Chapter 1

Vocab A net is a two-dimensional diagram that you can fold to form a three-dimensional figure.

Vocab Terms: POINT- has no dimension (no length, width, or thickness) represented by a dot Point C

Vocab Terms:

Vocab Terms: BCD

Vocab Terms:

Vocab Terms:

Postulate 1-1

Postulate 1-2

Postulate 1-3

Postulate 1-4

Congruent Angles - angles that have the same measure.

Types of Angles: Two angles are adjacent angles if they share a common vertex and side, but have no common interior points.

Angles 1 and 3 are vertical angles, so are angles 2 and 4 5 4 2 6 3 Vertical angles – two angles whose sides form two pairs of opposite rays. Vertical angles are congruent. Angles 1 and 3 are vertical angles, so are angles 2 and 4 Linear pair – two adjacent angles whose uncommon sides are opposite rays. Angles 5 and 6 are a linear pair and add up to 180°.

Complementary Angles: Two angles whose sum is 90°. Note that these two angles can be “pasted” together to form a right angle.

Supplementary Angles: Two angles whose sum is 180°. Note that these two angles can be “pasted” together to form a straight angle.

An angle bisector is a ray that divides an angle into two congruent angles. Its endpoint is at the angle vertex.

Midpoint The midpoint of a segment is the point that divides, or bisects, the segment into two congruent segments. Midpoint Formula: Find the midpoint between (-5,-3) and (1,-7).

Distance Formula -computes the distance between two points in a coordinate plane

Classifying Polygons Polygon – a closed plane figure formed by three or more segments. Each segment is called a side. Each endpoint of a side is called a vertex. Each segment intersects exactly two other segments at their endpoints. No two segments with a common endpoint are collinear.

Classifying Polygons You can classify a polygon by the number of sides.

Classifying Polygons A diagonal is a segment that connects two nonconsecutive vertices.

Circle r Tell students that we are now going to explore the different formulas to find perimeter, area, and circumference of different shapes. Please highlight that these are very important and they should definitely write them in their notes as well as what each letter represents. A circle uses circumference instead of perimeter to measure the outside length of the shape. r represents the radius of the circle d represents the diameter The area of a circle formula uses units squared! units units2

Square P = 4s units A = s2 units2 s Perimeter is 4 times the length of a side. Area is the length of a side squared.

Rectangle P = 2l + 2w units A = lw units2 l w Perimeter is length + width + length + width or (2)length + (2)width. Area is length times width. P = 2l + 2w units A = lw units2 w

Triangle P = a + b + c units A = ½bh units2 c a h b Perimeter adds up all the outside edges. Area is ½ the base of the triangle times the height. b

Chapter 2

Using Inductive Reasoning In the previous examples you used inductive reasoning to make a conjecture. Conjecture: an educated guess about what you think is true based on observations. What conjecture can you make about the twenty-first term in R, W, B, R, W, B, ...?

Finding a Counterexample Not all conjectures turn out to be true. You can prove that a conjecture is false by finding at least one counterexample.

Conditional Statements Type of logical statement Has 2 parts- a hypothesis and a conclusion Can be written in “if-then” form- the “if” part contains the hypothesis and the “then” part contains the conclusion.

Truth Value The truth value of a conditional is either true or false. To show it is true, every time the hypothesis is true, the conclusion must also be true. Example: If you live in the United States, then you live in North America. To show it is false, find only one case where the hypothesis is true and the conclusion is false. Example: If you live in North America, then you live in the United States.

Negation The negation of a statement is the opposite of the statement. The symbol “~p” is read “not p” Example: Statement: “The ocean is green.” Negation: “The ocean is not green.”

Related Conditional Statements

Equivalent Statements: when two statements are both true or both false. ORIGINAL- If angle A is 30°, then angle A is acute. INVERSE- If angle A is not 30°, then angle A is not acute. CONVERSE- If angle A is acute, then angle A is 30°. CONTRAPOSITIVE- If angle A is not acute, then angle A is not 30°. BOTH TRUE BOTH FALSE

Biconditional Statement: A biconditional is a single true statement that combines a true conditional and its true converse. A statement that contains the phrase “if and only if”. Conditional: If it is Sunday, then I am watching football. Converse: If I am watching football, then it is Sunday. It is Sunday, if and only if I am watching football. (p q).

Biconditional Statements: Can be either true or false. To be true, BOTH the conditional and its converse must be true. A TRUE Biconditional Statement is true both “forward” and “backward”. All definitions can be written as a biconditional statements.

Identifying a Good Definition A good definition is a statement that can help you identify or classify an object. Uses clearly understood terms. Is precise. Avoid words such as large, sort of, and almost. Is reversible. Can be written as a true biconditional. **One way to show it is NOT a good definition is to find a counterexample.

Symbolic Notation Conditional statement has a hypothesis and a conclusion. Written with symbolic notation: p represents hypothesis, q represents conclusion. If p, then q. Symbolic notation: p q p implies q

Converse? If q, then p. or Biconditional? p if and only if q or

Contrapositive? Inverse? If not q, then not p. or If not p, then not q. or Contrapositive? If not q, then not p. or

Checkpoint: Let p be “You are a pianist” and q be “you are a musician”. Write in words:

Laws of Logic: DEDUCTIVE REASONING – uses facts, definitions, and accepted properties in a logical order to write a logical argument. INDUCTIVE REASONING – uses examples and patterns to form a conjecture.

2 LAWS OF DEDUCTIVE REASONING LAW OF DETACHMENT: If is a true conditional statement and p is true, then q is true.

2ND LAW LAW OF SYLLOGISM: If and are true conditional statements, then is true.

Algebraic Properties

Theorem – a conjecture or statement that you prove true ***When writing a proof for a theorem, separate the theorem into a hypothesis and conclusion. The hypothesis becomes the “Given” statement and the conclusion is what you want to prove.

Chapter 3

Skew Lines – lines that do not intersect are not coplanar. Parallel Lines – two lines that are coplanar and do not intersect. Notation: // Skew Lines – lines that do not intersect are not coplanar. Parallel Planes – two planes that do not intersect.

TRANSVERSAL – a line that intersects two or more coplanar lines at different points. 1 2 3 4 5 6 7 8 Notice, angles 3, 4, 5, and 6 are interior angles (between the lines). Angles 1, 2, 7, and 8 are exterior angles (outside the lines).

CORRESPONDING ANGLES: occupy corresponding positions 1 2 3 4 6 5 7 8 Corresponding angles lie on the same side of the transversal, and in corresponding positions

ALTERNATE EXTERIOR ANGLES: lie outside the two lines on opposite sides of the transversal. 1 2 3 4 5 6 7 8

ALTERNATE INTERIOR ANGLES: lie between the two lines on opposite sides of the transversal. 1 2 3 4 5 6 7 8

SAME-SIDE INTERIOR ANGLES: lie between the two lines on the same side of the transversal. 3 5 1 2 4 6 7 8

Theorem 3-10: Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180°.

Theorem 3-11: Exterior Angle Theorem The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

SLOPE Definition Symbols Diagram The slope, m, of a line is the ratio of the vertical change (rise) to the horizontal change (run) between any two points. A line contains the points (x1, y1) and (x2, y2)

Equations of Lines: Y = mx + b SLOPE-INTERCEPT FORM slope Y-intercept :the y-coordinate of the point where the line crosses the y-axis. SLOPE-INTERCEPT FORM *only used for nonvertical lines

Point-Slope Form Standard Form

Checkpoint: Find the slope of a line that passes through the points (-3, 0) and (4,7).

Checkpoint: Find all three forms of the equation of the lines below. Line p, passes through (0, -3) and (1, -2). Line r passes through (-6, -1) and (3, 7).

Graphing Lines Identify the form of the equation. Identify/Graph the y-intercept Start at the y-intercept and use the slope to graph a couple points on the line Connect the points

Using Two Points to Write an Equation Write the equation of the line using the point(-2, -1) in point-slope form. Find the slope of the line Use the given values and plug into the point-slope form Write the equation of the line in slope-intercept form.

Writing Equations of Horizontal and Vertical Lines

Slopes of Parallel Lines When two line are parallel, their slopes are the same. **If two lines are parallel, their slopes will be the same, but they MUST have a different y-intercept! **Do all horizontal lines have the same slope? Explain.

Sage-n-Scribe

Slopes of Perpendicular Lines If two lines are perpendicular, their slopes are negative reciprocals. In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1 (opposite reciprocals). Vertical and horizontal lines are perpendicular.

Perpendicular Slopes: Opposite Reciprocals of each other.

Chapter 4

Congruent figures – have exactly the same size and shape Congruent figures – have exactly the same size and shape. (Congruent Corresponding Parts) *Corresponding angles and sides are congruent.

Theorem 4.1: Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. If and then

Triangles: figure formed by 3 segments joining 3 noncollinear points. CLASSIFICATION BY SIDES (# of congruent sides) Scalene Triangle-no congruent sides Isosceles Triangle-at least 2 congruent sides Equilateral Triangle-3 congruent sides

CLASSIFICATION BY ANGLES (all triangles have at least two acute angles, the third angle is used to classify) EQUIANGULAR TRIANGLE – 3 congruent angles (also acute)

Vertex – each of the three points joining the sides of a triangle. Vertex A Adjacent sides – two sides sharing a common vertex. Sides AB and AC Opposite side – the third side not sharing the vertex Side BC

Postulate 4-1: Side-Side-Side (SSS) Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

Postulate 4-2: Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of second triangle, then the two triangles are congruent.

Postulate 4-3: Angle-Side-Angle (ASA) Postulate If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of a second triangle, then the 2 triangles are congruent

Theorem 4-2: Angle-Angle-Side (AAS) Theorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the second triangle, then the two triangles are congruent.

Theorem 4.3 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent. Theorem 4.4 - Converse: If two angles of a triangle are congruent, then the sides opposite them are congruent

Right Triangles Hypotenuse: It is the side opposite the right angle in a right triangle. It is the longest side in a right triangle

AAS, SSS, SAS, ASA, and… For RIGHT triangles ONLY: Theorem 4.6: Hypotenuse-Leg (HL) Congruence If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

Chapter 5

Midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.

Theorem 5.1: Midsegment Theorem If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long. x ½ x

Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint. C A B M D Equidistant from two points – distance from each point is the same

Theorems: Theorem 5-4: Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Theorem 5-5: Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it lies on the bisector of the angle. B D A C

Perpendicular Bisectors of a Triangle A perpendicular bisector of a side of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.

Using Angle Bisectors of a Triangle An angle bisector of a triangle is a bisector of an angle of a triangle. The point of concurrency of the angle bisectors is called the incenter of the triangle. (Center of the inscribed circle.)It always lies inside the triangle.

Median of a Triangle A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. A median B C Every triangle has three medians.

Point of Concurrency for Medians The point of concurrency for the 3 medians is called the centroid. The centroid is always on the inside of the triangle. Centroid

Concurrency of Medians Theorem The medians of a triangle are concurrent at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. BG = (2/3)BE, AG = (2/3)AF, and CG = (2/3)CD E D F Note: The centroid of a triangle can be used as its balancing point.

Altitude of a Triangle The altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on, or outside the triangle.

Orthocenter of a Triangle Theorem 5.9: Concurrency of Altitudes Theorem: The lines that contain the altitudes of a triangle are concurrent. If segments AH, BH and CH are the altitudes of triangle DEF, the lines AH, BH, and CH intersect at a point H, the orthocenter. B A C

Summary

Writing an Indirect Proof Step 1: State as a temporary assumption the opposite (negation) of what you want to prove Step 2: Show that this temporary assumption leads to a contradiction Step 3: Conclude that the temporary assumption must be false and that what you want to prove must be true

Triangle Inequality Theorem Theorem. 5.12: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Hinge Theorem (SAS Inequality Theorem) Theorem 5.13: If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. V R RT > VX 80 100 W X T S