Geometry Readiness

Summer Packet University High School

This packet is designed for those who have completed Algebra I

and are enrolled in Geometry in the upcoming fall semester.

Summer Packet

Geometry

Welcome to Geometry. The topics in Geometry can be abstract but they have practical applications and serve as a foundation for

further study in mathematics and other college level mathematics. Mathematics is a series of building blocks. A student of

mathematics must have sufficient Algebra I skills to be proficient in Geometry. The majority of this packet reviews Algebra I skills

necessary for success in Geometry. In addition to Algebra I skills, students in geometry must possess some fundamental Geometry

skills learned in middle school mathematics. This packet also covers a few of the topics covered in middle school Geometry. The

summer packet is to be completed by the first day of school. Teachers will expect students to have completed all practice problems

within the packet. During the first few weeks of school, every student will take a test on the content in their class’s review packet.

The following is a list of websites to visit for additional Algebra help or practice material:

Khan Academy

Algebasics

Algebra-Class

Algebra Help

Help Algebra

College Cram

Interactive Mathematics

Math Expression

Purple Math

Part I: Simplifying Expressions and Combining Like Terms

Order of Operations Review:

When an expression contains more than one operation, the operations must be performed in a certain order.

I. Evaluate any expressions inside grouping symbols like ( ) or [ ]

II. Evaluate exponents

III. Perform multiplication and division in order from left to right

IV. Perform addition from left to right.

Many people remember this using either of the following acronyms:

PEMDAS

(Parenthesis, Exponents, Multiplication, Division, Addition Subtraction)

Some people use this saying to remember PEMDAS:

Please Excuse My Dear Aunt Sally

GEMS

(Grouping Symbols, Exponents, Multiplication & Division, Subtraction & Addition)

Examples:

The following symbols are also considered as grouping symbols when using the order of operations.

Evaluate each expression.

|1| –

|2| –

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(

)

Evaluate each expression using the values given.

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use and

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use and

Simplifying Expression with Variables Review:

Combining like terms:

Terms may only be combined (added/subtracted) if they are like terms.

Like terms may have different coefficients (the number multiplied in front

of the variable) but must have all of the same variable(s) and

all of the same exponents.

o Examples

In order to combine like terms, you might use some of the following properties:

Associative Property of Addition/Multiplication:

o If all terms share the same operation, the operation may be applied in any order with the same result.

o Examples:

Distributive Property:

o If there is a single term multiplied (outside parenthesis) to an expression inside parenthesis, the term may be

distributed and multiplied separately to each term inside the parenthesis.

o **Don’t forget that when a negative is outside of the parenthesis, it is equivalent to distributing a -1 to the

expression inside the parenthesis**

o Examples:

Simplify:

Simplify: – –

– – – –

Simplify:

o If there are expressions being multiplied that both have more than one term, you must be sure to distribute every

term in the first expression to every term in the second expression

When multiplying two binomials (expressions that each have two terms) many use the acronym F.O.I.L.

(first, outer, inner, last) to remember to multiply all terms.

o Examples:

Simplify:

Simplify:

Simplify each of the following expressions.

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|3| |4|

|5| |6| –

|7| |8| –

|9| |10| – –

|11| – |12| – –

|13| – – – – |14| – – –

|15| – |16| –

|17| |18|

|19| |20| –

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Simplifying Exponential Expressions

Exponent Rules:

To simplify expressions with exponents, you must follow these exponent rules:

Rule: Rule: Example:

Product Rule:

when terms with the same base are

multiplied, you add the exponents

Quotient Rule:

When terms with the same base are divided,

you subtract the exponents

(numerator exponent – denominator

exponent)

Power Rule:

When a term is raised to another exponent,

you multiply the exponents.

Power of a Product/Quotient Rule:

When a group of variables being multiplied

or divided is being raised to a power, you

may distribute the exponent and use power

rule for each variable.

(

)

(

)

Zero Exponent:

Anything raised to the zero power is equal

to one.

Negative Exponents:

A negative exponent is equivalent to taking

the reciprocal of the base of the exponent

and applying the absolute value of the

exponent.

(

)

(

)

(

)

(

)

(

)

**Remember that the Order of Operations still applies here – parenthesis must always be taken care of first**

Simplify each of the following expressions. Your answers should include only positive exponents.

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|5| |6| (

)

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Part II: Solving Linear Equations

The process behind solving equations with one variable follows the ideas of inverse operations.

To evaluate a numerical expression, you use order of operations (PEMDAS).

To solve an equation for a variable, you must undo the order of operations (reverse PEMDAS or SADMEP)

The rules for solving linear equations can be summarized as:

1. Distribute if necessary.

2. Combine like terms (Subtraction/Addition)

a. You might need to move terms to different sides of the equal sign in order to combine.

i. We do this by doing opposite operations with the variables. For example, if a variable is added on one side,

you would subtract it to move it to the other side. If a variable was subtracted, you would need to add it to

the other side.

3. Isolate the variable – get the variable alone on one side of the equal sign.

a. Add/subtract to move numerical terms to the other side.

b. Multiply/divide to remove coefficients. (Multiplication/Division)

Examples:

**Always check your answer by plugging the value into the original equation**

Solve each of the following equations.

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Solving a Proportion

A proportion is an equation that has fractions on both sides of the equal sign.

The key to solving a proportion is in “clearing the denominators”. Simply, all you need to do is multiply both sides of the equation by

each of the denominators, one at a time.

Example:

Solve: n Work:

After this, you solve the resulting equation like all other equations.

Solve each of the following.

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Solving for a variable in a formula

Solving for a single variable in a formula can make it easier to use that formula. This is often used with physics formulas to ensure

easier calculations.

The process is similar to that of solving multi-step equations: find the operations being performed on the variable you are solving for,

and then use inverse operations to isolate it.

Solve for the indicated variable.

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Part III: Linear Equations and Graphing

Graphing in the Coordinate Plane:

The Coordinate Plane is a grid that is defined by a horizontal axis called the x-axis and a vertical axis called the y-axis. It is a way of

graphing locations called coordinates, which look like (x, y), where x is the location on the x-axis and y is the location on the y-axis.

When two coordinates (often called points) are connected, they form a line that has what is called slope.

Slope is the measure of the “steepness” of a line, written as a fraction:

Given two points , you may find the slope of the line connecting them by graphing and counting the rise and run

or by using the formula:

Examples:

Every line can be written using an equation, which provides a map for any of the points that lie on the line.

There are many ways to write the equation of a line, but the most frequent is:

Slope-Intercept Form –

If an equation is written in slope-intercept form, the slope can be identified as the in the equation and the y-intercept (the point

where the line intersects the y-axis) can be identified as .

If an equation is in Slope-Intercept Form, the equation can easily be graphed in a coordinate plane by following the steps:

1. Identify the slope (m) and y-intercept (b) of the equation.

2. Plot the y-intercept on the graph

3. Count the slope (rise/run) away from your y-intercept to plot the next point,

4. Use the slope to plot a few more points for accuracy and connect the points to form a line.

If an equation is not written in slope-intercept form, you can always re-write the equation by solving for y.

Example:

The equations for horizontal and vertical lines cannot be written in slope-intercept form. This is because their slopes do not have both

a rise and a run. Instead, we use the following rules.

Vertical Lines: Horizontal Lines:

Equation: x = 3

Equation: y = -2

This graph has infinite rise and no run. A vertical line’s

slope is called undefined.

This graph has infinite run and no rise. A horizontal

line’s slope is zero.

Always take the form:

x = #

Always take the form:

y = #

Find the slope of the line that passes through each pair of points. Find the slope of the line shown in each grid.

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Graph each of the following equations in the provided grid.

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Writing the equations of lines

We can also use slope-intercept form to write the equations of lines by plugging in given information like the slope, y-intercept, and/or

other points on the line.

Examples:

Write the slope-intercept form of the equation of the line shown in each grid.

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|2|

Write the slope intercept form of the equation that satisfies each of the following requirements.

|3| Slope = -2 , y-intercept = 1 |4| Slope = -1 , y-intercept = -3

|5| Slope =

, passes through the point

|6| Slope =

, passes through the point

Write the following equations in slope-intercept form.

|7| |8| – |9|

slope

Parallel and Perpendicular Lines:

Two lines are parallel if they never intersect. In order to never intersect, parallel lines must have the same slope.

Two lines are perpendicular if they intersect at a 90 angle. In order to intersect this way, perpendicular lines must have slopes that are

reciprocals and the opposite signs of each other.

Examples:

slope same

slopes

Reciprocal,

opposite

sign slopes

State whether each pair of lines is parallel, perpendicular, or neither.

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Part IV: Solving Systems of Linear Equations

A system of equations is a set of two or more equations that contain the same variables.

The solution(s) to a system of equations is the set of values that are true in all of the equations in the system. These are values that

work when they are substituted into any of the equations in the system.

To solve a system of equations, there are three methods that may be applied. Each should give you the same solution(s).

The example below shows all three methods for a system of equations.

Solve:

By Graphing: By Substitution: By Elimination:

Step 1: Graph both equations in one

coordinate grid.

Step 2: Find the point(s) where they

intersect.

Step 1: Rearrange either/both equations

so that they are in the same form. Write

the equations so that their matching

variables line up.

Step 2: Multiply one or both equations by

numbers until one set of matching

variables have the same coefficients with

different signs.

Step 3: Add the equations (combining

like terms).

Step 4: Solve the resulting equation for

the remaining variable.

Step 5: Substitute this value into either of

the equations to solve for the other

variable.

Solve each of the following systems of equations.

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Part V: Quadratic Expression & Equations

Factoring Quadratic Expressions

There are many different ways that students learn to factor.

The idea behind factoring a quadratic is that you rearrange a quadratic expression into the multiplication of two binomial expressions.

Quadratic expressions are usually written in the form:

Essentially, all methods require you to find factors of either the last term of the quadratic ( ) or the last term multiplied by the first

term’s coefficient ( ). Then, you would test these factors to see which of them could add or subtract to form the middle term ( ). The

factors that work in this process become values in your binomial expressions. The process for when a = 1 is shown below.

Examples:

It is always a good idea to check your solutions by multiplying the binomials to ensure that you equation is still the same.

The following rules might also help you find the correct factors faster.

Factoring when a is not equal to 1 is more complicated. This time, you must consider how your values will multiply to each other in

your factors.

You should also always check to see if your equation has a GCF (greatest common factor) that can be factored out first.

Factor . Check your answer.

Factors of 3

Factors of 10

Outer + Inner

1 and 3 2 and 5

1 and 3 5 and 2

1 and 3 1 and 10 1 and 3 10 and 1

Check:

Factor each expression completely.

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Solving Quadratic Equations by Factoring:

Quadratic equations are solved by factoring by using what is called the Zero Product Property.

The Zero Product Property states that if any terms/expressions are multiplied and set equal to zero, then if any of the multiplied

terms/expressions are equal to zero, then the equation will be true.

Essential, this means that when any of the multiplied expressions are equal to zero, you have found a solution.

To use this, we simply factor our quadratic expression and then set each of the binomials equal to zero. This gives us two linear

equations to solve, which each give us a solution to our quadratic equation.

Solve each equation by factoring.

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Solving Quadratics Using the Quadratic Formula:

The quadratic formula is:

√

Each of the variables ( and ) in the formula are the coefficients of the three terms in a quadratic equation when the equation is in

standard form.

The standard form of a quadratic equation is:

Before using the quadratic formula, it is important that the quadratic equation is in standard form so that you can correctly identify the

coefficients and to use in the formula. Notice that the equation is equal to zero. Every quadratic equation must be set equal to

zero before the quadratic formula can be used.

After the quadratic equation is in standard form and you have identified the and , plug the values into the formula. Then,

simplify using the order of operations. You should simplify any square roots whenever possible. Your end result will be two solutions,

shown by the operation (one is with the +, one is with the – ).

Examples:

Steps: Example 1: Example 2:

Solve:

Solve:

Step 1: Identify a, b, and c. You

might need to rearrange the

equation so that it is in standard

form.

Step 2: Plug these values into the

quadratic equation

Step 3: Simplify

Step 4: Separate into two

solutions with the

Solve each of the following quadratic equations using the quadratic formula.

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Part VI: Introductory Geometry Review

Angle Relationships

When two parallel lines are intersected by another line called a transversal, there are special angle pairs that have

unique names and relationships.

These angles have the same measure

These angles are supplementary (their measures add to 180°)

These angles have the same measure

Corresponding Angles angles that are in the same relative location on each parallel

These angles have the same measure

when only two lines intersect, these angles are across from each other

These angles have the same measure.

Using angle relationships, find the measure of the indicated angle.

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Area, Perimeter, and Circumference:

Perimeter is defined as the length around a 2D figure.

Usually, perimeter is found by adding the lengths of all the sides of the figure.

The length around a circle is called circumference and has a unique formula.

Area is defined as the area inside a 2D figure.

For many shapes such as squares, rectangles, triangles, and circles, there are formulas for finding area.

Rectangle/Square: Triangle: Circle:

Visual:

b = base

h = height

b = base

h = height

**height must be a segment from a

vertex of the triangle perpendicular

to the opposite side**

r = radius

**radius is a segment from

the center of the circle to

any point on the circle**

Perimeter/

Circumference

Formula:

Perimeter/

Circumference

Example:

Area Formula:

Area Example:

Find the area and perimeter/circumference of each of the following figures. Round your answers to the nearest hundredth.

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5.3 6.2

3.7

9.2

Volume:

The volumes of 3D solids differ depending on the kind of solid. Each type has its own formulas for finding volume.

o A prism is a 3D solid that has two congruent and parallel bases

and rectangular sides. A prism usually named more specifically

by the name of its base.

The formula for volume of any prism is:

V = area of the base height OR

Example:

o A cylinder is a 3D solid that has two parallel and congruent circular bases.

The formula for volume of any cylinder is:

V = area of the base height OR

Example:

o A pyramid is a 3D solid that has one base, sides that are triangles,

and an apex (point at the top) where the sides meet.

The formula for volume of any pyramid is:

V =

area of the base height OR

Example:

Find the volume of each of the following solids. Round your answers to the nearest hundredth when necessary.

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Transformations:

Transformations are changes done to figures in the coordinate plane that can be described using mathematical operations.

Some transformations have names to describe them.

Transformations that do not change the shape or size of a 2D object are called Rigid Transformations.

Examples of two different types of Rigid Transformations are translations and reflections.

Translations: Reflections:

Translation: 5 units to the right and 1 unit up.

Reflection across the x-axis.

“Slide” or “shift” figures to the left, right, up and/or

down.

“Flip” or “mirror” a 2D figure across lines.

The mathematical operations of translations are

addition and subtraction.

You can add/subtract the movement from the vertices

of the figure.

To perform a reflection, count the distance between the 2D

figure and the line of reflection. The new 2D figure should

be the same distance from the line of reflection, but in the

opposite direction from the line.

Graph the image of each figure using the transformation given.

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Pythagorean Theorem:

A right triangle is a triangle that has one 90 angle. The sides of right triangles have unique relationships.

A right triangle has two sides that are called legs and one side that is called a hypotenuse. The hypotenuse is always the side of the

triangle that is opposite the 90 angle. It is also always the longest side of the triangle.

If you know the length of two sides of a right triangle, you can use the Pythagorean Theorem to find the third side length.

Pythagorean Theorem:

where a and b are the lengths of the two legs and c is the length of the hypotenuse.

Examples:

Find the missing side of each triangle. Round your answers to the nearest hundredths.

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Solutions:

Part I:

Order of Operations:

1. 22

2. 5

3. -8

4. 2

5. 19

6. -7

7. -6

8. 64

9. 8

10. -10

Simplifying Expressions:

1. 14m + 5

2. -6p + 5

3. 30x + 35

4. 52a – 54

5. -4m + 2

6. -18k – 21

7.

8.

9. 9n – 2

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

Simplifying Exponential Expressions

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Part II:

Solving linear equations

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Solving Proportions

1.

2.

3.

4.

5.

6.

7.

8.

Solving for a variable in a formula:

1.

2. – –

3.

4. –

5. –

6.

7.

8. –

9. –

Part III:

Graphing lines:

1.

2.

3.

4.

5. L

6. L

7. L

8. L

9. L

10. L

11. L

12. L

13. L

Writing the equations of lines

1.

2. –

3.

4. –

5.

6.

–

7.

8. –

9.

Parallel & Perpendicular:

1. parallel

2. perpendicular

3. neither

4. parallel

5. neither

6. perpendicular

Part IV:

Systems of Equations:

1. (8, 1)

2. (-1, 6)

3. (0, -6)

4. (-3, -1)

5. (-3, -2)

6. (-2, 0)

7. No Solution

8. (6, -4)

9. (-4, 8)

10. (0, 3)

11. (2, -7)

12. (-3, -4)

13. (-4, -2)

14. (-2, 1)

Part V:

Factoring:

1. – –

2.

3. – –

4. –

5. –

6. –

7. –

8. –

9. –

10. –

11. –

12.

13. –

14. –

15. –

16.

17. –

18. –

19.

20.

Solving by Factoring:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

Quadratic Formula:

1. L

2. L

3. L

4. L

5. L

6. L

Part VI:

Angle Relationships:

1. 120

2. 70

3. 133

4. 47

5. 129

6. 117

7. 110

8. 56

9. 132

Area, Perimeter, and Circumference

1.

2.

3.

4.

5.

6.

7.

8.

9.

Volume:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Transformations:

1. L

2. L

3. L

4. L

5. L

6. L

Pythagorean Theorem:

1. x = = 10.72

2. x = = 14.32

3. x = = 9.43

4. x = = 10.20

5. x = = 5.39

6. x = = 5.29

7. x = = 9.22

8. x = = 13.45