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Name _____________________________ Geometry Date _________ Period _________ 1) Get rid of parentheses by distributing 2) Combine like terms that are on the same side of the equal sign 3) Move all the variables to one side of the equation. 4) Move all the constants to the other side of the equation. 5) Solve like normal. Examples: 1) n+12 = 25 2) r- 14 = 13 5) 2y+1=15 6) 9) 3y-7y=28 h 3 15 3) 9x=72 4) 1 x3 5 2 7) -3(y-5)=12 8) 2(5x+4)=38 10) 4c+3(c-2)= -34 11) 6x=4x+18 12) 4n+5=6n+7 13) The length of a rectangle is 25 ft more than the width. The perimeter is 325 ft. 14) The width of a rectangle is 15 cm less than the length. The perimeter is 98 cm. Find the rectangle’s dimensions. 1 Name _____________________________ Geometry Date _________ Period _________ Coordinates – ordered pair ___________ – _______________________________ x – 1st coordinate (____________________________) y – 2nd coordinate (____________________________) Space – set of all points - contains all _______, _______, and _________ 2 Name _____________________________ Geometry Defining Geometric Vocabulary: Date _________ Period _________ 1) State the term 2) State the nearest classification 3) Describe what makes it unique. Examples: Collinear – Ex: C B Noncollinear – D A Ex: Coplanar – Ex: Noncoplanar – Ex: Points, Lines, & Planes: CLASS WORK Use this diagram for #1-5 1. Name three collinear points on line q and on line s H q I 2. Name 1 set of non-collinear points L 3. Name the opposite rays on line p and on line s 4. How many points are marked on line q? K J s 5. How many points are there on line q? 21. Name a point that is coplanar with A, E, and J A 22. Name a point that is coplanar with A, C, and I C 23. Name all the points that are noncoplanar with A, C, and D 24. Name all the points that are noncoplanar with F, H, and E F H 25. Where do plane ACH and plane IDC intersect? 26. Where do planes ACH, AFJ, and ACD intersect? E D 3 J I #21-32 Name _____________________________ Geometry Activity: Developing Definitions for One-Dimensional Figures Date _________ Period _________ Midpoint: Example: Counterexample: Parallel Lines: Example: Counterexample: Congruent Segments: Example: Counterexample: Bisector of a Segment Example: Counterexample: C Segment – Labeled as _________________________ - contains all points between the two end points. B A Q Measurement of a segment – distance between ________________ - measured in units of inches, feet, yards, meters, cm… etc.) #7) BD= #8) DA= #9) AC= #10) CD= 4 Name _____________________________ Geometry Parts of a Right Triangle: Date _________ Period _________ Hypotenuse Leg a and leg b Angle opposite hypotenuse Side opposite right angle Longest side ______________________________: In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse. Formula: c a b Note: _______ is always the ______________________ Example #1) Example #2) x 13 7 x 24 5 Using the Pythagorean Theorem to find the Distance in the Coordinate Grid: #1) #2) 5 Name _____________________________ Geometry #3) Date _________ Period _________ Distance Formula What is the formula? When do you use it? Examples: #1) Find the distance between W and Z, when W(1, 2) and Z(-4, -2) #2) Find WY. #3) Find XZ. A 6 Name _____________________________ Geometry Date _________ Period _________ How do you find the midpoint of a line? A B C D -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Example #3) Find the midpoint. a) BD b) AB c) DA d) CD Example #4) Midpoint Formula: What do you use it for? To find the midpoint of a line segment on the ______________________________________. Examples: #1) Find the midpoint of (-1, 7) and (2, 5) #2) Find the midpoint of (3, -5) and (7, 1) #3) Find the midpoint of (-1, 4) and (-1, 6) 7 Name _____________________________ Date _________ Geometry Period _________ #4) M is the midpoint of CD, where C(3,-2) and D(x, y) and M(-3, 1). Find the coordinates of D. #5) If N(4, -1) is the midpoint of ST and the coordinates of S are (3, -2), what are the coordinates of T? ____________________- How rapidly the line of an equation rises (positive slope) or declines (negative slope) Formula: m= Directions: Determine the slope of each line. a) b) c) d) 8 Name _____________________________ Geometry Date _________ Period _________ A ____________________ is a comparison of two quantities The ratio of a to b can be expressed as: Connor has a wallet with:1-$20 bill 2- $10 bills 1- $5 bill 8-$1 bills 1) What is the ratio of $1 bills to $10 bills? 2) What is the ratio of $10 bills to the total number of bills in the wallet? Point P divides AB in the ratio 3 to 1. 1. What does this mean? Example: A 32 foot long piece of rope has a knot tied to divide the rope into a ratio of 3:5. Where should the knot be tied? 9 Name _____________________________ Date _________ Geometry Period _________ Directed Line Segment: Tells the direction in which from which point to start and end. In this case, from Point A to Point B What does that tell you about the distance AP and PB in relation to AB? Example 1: Find the coordinate of point P that lies along the directed line segment from A(3, 4) to B(6, 10) and partitions the segment in the ratio of 3 to 2. 1. Find the rise and run for AB 2. Multiply the rise by the ratio from A to P, and the run by the ratio of A to P 3. Add/subtract these values to your starting point A 4. How can you use the distance formula to check that P partitions AB in the ratio of 3 to 2? Example 2: Find the point Q along the directed line segment from point R(–3, 3) to point S(6, –3) that divides the segment into the ratio 2 to 1 10 Name _____________________________ Date _________ Geometry Period _________ Example 3: Find the point Q along the directed line segment from point R(–2, 4) to point S(18, –6) that divides the segment in the ratio 3 to 7. Example 4: Find the coordinates of the point P that lies along the directed segment from A(1, 1) to B(7, 3) and partitions the segment in the ratio of 1 to 4 Example 5: Find the coordinates of point P that lies along the directed line segment from M to N and partitions the segment in the ratio of 3 to 2 11 Name _____________________________ Geometry Date _________ Period _________ Segment Postulate: If Q is between P and R, then ________________. - If PQ + QR = PR, then _____ is between P and R. Ex 1) Find x Given: A AB = 20 Z B AZ = 8 + x ZB = 5x 2) K, M, and P are collinear with P between K and M, PM = 2x + 4, MK = 14x - 56, and PK = x + 17 Solve for x and then find MK. 3) P, B, L, and M are collinear and are in the following order: P is between B and M, L is between M and P Draw a diagram and solve for x and then find PL, given: ML = 3x +16, PL = 2x +11, BM = 3x +140, and PB = 3x + 13 12 Name _____________________________ Geometry What notation do I use for angles? Date _________ Period _________ S vertex Labeled as: 1) 1 P 2) T vertex: _______________________ 3) Why can’t I use one letter? R To call it < B is too confusing!! S ___________________ – they’re 2 different <’s! Sides of an angle – rays of angle B T Ex. 1) a.) What number name is <ABD? D A 1 b.) What is the vertex of < 3? 2 3 C B c.) What are the sides of < 2? 3 parts of an angle: L 1) 2) W 3) M X N 13 Name _____________________________ Geometry Date _________ Period _________ Measure of an angle – ____________ Use a protractor to measure an angle Ex 1: A is in the interior of <BNJ. If m<ANJ=7x+11, m<ANB=15x+24, and <BNJ=9x+204. Solve for x and find <ANJ. 14 Name _____________________________ Geometry Example #3) Use a protractor to find the measure of each numbered angle. 3 2 Date _________ Period _________ 1 4 ______________ – form straight line *****Opposite rays form an angle that is ______° Example #2) In the figure, KZ and KX are opposite rays. Find the value of x and the measure of the indicated angle. m< XKP = 5x + 2, m<PKQ = 3x + 4, m< XKQ = 150; m< XKP Q P X K Z Examples: #4) Find <RXT if <PXR=3x and <RXT=5x+20 S Q R P X T 15 Name _____________________________ Geometry #5) Find <QXT if QXT 2 x 75 and QXP 9 x 6 Date _________ Period _________ S Q R P X T What is a midpoint? Midpoint – _____________ def: the midpoint, M, of PQ is the point between P and Q such that ___________. P M Q Congruent– exact same _________________ and ___________________ H _______ is the midpoint of GH G J GJ JH B Example #1) 4 ft A X 4 ft Y Ex #2) 16 Name _____________________________ Geometry Ex #3) Date _________ Period _________ What happens if you cut an angle in ½? E Get ________________________ This is called the _______________________ D F ___________________ – 2 congruent segments (midpoint) ___________________ – 2 congruent angles (bisecting ray must be in the interior of the angle) G DF bisect <EDG <EDF = <FDG Example #1) Point D is in the interior of <ABC. Find <ABC if <ABD <DBC, <ABD=11x-13, and <DBC=5x+23. Example #2) In the figure AM bisects <LAR. Find LAR if <MAR=2x+13 and <MAL=4x-3 R S A M L 17 Name _____________________________ Date _________ Geometry Period _________ Example #4) BS bisects <ABT and AB bisects <RBT. Find the value of x and the measure of the indicated angle. a.) <RBA = 2x + 13, <TBA = 4x – 3; <RBT A R S B T b.) <TBS = 25 - 2x, <SBA = 3x + 5; <TBS A R S B T Adjacent angles – common __________________, common _______________, no same interior points (next to) Vertical angles – 2 nonadjacent angles formed by intersecting lines (opposite) Vertical Angles: Adjacent Angles: R 1 4 Y T 2 3 P S Vertical angles are ___________________!!!! 18 Name _____________________________ Date _________ Geometry Period _________ Linear pair – two adjacent angles that form a _________________________________. R ___________________________________ are Y a linear pair. T Sum of the measures of the angles in a linear P S pair is ____________ Supplementary angles – sum of measures of 2 angles is _________ C 180 - x x A B D Ex: Two angles are supplementary. One angle is twice as big as the smaller angle. What is the measure of both angles? Complementary angles – sum of measures of 2 angles is ____________ F G x 90 - x E H Ex: An angle is 68˚ more than its complement. What is the angle’s measure? More Examples: G #1) GH and LM intersect at Q. Find the value of x and the measure of <LQH. L Q 3x + 22 6x - 8 H M 19 Name _____________________________ Geometry #2) If <NOP and <POT are supplementary, find m<POS and m<SOT. P x + 28 90 N Date _________ Period _________ S 6x - 15 O T #3) If <NPO and <NPM are complementary, find m<NPM and m<NPO. M N 26 + x x O P #4) In the figure, line MN and line OP intersect at D. Find the value of x and <ODN. M O 9x-4 P D 4x-11 N #5) If <BCE and <ECD are supplementary, find <ECF and <FCD. F E 63° B 11x+2 5x-13 c D 20