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Welcome to LC Math 2 Adv! • Syllabus • Supplies • NB: Learning Logs Definitions/Postulates Notes/CW Homework Assessments • Grading Policy • Remind 101 MyLifeUnit Compass $7 on Amazon Prime • Class Info • Exploration and Investigation • Geometry (infused with Algebra) and Probability • Unit 1 Newsletter • Unit 1 Learning Log Points, Lines and Planes In geometry, several terms are accepted as intuitive ideas, and are therefore not defined. They are: POINT LINE PLANE Point Line Plane Collinear Points If points are collinear, a single line can be drawn through them. Coplanar Points If points are coplanar, they can be contained in a single plane. Sample Coplanar Points: Sample Noncoplanar Points: Intersection of Geometric Figures The intersection of two figures is the set of points that are in both figures. Note: Dashes in diagrams indicate parts hidden from view in figures in space. Exercises a. Give two other names for PQ and for plane R. b. Name three points that are collinear. c. Name points that are noncollinear. d. Name four points that are coplanar. e. Name four points that are noncoplanar. Exercises a. Sketch line l contained in plane M. b. Sketch line l and plane M, which do not intersect.. c. Sketch plane M that intersects line l at point P. a. b. c. P Exercises Sketch two planes that intersect in a line. STEP 1 Draw: a vertical plane. Shade the plane. STEP 2 Draw: a second plane that is horizontal. Shade this plane a different color. Use dashed lines to show where one plane is hidden. STEP 3 Draw: the line of intersection. Exercises Sketch lines n and m that intersect plane P at point Q. Exercises Name the intersection of PQ and line k. Point M Name the intersection of plane A and plane B. Line k Name the intersection of line k and plane A. Line k Learning Log Summary Overarching – I can define and use appropriate notation to describe or identify aspects of a geometric diagram or interpret the notation provided. I can sketch a geometric diagram using given information and use a sketch to glean information. Closure Homework Sign and Return Syllabus Bring Supplies (due Monday 8/22) pg. 7-9 ~ 2-26 (even), 27-36 Additional Vocabulary Additional Vocabulary Additional Vocabulary Opposite rays - two collinear rays with a common endpoint. Additional Vocabulary Length - The length of a segment is the distance between the endpoints. DE = 2cm Finding the Length of a Segment Show the work used to find: AB = BA= BC = CB = PQ x y Use the examples above to write a formula to find the distance between any points P and Q. Postulate Using number lines involves following basic assumptions. Statements such as these, that are accepted without proof, are called postulates or axioms. Segment Addition Postulate Implies that B is collinear with A and C. If B is between A and C, then AB+BC=AC. Additional Vocabulary Congruent - Two objects that have the same shape and size. Congruent segments have equal length. AB CD EF GH Additional Vocabulary Midpoint of a Segment - The point that divides the segment into two congruent segments. AM MB AM MB M is the midpoint of AB Bisector of a Segment The bisector of a segment is a line, segment, ray, or plane that intersects the segment at its midpoint. AB is the bisector of PQ Bisector of a Segment Using the diagram to the right… Is M the midpoint of PQ ? Explain. Make 3 statements about what geometric figures bisect PQ. True or False: JM bisects KM True or False: JM KM True or False: PM MQ Learning Log Summary LT 1 - I can state the Segment Addition Postulate and use it in proof and problem solving contexts. The Segment Addition Postulate says… If two parts of one segment are known… Closure Homework pg. 15 ~ 1-45 (odd), 46-48 Learning Log Summary LT 2 - I can justify the formulas for calculating the distance between two points and the midpoint of a segment on the coordinate plane and apply them to solve problems. The Distance Formula is… It makes sense because… The Midpoint Formula is… It makes sense because… Learning Log Summary LT 3 - I can state and explain the Protractor Postulate and Angle Addition Postulate. I can apply the Angle Addition Postulate in proof and problem solving contexts. The Protractor Postulate in my own words is… A diagram and statement that illustrates the Angle Addition Postulate is… Deductive Reasoning Deductive Reasoning Inductive Reasoning • Based on definitions, postulates, and given information. • Based on patterns and observations. • Requires “proof”. • Doesn’t require “proof” • Conclusion must be true. • Conclusion is probably true. Joe observes that spaghetti is on the school lunch menu for three Wednesdays in a row. He concludes that the school always serves spaghetti on Wednesdays. Jane knows that the square of any real number is positive. So she concludes that if x>0, then x2>0. Learning Log Summary LT 5 - I can describe the use of inductive and deductive reasoning in mathematical sense making. Deductive reasoning is… Inductive reasoning is… Examples of inductive vs. deductive reasoning are… Deductive Reasoning “If it is raining after school, then I will give you a ride home.” Hypothesis Conclusion This is an example of an if-then statement. They are also called conditional statements (or just conditionals) If p, then q. Deductive Reasoning The converse of a conditional is formed by interchanging the hypothesis and conclusion. Statement: Converse: If p, then q. If q, then p. Ex) Find the converse of the statement. Statement: If Ted lives in California, he lives in the USA. Deductive Reasoning Conditionals are not always written in the same way. Here are some general forms of conditional statements. Determine which letter represents the hypothesis and conclusion. If p, then q p implies q P if q p only if q Learning Log Summary LT 6 - I can describe the components to a conditional statement and write its converse. The components of a conditional are… Conditionals can be written as… The converse to a conditional is… Deductive Reasoning A counterexample is an example for which the hypothesis is true but the conclusion is false. To disprove a statement, only one counterexample must be found. Statement: If Ted lives in California, he lives in the USA. Learning Log Summary LT 7 - I can determine the truth of a statement and provide a counterexample if false. A counterexample is… An example of a false statement and possible counterexample is… Deductive Reasoning If a conditional and its converse are both true, they can be combined into a single statement by using the words “if and only if”. This single statement is known as a biconditional. Conditional: If line segments are congruent, they have equal lengths. Converse: Biconditional?: Learning Log Summary LT 8 - I can determine whether a statement and its converse can be written as a biconditional. A biconditional is… An example of a true conditional and converse that can be rewritten as a biconditional is… Closure Homework pg. 35 ~ 1-7 (odd), 17-27 (odd) and pg. 35 ~ 9,10, 18-28 (even), 29-30 Deductive Reasoning Practice Identify the hypothesis and conclusion in each conditional: • 𝐴𝐵 = 𝐵𝐶 if B is the midpoint of 𝐴𝐶. • We will go to the beach only if it is sunny. • 𝑥 = −2 implies 𝑥 2 = 4 Deductive Reasoning Practice Determine if the statement is true or false. Then write its converse and determine if that statement is true or false. • If two angles are right angles, then they are congruent. • 𝑥 > 7 implies 𝑥 > 2. • An animal is a penguin only if it is a bird. • If a number is divisible by 2, then it is divisible by 4. Deductive Reasoning Practice Determine if the conditional can be rewritten as a biconditional. If so, re-write it as such. • If an angle has a measure of 90°, then it is a right angle. • 𝐷𝐸 = 𝐹𝐺 only if 𝐷𝐸 = 𝐹𝐺 • Two angles being adjacent implies that they have a common vertex. Theorems Learning Log Summary LT 10 - I can state and prove the Midpoint Theorem and Angle Bisector Theorem and use them in proof and problem solving contexts. The Midpoint Theorem is… The Angle Bisector Theorem is… To prove the Midpoint or Angle Bisector Theorem… Special Pairs of Angles Complementary angles are two angles whose measures have a sum of 90°. Each angle is called a complement of the other. A C B STA and ATR are complementary C is a complement of A Special Pairs of Angles Supplementary angles are two angles whose measures have a sum of 180°. Each angle is called a supplement of the other. B and Q are supplementary ABC is a supplement of CBD These angles are a linear pair. Special Pairs of Angles Ex) A supplement of an angle is three times as large as a complement of that angle. Find the measure of the angle. Learning Log Summary LT 11 - I can apply the definitions of complementary and supplementary angles in problems and proofs. Complementary angles… Supplementary angles… If x is an angle, its supplement/complement is… Special Pairs of Angles Vertical Angles are two angles such that the sides of one angle are opposite rays to the sides of the other angle. When two lines intersect, they form two pairs of vertical angles. 1 and 3 are vertical angles 2 and 4 are vertical angles Special Pairs of Angles What do you notice about the measure of vertical angles? Special Pairs of Angles Given: 1 Prove: 1 3 and 3 Statements are vertical angles Reasons Learning Log Summary LT 12 - I can define vertical angles and prove that they are congruent. Vertical angles… To prove vertical angles are congruent… Perpendicular Lines Perpendicular Lines are two lines that intersect to form right angles. The definition of perpendicular lines can be used in two ways: If 𝐽𝐾 is perpendicular to 𝑀𝑁, written JK MN then each of the numbered angles is a right angle. If any one of the numbered angles is a right angles, then JK MN . M J 2 1 3 4 N K Learning Log Summary LT 13 - I can apply the definition of perpendicular lines in problems and proofs. Perpendicular lines… To prove an angle is a right angle… To prove that two lines are perpendicular… Supplementary Angle Theorem If two angles are supplementary to the same angle, then they are congruent. Complementary Angle Theorem If two angles are complementary to the same angle, then they are congruent. Learning Log Summary LT 14 - I can prove theorems involving complementary and supplementary angles. If two angles are complementary/supplementary to the same angle… Common strategies in proofs involving complements/supplements are… Unit 1 Test • 12 Multiple Choice Questions • Definitions • Postulates/Theorems • Properties • 6 Free Response Questions • Proving Known Theorems • Proving New Statements • Applying Knowledge to Problems