Distance and Midpoint Objective: To use two dimensional coordinate systems to represent points, lines, line segments and figures. To derive and use formulas involving length and midpoint

Measuring Angles Objectives: To identify and classify different angles

MEASURING SEGMENTS Objective: To use a one-dimensional coordinate system to represent points, lines and line segments.

MEASURING SEGMENTS Objective: To use a one-dimensional coordinate system to represent points, lines and line segments.

MEASURING SEGMENTS Objective: To use a one-dimensional coordinate system to represent points, lines and line segments.

PATTERNS AND INDUCTIVE REASONING Objectives: To find counterexamples to disprove statements that are false To use inductive reasoning to formulate a conjecture

Points, Lines and Planes Objective: To learn to identify, classify and name points, space, line, collinear points, plane, coplanar, postulate and axiom Point: has no size and represented by a small dot and named with a capital letter. Ex: Space the set of all points Line a series of points that extend in two opposite direction. Can be named by any two points in the line or with a single lower case letter. Ex: Collinear points that lie on the same lines Point D Points A, B & C are collinear

Plane is a flat surface that has no thickness. It contains many lines and extends without end. You can name a plane by either a single capital letter or by at least three of it non collinear points. Ex: Coplanar Points and lines in the same plane are coplanar. Points V, X, S, Y and U are coplanar Point Z and Point U are non coplanar

Basic Postulate of Geometry Postulate or Axiom – an accepted statement of fact

Segments, Ray, Parallel Lines and Planes Objective: To identify segments and rays To recognize parallel lines and planes Line Segment The part of a line consisting of two endpoints and all points between them Ray Is the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint Opposite Rays Two collinear rays with the same endpoint. Opposite rays always form a line Parallel Lines Coplanar lines that do not intersect Skew Lines Non coplanar, therefore they are not parallel and do not intersect Parallel Planes Planes that do not intersect. A line and a plane that do not intersect are also parallel Plane ABCD II EFGH Take Home Quiz: Get One Per Page At The Front Desk / Copy Notes Below

Conditional Statements Objective: G.3-A: To determine the validity of a conditional statement and its converse G.3-C: To find counterexamples to disprove statements that are false. Conditional Statement Another name for an if-then statement. Every conditional has two parts; the part following if is the hypothesis and the part following then is the conclusion. Example: Truth Value Conditional have a truth value of true or false. To show that a conditional is true, the hypothesis is true, the conclusion is also true To show that the conditional is false, you need to find one counterexample that makes the hypothesis true and the conclusion false.

Converse of a conditional switches the hypothesis and the conclusion Example: Conditional If two lines intersect to form right triangles, then they are perpendicular. Converse: If two line are perpendicular, then they intersect to form right triangles.

BICONDITIONAL AND DEFINITIONS Objective: To develop an awareness of the structure of a mathematical system, connecting definitions and logical reasoning. To determine the validity of a conditional statement and its converse. Biconditional When a conditional and its converse are true, you can combine them as true biconditional. This is the statement you get by connecting the conditional and its converse with the word “and”. You can write a biconditional more concisely, however by joining the two parts of each conditional with the phrase “if and only”.

Deductive Reasoning Objective: G.3-C: To use logical reasoning to prove statements are true G.3-E: To use deductive reasoning to prove a statement Deductive Reasoning Or logical reasoning, is the process of reasoning logically from given statements to a conclusion. If the given statements are true, deductive reasoning produces a true conclusion. Law of Detachment

Law of Syllogism Another law of deductive reasoning. It allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other. Not possible because we do not know that the hypothesis is true

If a number ends in 0, then it is divisible by 5 Not possible, the conclusion of one statement is not the hypothesis of the other statement The Volga River is less than 2300 miles long The Volga River is not one of the world’s ten longest river

Reasoning in Algebra Objective: To develop an awareness of the structure of a mathematical system connecting postulates, logical reasoning and theorems

Justifying Steps in Solving an Equation Substitution Property Subtracting Property of Equality Division Property of Equality

Proving Angles Congruent Objective: G.1-A: To develop an awareness of the structure of a mathematical system, connecting logical reasoning and theorems. G.1-B: To make conjectures about angles and lines and determine the validity of the conjectures. G.1-E: to use deductive reasoning to prove a statement THEOREM - an idea that has been demonstrated as true or is assumed to be so demonstrable. In the proof of a theorem. A GIVEN list shows you what you know from the hypothesis of the theorem. You prove the conclusion of the theorem. A diagram records the given information visually. Example:

Example 1: Find the value of x

90 o Substitution. mL3mL3

Inverses, Contrapositives and Indirect Reasoning Objective: G.3-A To determine the validity of a conditional statement, its converse, inverse and contrapositive G.3-C To use logical reasoning to prove statements that are true and find counterexamples to disprove statements that are false G.3-E To use deductive reasoning to prove a statement Negation – statement that has the opposite truth value. Ex: Knoxville is the capital of Tennessee (false) The Negation: Knoxville is not the capital of Tennessee (true) Inverse – in a conditional statement, it negates the hypothesis and conclusion Contrapositive - it switches the hypothesis and the conclusion and negates both Ex:

Properties of Parallel Lines Objectives: G.3-C: To use logical reasoning to prove statements are true G.3-E: To use deductive reasoning to prove a statement G.4-A: To select an appropriate representation in order to solve problems Transversal Line – a line that intersect two coplanar lines at two distinct points. Thee diagram shows the eight angles formed by a transversal line “t” and two lines “l” and “m”

Assignment: P. 132 Nos (Even)

PROVING LINES PARALLEL Objective: G.3-C: To use logical reasoning to prove statements are true G.3-E: To use deductive reasoning to prove a statement G.4-A: To select an appropriate representation in order to solve problems

SLOPES

Calculating Slope With Two Points (x 1, y 1 ) and (x 2, y 2 ) Given Two Points: 1) A ( 2, -4) & B (-7, -2) Slope Formula

Lines in the Coordinate Plane Objective: To use a two-dimensional coordinate system to represent points andlines To use the slope formula

Parallel Lines and Triangle Angle-Sum Theorem Objective: To learn the Triangle Angle-Sum Theorem and Identify exterior angle of a polygon and remote interior angles and review classifications of triangles Classification of Triangle By Side By Angle

The Polygon angle-Sum Theorem Objective: Learn to identify different classifications of polygons and identify missing interior angles

Congruent Figure Objective: Learn to identify and prove corresponding sides and angles of congruent polygons Congruent Polygons – polygons that have congruent corresponding parts --- their matching sides and angles. Matching vertices are corresponding vertices. Example:

Triangle Congruence by SSS and SAS Objective: Learn to prove congruent triangle by using the SSS and SAS Postulate

Proof: Given GH HK by definition of midpoint By SSS

No, you don’t know that L E is congruent to L DBC or that line AB is congruent to line DC

Isosceles and Equilateral Triangles Objective: Learn and apply the properties of isosceles triangle and equilateral triangles to solve problems, use deductive reasoning to prove statements are true and apply triangle congruence relationships. BASE VERTEX ANGLE LEGS BASE ANGLES Legs – the congruent sides of the isosceles triangle Base – the third side of the isosceles triangle Vertex Angle – the angle between the two congruent sides Base Angles – the two congruent angles Isosceles Triangle is a triangle with two congruent sides

Corresponding Parts of Congruent Triangles are Congruent

A B C

Corollary a statement that follows immediately from a theorem, Equilateral Congruent Side Equiangular Congruent Angles

X Y X YZ X Y Z

Example: Solve for y y = 180Simplify

Triangle Congruence by ASA and AAS Objective: Use ASA and AAS to prove congruent trangles.

Using Congruent Triangles: CPCTC Objective: To apply triangle congruence relationship CPCTC- Corresponding Parts of Congruent Triangles are Congruent - CPCTC as a reason is used once you have proven the triangles congruent by SSS, SAS, ASA and AAS Example:

Congruence of Right Triangle Objective: To construct and justify statements about triangles and their properties

Midsegments of Triangles Objective: Use the formulas involving length, slope and midpoint

Perpendicular Bisector Objective: To formulate and test conjectures about the properties and attributes of triangles and their component parts based on explorations and concrete models 5-2

Inequalities in Triangles Objective: Identify the properties of inequalities in triangles

Concurrent Lines, Median and Altitude Objective: To identify concurrent lines, median and altitude of a triangle

Squares, Square Roots and Radicals Objective: Learn to find square roots of perfect square, estimate square root of non perfect square and simplify radicals.

Simplifying Square Roots (Radicals) Simplifying square root is the process of breaking the radical apart such that we are left with the product of an integer times the square root of a prime number Procedure: 1)Split the radical of a product into the product of the radicals. Find the factors that are perfect squares. 2)Solve the square root of one or two of the factors that are perfect squares 3)Multiply the square roots of the perfect squares 4)Represent the answer as the product of the perfect square/s and the radical Example:

The Pythagorean Theorem and its Converse Objective: Use the Pythagorean Theorem and its converse to solve for the missing parameter of a right triangle, to identify if the parameters provided form a right triangle, obtuse triangle or acute triangle

Example:

Special Right Triangle Objective: Learn the properties of special right triangles ( and )

The Tangent Ratio Objective: To develop, apply and justify triangle similarity relationships: Tangent Ratio

Using Tangent Ratio to find missing measurement in a Right Triangle is the angle, 54 o is the side opposite the angle, w a is the side adjacent to the angle, 10

Sine and Cosine Ratio Objective: To develop, apply and justify triangle similarity relationships: the sine and cosine ratios

Angles of Elevation and Depression Objective: To select an appropriate representation in order to solve problems To apply trigonometric ratios Angle of Elevation - An angle of elevation is the angle formed by a horizontal line and the line of sight to an object above the horizontal line. Angle of Depression – An angle of depression is the angle formed by a horizontal line and the line of sight to an object below the horizontal line.