Chapter 1: Lines in a PlaneLessons:1. The Basics2. Algebraic Relations3. Laws of Logic4. Styles of Proofs5. Parallel Line Properties6. Parallel Line Proofs7. Curves in a Plane

Did you know:– Lines are infinite in length?– Linear equations use x?– Quadratic equations use x2?– Cubic equations use x3? – The linear coordinate plane is named after French

mathematician and philosopher René Descartes (Latin: Cartesius)?

– You will learn all of this and more in this book!

Trey DeitchJoey CaudillIan TanguyXin Wei

Lesson 1: The Basics

• Parallel Lines are coplanar lines that do not intersect.

• Intersecting Lines are coplanar lines that have exactly one point in common.

• Perpendicular Lines are intersecting lines that meet at a right angle.

• Oblique Lines are intersecting lines that do not meet at a right angle.

• Skew Lines are two lines that do not lie in the same plane.

• Vertical angles are two angles whose sides form two pairs of opposite rays.

∠1 and ∠2 are vertical angles.

• A Linear Pair is a pair of two adjacent angles whose non-common sides are opposite rays.

∠1 and ∠2 are a linear pair. 1 2

1 2

• Parallel Planes are planes that do not intersect.

• Perpendicular Planes are intersecting planes that meet at a right angle.

Theorem Corner:Theorem 1.1 – Transitivity of Parallel Lines – If two lines are parallel to the same line, then they are parallel to each other.

If line I is parallel to lines AN, then the three lines are parallel.

I A N

Theorem 1.2 – Property of Perpendicular Lines – If two coplanar lines are perpendicular to the same line , then they are parallel.

If line X is perpendicularto lines I and N, then linesI and N are parallel X

I N

Example One: The Transitivity of Parallel Lines

l1 l2 l3

Say that the three lines to the left are coplanar. Show that if l1 is parallel to l2 and l2 is parallel to l3 than l1 is parallel to l3.

Given: l1 is parallel to l2 and l2 is parallel to l3

l1 l2 l3 Prove: l1 is parallel to l3.

m1 is the slope of line l1, m2 is the slope of line l2, and m3 is the slope of line l3.

Statements Reasonsm1 = m2 Parallel Lines have equal slopes.m2 = m3 Parallel Lines have equal slopes.m1 = m3 Transitive Property of Equality

l1 and l3 are parallel because they have the same slope.

Example Two: Property of Perpendicular LinesSay that the three lines above are coplanar. Show that if l1 is parallel to l2 and l2 is parallel to l3 than l1 is parallel to l3.

Given: l1 is perpendicular to l2 and l2 is perpendicular to l3

Prove: l1 is parallel to l3.

m1 is the slope of line l1, m2 is the slope of line l2, and m3 is the slope of line l3.

Statements Reasonsm1 · m2 = -1 The product of the slopes of perpendicular lines is always -1.m2 · m3 = -1 The product of the slopes of perpendicular lines is always -1.m1 · m2 = m2 · m3 Transitive Property of Equality.m1 = m3 Divide both sides by m2.

l1 and l3 are parallel because they have the same slope.

Lesson 2: Algebraic Relations

Two non vertical lines are parallel if and only if they have the same slope.

No Solution One Solution Many SolutionsParallel Lines Intersecting Lines Coincident Lines

Facts to Know:• Two non vertical lines are perpendicular if and only if they have the negative reciprocal slope.• Vertical lines have undefined slope• Horizontal lines have no slope.• Two intersecting lines have one solution.• Two parallel lines have no solution.• Two coinciding lines have infinite solutions.• Slope intersect form: y=mx+b, where m is the slope and b is the y-intercept. • Standard Form: ax+by=c slope is -a/b • You can solve linear systems using substitution.

Postulate Palace:• Postulate 12 – If two distinct lines intersect, then their intersection is exactly one point.• Postulate 13 – Parallel Postulate – I f there is a line and a point not on the line, then there is

exactly one line through the point parallel to the given line.• Postulate 14 - Perpendicular Postulate – I f there is a line and a point not on the line, then there

is exactly one line through the point perpendicular to the given line.

Example One:

Solve the system:1. 2y – 3x = 122. 2y – x = 8

By substituting your answer (y = 3) back into the first equation, you obtain x = -2. Check this solution bu substituting x = 18 and y = 2 into both of the original equations. You can also check the solutions by graphing the lines.

Example Two:Identify the slope, y-intercept, and the x-intercept of the line y=6x-7.

In the form y=mx+b, m is the slope and b is the y intercept. Therefore, the slope is 6 or 6/1 and the y intercept is -7.The x-intercept is the point where the line crosses the x-axis. When a line crosses the x-axis, y = 0.We can set up the equation 0=6x-7 to find that the x-intercept is 7/6.We have concluded that the slope of the line is six, its y intercept is -7, and its x intercept is 7/6.

x = 2y – 8 Solve Equation two for x.2y – 3(2y – 8) = 12 Substitute Equation two into equation one.2y – 6y + 24 = 12 Distribute.-4y + 24 = 12 Simplify.-4y = -12 Subtract 24 from both sides.y = 3 Divide both sides by -4.

Lesson 3: Laws of Logic

• Conditional: if p then q(p→q) A conditional statement is a statement that has a hypothesis and a conclusion can be put in the form if A, then B where A is called the, hypothesis (or premise or antecedent) and B is called the conclusion (or consequent). For example, If I go play tennis, then the weather is good.

• Converse: if q then p (q→p)A converse statement is the conditional statement reversed in the form if B, then A and it is not always true. For example, If the weather is good, then I go play tennis.

• Inverse: if not p then not q (−p→−q)An inverse statement is the opposite of the conditional statement in the form of if not A, then not B and it is not always true. For example, If I do not go play tennis, then the weather is not good.

• Contrapositive: if not q then not p (−q→−p)A contrapositive statement is a conditional statement that is the inverse of the converse. For example, If I the weather is not good, then I will not go play tennis.

• Biconditional: p if and only if q (p↔q)A biconditional statement is the conditional statement that is the inverse of the contrapositive. For example, I will go play tennis if and only if the weather is good.

Note that the inverse and converse of the same statement are each other’s contrapositive and are, therefore, both true or both false conditional statements. There are two types of converses, true converses and false converses. A true converse is when both the conditional statement and its converse are correct. A false conditional statement occurs when a statement is correct, but its converse is false. This same fact holds true for the contrapositive.

Symbols• The − symbol is the negation symbol. This symbol means that something is not.• The → symbol means “then” or “therefore”. For example, a→b.• The ↔ symbol means “if and only if”. This symbol is used in biconditional statements to show

that something will only occur if something else occurs. This is represented by saying p↔q.

Real-life ApplicationConditional statements are very important in computer programming. They are used to tell the machine when and how to execute the program. For example, if X=10, then Display 'Hello'” The format is used most notably by Javascript and by graphing calculators. You can try and write the program at right on your calculator.

PROGRAM:IFTHEN:1 X→:10 Y→:If X<10:Then:2X+3 X→:2Y-3 Y→:End:Disp X,Y

Lesson 4: Styles of ProofsParagraph Proof: A proof written in the form of a narrative.

Example:Given: ∠D and ∠I are right anglesProve: m∠H and m∠T are equal

By using Theorem 1.4 you know that ∠ D is equal to ∠I. Using vertical angles you can prove that ∠D is equal to ∠H. Using Linear Pair Postulate ∠I is equal to ∠T. And since ∠D equals ∠I you can conclude, using transitive property that m∠H and m∠T are equal. Flow Proof: Uses arrows to show the flow of the logical argument.

Example:Given: ∠D and ∠I are right anglesProve: m∠H and m∠T are equal

Given Vertical Angles

Theorem 1.4 Given Linear Pair

Two-Column Proof: a sequence of statements, each written with a reason.

Example:Given: ∠D and ∠I are right anglesProve: m∠H and m∠T are equal

Statements Reasons

1. ∠D and ∠I are right angles 1. Given2. m∠D = m∠I 2. Theorem 1.43. m∠D = m∠H 3. Vertical Angles4. m∠I = m∠T 4. Linear Pair5. m∠H = m∠T 5. Transitive

Theorem Corner:Try writing proofs for the following theorems:

• Theorem 1.3 – If two lines are perpendicular, then they intersect to form four right angles.• Theorem 1.4 – All right angles are congruent.• Theorem 1.5 – If two lines intersect to form a pair of adjacent congruent angles, then the lines

are perpendicular.

∠D is a rt. angle

D E I T C H

∠I is a rt. angle

m∠D = m∠I

m∠I = m∠T

m∠D = m∠H m∠H = m∠ T

Transitive

Lesson 5: Parallel Line Properties

Definition Depot:• Two angles are Corresponding Angles if they occupy corresponding positions. • Two angles are Alternate Interior Angles if they lie between l and m on opposite sides of t. • Two angles are Alternate Exterior Angles if they lie outside l and m on opposite sides of t. • Two angles are Consecutive Interior Angles if they lie between l and m on the same side of t. • Two angles are Consecutive Exterior Angles if they lie outside l and m on opposite sides of t.• If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Postulate Palace:• Postulate 15 – Corresponding Angles Postulate – If two parallel lines are cut by a transversal,

then the pair of corresponding angles are congruent.

Theorem Corner:• Theorem 1.6 – Alternate Interior Angles Theorem – If two parallel lines are cut by a transversal,

then the pair of alternate interior angles are congruent.• Theorem 1.7 – Consecutive Interior Angles Theorem – If two parallel lines are cut by a

transversal, then the pair of consecutive interior angles are supplementary.• Theorem 1.8 – Alternate Exterior Angles Theorem – If two parallel lines are cut by a

transversal, then the pair of alternate exterior angles are congruent.• Theorem 1.9 – Perpendicular Transversal Theorem – If a transversal is perpendicular to one of

two parallel lines, then it is perpendicular to the second.

Example: How is ∠7 related to the other angles? W E

I

∠7 and ∠6 are a linear pair. So are ∠7 and ∠8.∠7 and ∠5 are Vertical Angles.∠7 and ∠3 are Alternate Interior Angles∠ 7 and ∠ 4 Corresponding Angles

Proof of The Alternate Interior Angles TheoremGiven: l1 ║l2 Prove: ∠1 ≅ ∠2

Statements Reasons1. l1 ║l2 1. Given2.∠1 ≅ ∠3 2. Vertical angles are ≅ 3.∠3 ≅ ∠2 3. 2 lines ║ ⇒ corr. ∠s are ≅ 4 ∠1 ≅ ∠2 4. Transitive Property of Congruence

1

243

6

87 5

1 3

2

t l 2

l 1

Lesson 6: Parallel Line Proofs

In order to used what you have learned in the previous lesson, you need to know that lines are parallel. You should notice that each of the proofs has as part of its hypothesis “parallel lines”. The theorems and postulates that follow are the converses of those in the previous lesson. Since the converse of a statement is not necessarily true, each of the following must either be accepted as a postulate or proven true.

Postulate Palace:• Postulate 16 - Corresponding Angles Converse – If two lines are cut by a transversal so that

corresponding angles are congruent , then the lines are parallel.

Theorem Corner:• Theorem 1.10 - Alternate Interior Angles Converse – If two lines are cut by a transversal so that

alternate interior angles are congruent, then the lines are parallel.• Theorem 1.11 – Consecutive Interior Angles Converse – If two lines are cut by a transversal so

that consecutive interior angles are congruent, then the lines are parallel.• Theorem 1.12 – Alternate Exterior Angles Converse – If two lines are cut by a transversal so

that alternate exterior angles are congruent, then the lines are parallel.

Proof of Theorem 1.10Given: ∠1 ≅ ∠2 Prove: l1 ║l2

Statements: Reasons:1. ∠1 ≅ ∠2 1. Given2.∠1 ≅ ∠3 2. Vertical angles are ≅ 3.∠3 ≅ ∠2 3. Transitive Prop. of Congruence4. l1 ║l2 4. corr. ∠s are ≅ ⇒ 2 lines ║

1 3 2

Lesson 7: Curves in a Plane

Commonly used in topography, a curve is the idea of a geometrical one-dimensional continuum.

There are three types of curves:• A plane curve is a curve that lies in a single Euclidean plane. • A space curve is a curve in a three dimensional space, usually Euclidean space• A skew curve is a space curve which lies in no particular plane.

A plane curve may be closed or open.Some of the most common closed curves are the circle and the ellipse.Some of the most common open curves are the parabola and the hyperbola.

CirclesA circle is the set of points in a plane that are equidistant from a given point (center)

A circle is commonly represented by the formula

The circumference of a circle is π2r The area of a circle is π r2

Ellipse – the locus of points on a plane where the sum of the distances from any point on the curve to two fixed points (foci) is constant.

It is represented by the formula:

The area enclosed by an ellipse is πab, where 'a' and 'b' are the semi-major and semi-minor axis. If a=b then it is a circle.

Closed Curve Closed Curve Open Curve Open Curve

HyperbolaThe locus of points where the difference in the distance to two fixed points (foci) is constant.

Formulas:

East-west hyperbola centered at (h,k):

North-south opening hyperbola centered at (h,k):

Parabola (Quadratic Equation)Locus of points in a plane which are equidistant from a given point (focus) and a given line (directrix). The quadratic equation is a second degree polynomial.

The quadratic equation can be written:

The quadratic formula can be written:

Discriminant:• b2 – 4ac is the discriminant.• If b2 – 4ac = 0 then the equation has one real root called the double root which is counted twice

and whose value is

• If b2 – 4ac > 0 then the equation has two unequal real roots.• If b2 – 4ac < 0 then the equation has two unequal imaginary complex roots which are complex

conjugates of each other. An imaginary number is a number that includes the square root of -1, called i.(This is taught in Algebra II)

• the roots are distinct, if and only if the discriminant is non-zero• the roots are real, if and only if the discriminant is non-negative.

The sum of the roots equals

The product of the roots equals

The sum of the squares of the roots equals

Formulas:

Vertical Parabola

Horizontal Parabola

Cubic functionA function of the form where a is nonzero; or in other words, a polynomial of degree three.

The formula for Cubic Function is

The cubic equation has at most three roots.

The sum of the roots equals

The sum of the roots taken two at a time equals

The product of the roots equals