________________________________. REAL WORLD EXAMPLES Examples: Determine the next two terms in the sequence. 1.) 2,4,6,8,10,…….. 2.) 5, 10, 20, 40,……. 3.) 20, 16, 12, 8, …….. 2.) 1,1,2,3,5,8,…….. 3.)1, 2, 3, 3, 6, 18, 54, ….. Helpful Hints: 1.) 2.) 3.) 4.) Harder Ones: 1.) 3,-6, 12, -24, ……. 4.)F, S, T, F, F, …….. 5.) M, T, W, T,……. 6.) A, B, D, E, G, H, … More Practice: 7.) 5, 11, 18, 26, ….. 8.) 1,1,2,6,24,120,….. 1 1 1 10.) 1, , , ,...... 4 9 16 11.)T,F,S,E,T,T,F, S,…… 9.) 30, 20, 11, 3, ….. 12.) NOW IT IS TIME TO THINK OF YOUR OWN….. 1.) 2.) PROBLEMS WITH INDUCTIVE REASONING: Counter Example: _____________________________________________________________________________________ ________________________________________________________________________________. 1.) A triangle can be created if you are given 3 points. 2.) When you multiply a number by 2 it becomes larger. Real World: 1.) Geometry Section 2.5 __________________________________________________ Start thinking: Prove the following with the given information and other facts that you can think of. You must justify everything you state in your reasoning process. Background Information: You overhear your aunt talking about how she wants to go to Vancouver, Canada but she does not know if she is allowed to. You do know that she does have a passport. Prove that she is legally allowed to go to Vancouver Canada. Reasoning Put Into Words Two column Proof PROPERTIES: Property Addition Property: Subtraction Property: Multiplication Property: Reflexive Property: Example Property Example Symmetric property: Transitive Property: Substitution Property: Distributive Property: PRACTICE: Name the property that justifies each of the following. 1.) 2x + 1 = 11 2x = 10 2.) 3x = 9 x=3 3.) 3(x + 1) 3x +3 4.) x = -2 and y = 7x +1 5.) AB AB 6.) x 10 2 x = 20 y = 7 (-2) +1 Examples: Write a two-column proof of the following. Given : 2x +1 =9 Prove: x = 4 Example 1: Fill in the missing parts of the proof. Example 2: Example: Fill in the missing parts of the following proof Geometry Section 2.6 ________________________________________________________ Start thinking: Solve for x in the following. 1.) Theorem 2.1: Vertical Angles Theorem: ______________________________________________________________ ________________________________________________________________________________. Complimentary Angles: Look at the figures to the right. Notice that <4 and <5 are complimentary and <5 and <6 are complimentary. What can you conclude about <4 and <6? Theorem 2.2: If two angles are compliments of the same angle then the two angles are ________________ Supplementary Angles: Look at the figures to the right. Notice that <1 and <2 are supplementary and <2 and <3 are supplementary. What can you conclude about <1 and <3? Theorem 2.3: If two angles are supplements of the same angle then the two angles are ________________ PROOF: Given: Figure above and that <1 and <2 are supplementary and <3 and < 2 are supplementary. Prove: 1 2 OTHER THEOREMS: Theorem 2.4: All right angles are ____________________________________. PROOF: GIVEN: Angle 1 and Angle 2 are right. Prove : Angle 1 is congruent to Angle 2 Theorem 2.7: Congruent and Supplementary Angles: If two angles are both congruent and supplementary then the two angles are ____________________________________________________. Ex3: Use the Congruent Supplements Theorem Given 1 and 2 are supplements. 1 and 4 are supplements. m2 = 45° Prove m4 = 45° Statements 1. 1 and 2 are supplements. 1 and 4 are supplements. Reasons 1) 2. 2) Congruent Supplements Theorem 3. m2 = m4 3) 4. m2 = 45° 4) 5. 5) Substitution Property of Equality Geometry Section 3.1 _____________________________________________ Start Thinking: Come up with a correct mathematical definition of parallel lines. Vocabulary: Terms/Definition Notation and Example Parallel lines: Skew lines: Parallel Planes: Ex1: Identify relationships in space Think of each segment in the figure as part of a line. Which line(s) or plane(s) in the figure appear to fit the description? a.Line(s) parallel to AF b. Line(s) skew to AF c. Line(s) perpendicular to and containing point E and containing point E AF and containing point E d. Plane(s) parallel to plane FGH and containing point E EX2: Identify parallel and perpendicular lines Use the diagram at the right to answer each question. a. Name a pair of parallel lines. b. Name a pair of perpendicular lines. c. Is AB BC ? Explain. Definition: TRANSVERSALS ______________________________________________________________________________ TRANSVERSALS create FOUR special angle pair relationships l 2 1 _____________________________________ 3 4 5 6 8 m 7 _____________________________________ l 2 1 3 4 5 8 6 m 7 _____________________________________ l 2 1 3 4 5 8 6 m 7 _____________________________________ l 2 1 4 5 8 6 7 3 m Ex 3: Identify the relationship between the angles given: a. 1 and 9 ____________________________________ b. 8 and 13 ____________________________________ c. 6 and 16 ____________________________________ d. 4 and 10 ____________________________________ e. 8 and 16 ____________________________________ f. 10 and 13 ____________________________________ PRACTICE: 1) Identify all pairs of all of the angles that have the desired relationship to the given angle. a.) AIA with <20 b.)SSIA with <30 c.)CA with <19 d.) SSIA with <17 e.)AIA with <29 2.) Think of each segment in the diagram as part of a line. Complete the statement with parallel, skew, or perpendicular. a. WZand ZR are ___________________. WZand ST are ___________________. c. QT and YS are ____________________. b. d. Plane WZR and plane SYZ are ______________________. e. Plane RQT and plane YXW are ______________________. 2) Think of each segment in the diagram as part of a line. Which line(s) or plane(s) appear to fit the description? a. Line(s) parallel to EH b. Line(s) perpendicular to EH c. Line(s) skew to CD and containing point F d. Plane(s) perpendicular to plane AEH e. Plane(s) parallel to plane FGC 3) Complete the statement with sometimes, always, or never. a. If two lines are parallel, then they __________________ intersect. b. If one line is skew to another, then they are ___________________ coplanar. c. If two lines intersect, then they are ______________________ perpendicular. d. If two lines are coplanar, then they are ___________________________ parallel. Geometry Section 3.2 ______________________________________________________ Start Thinking: Give an example of the following types of angles on the figure to the left. Alternate Interior Angles: __________________ Same Side Interior Angles: _________________ Alternate Exterior Angles: _________________ Corresponding Angles: _____________________. In the example above the transversal is intersecting two parallel lines. This is a special case, which results in special relationships between the different types of angles. Using a piece of tracing paper trace one of the angles and see what the relationship is. Postulates / Theorems: Corresponding Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are ____________________________________________. Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are____________________________________. Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are ________________________________. Same side Interior Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of same side interior angles are ___________________________________. EXAMPLES: Ex 1: Find the value of x. Ex 2: Solve for x EX 3: Solve for x EX 4: Solve for X and Y PROOF of ALTERNATE INTERIOR ANGLES THEOREM using SSIA Postulate. n Given: m is parallel to n Prove: 4 6 m Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 4 6 3 Practice: 1) If m7 = 75°, find m1, m3, and m5. Tell which postulate or theorem you use in each case. DO NOT USE VERTICAL ANGLES. m 3 = 75°, _______________________________________ m 5 = 75°, _______________________________________ m 1 = 75°, 2) _______________________________________ Find all the angle measures. 2 1 8 4 6 5 3 105° 7 3.) Application: You are building a wood shed. You have already put up both walls. One is 6 feet tall and the other is 10 feet tall. You then use a metal brace, which is bent at a 130 degree angle to attach the roof to the smaller wall. You want to create another brace to attach the roof to the back wall. What angle should the brace be bent at? Geometry Section 3.3 ______________________________________________ Start Thinking: IF two parallel lines are cut by a transversal THEN……. Alternate Interior Angles are ______________________________. Alternate Exterior Angles are ______________________________. Corresponding Angles are __________________________________. Same Side interior Angles are ________________________________. CONDITIONAL STATEMENT: __________________________________________________________. CONVERSE: _______________________________________________________________________. DIAGRAM: THEOREMS *Converse of Alternate Interior Angles Theorem: - IF __________________________________________________________ THEN ________________________________________. *Converse of Corresponding Angles Theorem: - IF __________________________________________________________ THEN ________________________________________. *Converse of Same Side Interior Angles Postulate: - IF __________________________________________________________ THEN ________________________________________. *Converse of Alternate Exterior Angles Theorem: - IF __________________________________________________________ THEN ________________________________________. PRACTICE: Determine what lines, if any, must be parallel given each piece of information. Justify your reasoning. Write NEI if there is not enough info. a.) < 7 and <17 are congruent b.) <21 and <20 are congruent c.) <20 and <9 are congruent d.)< 14 and <3 are supplementary e.) <22 and <7 are congruent. f.) <19 and <11 are congruent g.) m<14 + m<4 = 180 h.) <10 and <7 are congruent i.) m<17 = 40 and m<20 = 40 j.) m<23 = 120 and m<6 = 120 Hint Use Converse of Corresponding Angles THM 2.) Complete a 2 column proof using the Converse of the Corresponding Angles Thm. 3.) 4.) A parallelogram is defined as a quadrilateral that has two pairs of opposite parallel sides. m<A and m<C = 105. M<B = m<D. Is the figure a parallelogram? Justify your answer. 4) 5.) GIVEN: g || h, 1 2 PROVE: p || r Statements Reasons g || h 1. ___________________________________________ 1 3 2. ___________________________________________ 1 2 3. ___________________________________________ 2 3 4. ___________________________________________ p || r 5. ___________________________________________ Geometry Section 3.4 _____________________________________________ Start Thinking: You have a 1 inch wide and 6 foot long piece of lumber. You are trying to create a line with a piece of chalk that is 8 inches away from a wall so that the line is parallel to the wall. Describe how you could do this. Theorem 3.8 Transitive Property of Parallel Lines: _______________________________________ ____________________________________________________. Example: Theorem 3.9: If two lines are perpendicular to the same line then they are ____________________. STATEMENTS REASONS PRACTICE: 1.) 2.) PROOF: GIVEN: <1 is congruent to < 2 < 3 and < 4 are supplementary PROVE: Line L is parallel to Line N THEOREM: Theorem 3-10: In a plane, if a line is perpendicular to one of two parallel lines then.. ___________________________________________________________________________________ _____ Example of why it has to be “IN A PLANE” PROOF: 1.) 2.) Write a flow proof. Given: PQS and QSR are supplementary. Prove: Geometry Section 3.5 ______________________________________________ Activity: Using a straight edge, and either a pair of scissors or careful tearing create a triangle out of a piece of paper. Now rip off all of the angles of the triangle. Place then next to one another so that all of the vertexes are at the same point. What do you notice? What does this tell you about the sum of the interior angles of a triangle? Triangle Angle Sum Theorem: ________________________________________________________________________ __________________________________________________________________________. Postulate 3.2: Parallel Line Postulate: ________________________________________________________________ __________________________________________________________________________________. Solve the following for x: 1.) 2.) MORE PRACTICE: 1.) 2.) DEFINITION: Exterior Angles of a Triangle: ___________________________________________________________ _____________________________________________________________ Remote Interior Angles: __________________________________________________________________ __________________________________________________________________ PROOF OF THEOREM 3-12: GIVEN: Diagram to right Prove: m<2 + m<3 = m<1 PRACTICE PROBLEMS : 1.) 2.) ONE MORE PROOF: GIVEN: Diagram to the right PROVE: m<4+ m<5 +m<6 = 360 Geometry Section 3.7/3.8 ___________________________________________________ Start Thinking: You are going to go with a bike ride with a buddy. You are a little concerned because he says that there is a hill. What are some questions you might ask him about the hill? PRACTICE: Calculate the following slopes. 1.) (3, 10) and (7, 1) 2.) (-1, 6) and (4,-4) 3.) SPECIAL SLOPES: 1.) 2.) 3.) (3, 1) (3.7) 4.) (3,1) (5,1) EQUATIONS GRAPHS GRAPH EQUATION PRACTICE: Graph the following lines. 2.) y 1.) y = 1/3x – 4 2x 1 3 3.) y = x PRACTICE: Write the equation of the following lines. 1.) 2.) 3.) WRITING THE EQUATION OF A LINE GIVEN A POINT AND SLOPE. EX: Slope = 3 and point (1,5) METHOD 1 (SOLVING FOR B) METHOD 2 (POINT SLOPE) PRACTICE: 1.) Slope= -2 and (3, -6) 2.) Slope= 2/3 and (9 , 2) 3.) Slope= Undefined and (2,3) WRITING THE EQUATION OF A LINE GIVEN TWO POINTS. 1.) (1, 3) and (3, 13) 2.) (-1,0) and (2, 6) 3.) (2, 7) and (5,7) Start Thinking: Look at the following lines. How are they related? How would you relate their steepness? What do we call the “steepness of a line”? EX1: Determine if the following lines are parallel? EX2: WRITE THE EQUATION FO THE LINE WITH THE FOLOWING CHARACTERISTICS. 1.) Parallel to y = 3x+5 and through (4,2) 2.) Parallel to x = 2 through (-4,7) EX1: Are the following lines perpendicular? EX 2: What is the equation of a line perpendicular to y = 2/3x +1 through (10, 5)? EX3: Determine if the figure with the following vertices is a parallelogram. A(0,2) B (3,4) C (2,7) D (-1,5) CONCEPT USING VARIABLES: EX 1: What is the equation of a line that contains the following points (0,0) and (a,v). EX 2: Is the line from EX1 parallel, perpendicular, or neither with the following line? Explain. y b xd a EX 3: Is a line that goes through the point (0,0) and (a,b) parallel to a line that goes through (1.0) and (a+1,b)?