Geometry – Pre-requisite Skills Mr. Rosilez

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 𝑐≠0, then 𝑎 𝑐 = 𝑏 𝑐 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. 4 𝑥 = 12 30 →12𝑥=120→𝑥=10 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.

Solving Equations - Examples 3. 2(𝑥−3) 5 = 𝑥 4 8 𝑥−3 =5𝑥 8𝑥−24=5𝑥 −24=−3𝑥 8=𝑥 4. −2 𝑥−3 ≤12 −2𝑥+6≤12 −2𝑥≤6 𝑥≥−3 1. 2 𝑥−6 =5𝑥+9 2𝑥−12=5𝑥+9 −12=3𝑥+9 −21=3𝑥 −7=𝑥 2. 1 3 𝑥+ 3 4 =2𝑥− 1 2 12( 1 3 𝑥+ 3 4 =2𝑥− 1 2 ) 4𝑥+9=24𝑥−6 15=20𝑥 𝑥= 15 20 = 3 4

Variables and Exponents Variables may only be combined if like terms. Only add or subtract the coefficients, not the variables! 3 𝑎 2 +5 𝑎 2 =8 𝑎 2 An exponent tells you how many times to multiply a base. The expression 6 3 is called a power with base 6 and exponent 3. 6 3 =6×6×6=216 Rules for exponents: Product of Powers 𝑎 𝑚 ∙ 𝑎 𝑛 = 𝑎 𝑚+𝑛 Power of a Product 𝑎∙𝑏 𝑚 = 𝑎 𝑚 ∙ 𝑏 𝑚 Power of a Power 𝑎 𝑚 𝑛 = 𝑎 𝑚𝑛 Quotient of Powers 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 , 𝑎≠0 Power of a Quotient 𝑎 𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚 , 𝑏≠0 Negative Exponent 𝑎 −𝑛 = 1 𝑎 𝑛 , 𝑎≠0 Zero Exponent 𝑎 0 =1, 𝑎≠0

Examples – Solving Variables & Exponents 1. 3𝑥 8 ∙2 𝑥 4 =6 𝑥 12 2. 𝑚 −2 −3 = 𝑚 6 3. 2 𝑛 2 4 ∙ 3𝑛 2 16 𝑛 8 ∙9 𝑛 2 =144 𝑛 10 4. 16 𝑥 2 𝑦 2𝑥𝑦 =8𝑥 5. 3 𝑟 −3 𝑠 2 10𝑠 9 𝑟 −6 𝑠 2 10𝑠 = 9𝑠 10 𝑟 6 6. 3 𝑎 2 𝑏 0 𝑐 21 𝑎 −3 𝑏 4 𝑐 2 = 𝑎 5 7 𝑏 4 𝑐 7. −6𝑟+3𝑠−5𝑟+8 −11𝑟+3𝑠+8 8. 3𝑟 5𝑟+2 −4(2 𝑟 2 −𝑟+3) 15 𝑟 2 +6𝑟−8 𝑟 2 +4𝑟−12 7 𝑟 2 +10𝑟−12

Working with Square Roots A square root of a number “n” is a number “m” such that 𝑚 2 =𝑛. 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: 𝑎𝑏 = 𝑎 ∙ 𝑏 𝑎 𝑏 = 𝑎 𝑏 = 𝑎𝑏 𝑏 Examples: 1. 121 =11 2. 48 = 16 ∙ 3 =4 3 3. 175 − 28 25 ∙ 7 − 4 ∙ 7 5 7 −2 7 =3 7 4. 5 18 ∙2 2 10∙ 36 =10∙6=60 5. 12 6 = 12 6 ∙ 6 6 = 12 6 6 =2 6

Linear Equations The following are the three methods of writing linear equations: 1. Standard Form 𝐴𝑥+𝐵𝑦=𝐶 2. Slope-intercept Form 𝑦=𝑚𝑥+𝑏 3. Point-slope Form 𝑦− 𝑦 1 =𝑚(𝑥− 𝑥 1 ) When graphing lines, it is best to convert to slope intercept form. Slope (m) – The steepness of a line. 𝑚= 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 = ∆𝑦 ∆𝑥 = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 where 𝑥 1, 𝑦 1 and 𝑥 2, 𝑦 2 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.

Examples for Linear Equations Find the slope for each: 1. 4, 2 (6, −3) 𝑚= −3−2 6−4 = −5 2 2. 3, 6 (3, 4) 𝑚= 4−6 3−3 = −2 0 =𝑢𝑛𝑑𝑒𝑓. Find the intercepts for: 2𝑥−6𝑦=12 X-int : 2𝑥−6 0 =12 2𝑥=12→𝑥=6 Y-int : 2 0 −6𝑦=12 −6𝑦=12→ 𝑦=−2 Graph the following:: 𝑦=2𝑥−4 2𝑥−3𝑦=6 𝑦=−4 𝑥=−3 𝑦−4=− 1 2 (𝑥+4)

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 2𝑥+3𝑦=4 3𝑥+5𝑦=8 3 2𝑥+3𝑦=4 2 3𝑥+5𝑦=8 6𝑥+9𝑦 =12 − 6𝑥+10𝑦=16 −𝑦=−4 𝑦=4 Substitute y = 4 to solve for x. 2𝑥+3 4 =4 2𝑥+12=4 2𝑥=−8 𝑥=−4

Factoring and Solving Equations Rules for Factoring: 1. Pull out the GCF. 4 𝑥 2 −8𝑥+12=4( 𝑥 2 −2x+3) 2. For equations in the form 𝑎 𝑥 2 +𝑏𝑥+𝑐 𝑤ℎ𝑒𝑟𝑒 𝑎=1, find 2 numbers such that when you multiply = ac, when you add = b. 𝑥 2 +4𝑥+3 𝑎𝑐=3, 𝑏=4, 𝑠𝑜 3, 1 (𝑥+3)(𝑥+1) 3. If 𝑎≠1, try grouping 6 𝑥 2 +13𝑥−5 𝑎𝑐=−30, 𝑏=13, 𝑠𝑜 15, −2 6 𝑥 2 +15𝑥−2𝑥−5 3𝑥 2𝑥+5 −1 2𝑥+5 (2𝑥+5)(3𝑥−1) 4. For difference of squares 4 𝑥 2 −16 𝑦 2 4 𝑥 2 =2𝑥 16 𝑦 2 =4𝑦 (2𝑥+4𝑦)(2𝑥−4𝑦)

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. 3 𝑥 2 −12𝑥=0 3𝑥 𝑥−4 =0 3𝑥=0 𝑥−4=0 𝑥=0,4 2. 𝑥 2 −6𝑥+8=0 𝑥−4 𝑥−2 =0 𝑥−4=0 𝑥−2=0 𝑥=4, 2 If you can’t factor, then use the quadratic formula to solve 𝑎 𝑥 2 +𝑏𝑥+𝑐. 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 Ex: 3 𝑥 2 −5𝑥−9 𝑎=3, 𝑏=−5, 𝑐=−9 𝑥= −(−5)± (−5) 2 −4(3)(−9) 2(3) 𝑥= 5± 25+108 6 𝑥= 5± 133 6

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 . . C D . . B A

Key Geometric Concepts Complementary Angles – Two angles who sum is 90°. Supplementary Angles – Two angles whose sum is180°. 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 1 2 3 4 5 6 7 8

Key Geometric Concepts - Continued Pythagorean Theorem – Used to find the third side of a right triangle, given the other two. The formula is 𝑎 2 + 𝑏 2 = 𝑐 2 where “c” is the hypotenuse. Basic Examples 1. 2. Find the measure of each. If ∠𝐴𝐵𝐶=55°and 𝑎=180°−75°=105° ∠𝐴𝐵𝐷=135°, find ∠𝐶𝐵𝐷. 𝑏=75° 𝑎=𝑐=105° ∠𝐶𝐵𝐷=135°−55°=80° 3. Find the complement and 4. 8 2 + 𝑏 2 = 12 2 and supplement of a 40°angle. 64+ 𝑏 2 =144 𝐶=90°−40°=50° 𝑏 2 =80 𝑆=180°−40°=140° 𝑏= 80 =4 5 B D A C a 75 b c 12 b 8

Perimeter (Circumference) and Area Square 𝑃=4𝑠 𝐴= 𝑠 2 Rectangle 𝑃=2𝑙+2𝑤 𝐴=𝑙𝑤 Triangle 𝑃=𝑎+𝑏+𝑐 𝐴= 1 2 𝑏ℎ Trapezoid 𝑃=𝑠𝑢𝑚 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 𝐴= 1 2 𝑏 1 + 𝑏 2 ℎ Circle 𝐶=2𝜋𝑟 𝐴=𝜋 𝑟 2 𝜋=3.14 1. 2. 3. 4. 5. 𝑃=4 6 𝑃=2 12 +2 6 𝑃=8+8+11 𝑃=2 8 +10+16 𝐶=2(3.14)(4) 𝑃=24 𝑃=36 𝑃=27 𝑃=42 𝐶≈25.12 𝐴= 6 2 𝐴=12 6 𝐴= 1 2 11 6 𝐴= 1 2 10+16 6 𝐴=3.14( 4 2 ) 𝐴=36 𝐴=72 𝐴=33 𝐴=78 𝐴≈50.24 6 12 10 8 4 6 8 6 6 11 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. 𝑆𝐴=12 3 2 +3 2 2 +12 2 (2) 𝑆𝐴=72+12+48=132 𝑖𝑛 2 2. 𝑆𝐴≈2 3.14 4 2 +2 3.14 4 12 𝑆𝐴≈100.48+301.4440≈1.92 𝑖𝑛 2 2 in 3 in 12 in 4in 12 in

Volumes of Prisms, Cones, Cylinders and Spheres 𝑉=𝜋 𝑟 2 ℎ 𝑉=(3.14)( 3 2 )(10) 𝑉≈282.6 𝑖𝑛 3 Sphere: 𝑉= 4 3 𝜋 𝑟 3 𝑉= 4 3 (3.14)( 6 3 ) 𝑉≈904.3 𝑚 3 Prism: 𝑉=𝑙𝑤ℎ 𝑜𝑟 𝑉=𝐵ℎ 𝑉= 10 8 6 𝑉=480 𝑖𝑛 3 Cone: 𝑉= 1 3 𝜋 𝑟 2 ℎ 𝑉= 1 3 (3.14)( 5 2 )(8) 𝑉≈209.3 𝑚 3 3 in 8 in 10 in 6 in 10 in 6 m 8 m 5 m