Geometry – Pre-requisite Skills Mr. Rosilez

1. Geometry – Pre-requisite SkillsMr. Rosilez

2. Solving Equations – Key Concepts Addition Property – If a = b, then a + c = b + c Subtraction Property - If a = b, then a – c = b – c Multiplication Property – If a = b, then ac = bc Division Property – If a = b, and 0, then Distributive Property – a(b c) = ab ac Substitution – If a = b, then b may replace a Proportion – A statement that two ratios are equal. To solve, cross multiply. Ex. Inequalities – Solving inequalities work the same way as solving equations except for one difference. Whenever you multiply or divide by a negative number, you must change the direction of the sign.

3. Solving Equations - Examples 1. 2. ) 3. 4.

4. Variables and Exponents • Variables may only be combined if like terms. Only add or subtract the coefficients, not the variables! • An exponent tells you how many times to multiply a base. The expression is called a power with base 6 and exponent 3. • Rules for exponents: Product of Powers Power of a Product Power of a Power Quotient of Powers Power of a Quotient Negative Exponent Zero Exponent

5. Examples – Solving Variables & Exponents 1. 2. 3. 4. 5. 6. 7. 8.

6. Working with Square Roots • A square root of a number “n” is a number “m” such that Every positive number has two roots, a “+” and a “-”. We will focus on “+” roots. Negative numbers have no square roots. The square root of “0” is “0”. • Properties: • Examples: 1. 2. 3. 4. 5.

7. Linear Equations • The following are the three methods of writing linear equations: 1. Standard Form 2. Slope-intercept Form 3. Point-slope Form • When graphing lines, it is best to convert to slope intercept form. • Slope (m) – The steepness of a line. where and are points on the line. Horizontal lines have zero slope, vertical lines have undefined slope. • X-intercept – Where the graph crosses the x-axis. To solve, substitute zero for “y” in the equation & solve. • Y-intercept – Where the graph crosses the y-axis. To solve, substitute zero for “x” in the equation & solve.

8. Examples for Linear Equations • Find the slope for each: 1. 2. • Find the intercepts for: X-int : Y-int : • Graph the following::

9. Systems of Equations • There are three basic methods for solving systems of equations (Graphing, substitution, and elimination). We will focus on elimination. • The goal of elimination is two get rid of one of the variables in the two equations by multiplying one or both equations, then either adding or subtracting the equations to eliminate a variable in order to solve for the other. You then substitute this value into one of the original equations to solve for the eliminated variable. • Example Substitute y = 4 to solve for x.

10. Factoring and Solving Equations • Rules for Factoring: 1. Pull out the GCF. 2. For equations in the form find 2 numbers such that when you multiply = ac, when you add = b. 3. If try grouping 4. For difference of squares

11. Factoring and Solving Quadratic Equations • In order to solve quadratic equations, try factoring them first. Once factored, set each part equal to zero and solve. 1. 2. • If you can’t factor, then use the quadratic formula to solve . Ex:

12. Undefined Terms, Points, Lines, and Planes • Undefined terms – Words that don’t have formal definitions but there is an agreement upon. • Point – Has no dimension. Its represented by a dot. . • Line – A line has one dimension and extends without end. • Plane – Has two dimensions and extends without end. • Line Segment – Part of a line with specific endpoints. • Ray – Part of a line with a specific endpoint. A • . • . m X Y • . • . M • . J H I . . D C • . • . B A

13. Key Geometric Concepts • Complementary Angles – Two angles who sum is • Supplementary Angles – Two angles whose sum is • Adjacent Angles – Two angles that share a common vertex and side but do not overlap. • Vertical Angles – Angles that are formed when two lines intersect. Angles 1 & 2 are vertical. • Parallel Lines – Two lines that never intersect. • Transversal – A line that intersects two or more lines. Angles 1 & 5 are corresponding, 4 and 5 are alternate interior, 2 & 7 are alternate exterior, and 4 & 6 are consecutive interior. • Linear Pair – Two adjacent angles whose non- common sides are opposite rays. 1 2 2 1 4 3 5 6 7 8

14. Key Geometric Concepts - Continued • Pythagorean Theorem – Used to find the third side of a right triangle, given the other two. The formula is where “c” is the hypotenuse. Basic Examples 1. 2. Find the measure of each. If and , find 3. Find the complement and 4. and supplement of a angle. D B A C a 75 b c 12 b 8

15. Perimeter (Circumference) and Area • Square • Rectangle • Triangle • Trapezoid • Circle 1. 2. 3. 4. 5. 10 12 6 8 4 8 6 6 6 11 16

16. Surface Area • Surface area is the sum of the areas of the faces of a polyhedron or other solid. Sometimes it’s easier to make a net of the figure and find each individual area. = • Find the surface area of each. 1. 2. 2 in 3 in 12 in 4in 12 in

17. Volumes of Prisms, Cones, Cylinders and Spheres • Prism: • Cone: • Cylinder: • Sphere: 3 in 10 in 8 in 6 in 10 in 6 m 8 m 5 m