Summer Dear Geometry Students and Parents:

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Wendy Byrd
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1 Summer 2018 Dear Geometry Students and Parents: Welcome to Geometry! For the school year, we would like to focus your attention to the prerequisite skills and concepts for Geometry. In order to be successful in Geometry, a student must demonstrate a proficiency in: Solving Quadratic Equations (factoring and quadratic formula) Systems of Equations Radicals Geometric Notation and Definitions Segments ngles Parallel Lines Cut by a Transversal Polygons and Their Measures rea and Perimeter 3D Figures s prerequisite skills, these topics are not re-taught in the Geometry course. To ensure that all students demonstrate the basic skills to be successful, these topics will be assessed throughout the school year by different methods, such as weekly assignments and spiral review questions on assessments. Students are encouraged to seek etra help before or after school from their teacher for any topics requiring more personal in-depth remediation. It is epected that each student will fully complete the review questions. The teacher will check the packet for completion and effort for homework credit. If you have any questions, please do not hesitate to contact your child s teacher. Geometry Summer Review Packet Page 1

2 Quadratic Equations Standard Form of a Quadratic: Solving Quadratic Equations: Factoring Quadratic Formula Guided Practice: Solve by Factoring, where. Solve by Factoring: 2 set equal to zero, and in standard form, if not already done. look for a GCF and factor out look at the signs to see what the ( )( ) signs of our factors will be a = 1 c = 21 ac = 21 find the product of ac factors product sum -21, , Second sign positive, both first sign. Second sign negative, signs different. list the factors of ac (in this case, both negative) find the sum of the factors. choose the one that equals b 2( 7)( 3) = 0 put factors into parenthesis prime prime check to see if factors can be factored further. If so, factor again. ( 7) = 0 or ( 3) = 0 set each linear factor equal to zero = 7 or = 3 solve each linear factor Tips: If the a is negative, start by factoring out a 1 If there are no factors of c that add up to b, then the trinomial is prime. Remember: If c is negative, the factors will have different signs (one positive and one negative). If c is positive, then the factors will both have the same sign as b. The ac method is only one of many types of factoring (GCF, Grouping, Difference of Perfect Squares, Perfect Square Trinomial, Slide and Divide). Geometry Summer Review Packet Page 2

3 Guided Practice: Solve using the Quadratic Formula. Quadratic Formula: Solve using the Quadratic Formula: = 0 set equal to zero, and in standard form, if not already done. a = 3, b = 15, c =18 Identify a, b and c. Substitute into the quadratic formula Simplify (remember the order of operations!) or Split up the ± or Simplify each answer = -2 or = -3 Geometry Summer Review Packet Page 3

4 System of Equations Ways to Solve a System of Equation: Graphing Substitution Elimination Guided Practice: Solve a System using Elimination. Solve the system of equations: { { Line up the variables on one side and the constants on the other side. + eliminate Determine a variable you want to eliminate you want the coefficients to be the same number with opposite signs. If necessary, multiply one or both equations by a number to force the coefficients to be the same number with opposite signs. dd the equations together so that one variable cancels out. Solve the equation. Substitute the value of variable into one of the original equations to determine the value of the second variable multiply second equation by 2 (-3, -2) Write your answer as an ordered pair. Tips: You may have to multiply by a negative number to get the coefficients to have opposite signs. To solve by graphing, graph each equation and identify the point of intersection. To solve by substitution, solve one equation for one variable and substitute it into the other equation. Then solve for the other variable. Substitute that back into the first equation to find the remaining variable. Geometry Summer Review Packet Page 4

5 Radicals Tips: You will need to know all your perfect squares through. Radical symbol Guided Practice: Simplify Radicals Simplify: split the radicand into a factor pair, one of which is a perfect square put each factor under its own radical symbol find the square root of the perfect square factor simplify Guided Practice: dd/subtract Radicals Tips: In order to add or subtract radicals, the inde and the radicand must be the same! You combine radicals as if you were combining like-terms add/subtract their coefficients ONLY. Sometimes it may be necessary to simplify the radicals before you add or subtract. Simplify: Simplify each radical separately, if necessary (as above) Combine like radicals Geometry Summer Review Packet Page 5

6 Guided Practice: Multiplication of Radicals Tips: This is NOT the only way to simplify these problems Simplify: Simplify each radical, if possible 2 8 Group coefficients together, group radicands together 16 simplified Multiply coefficients and radicands Simplify radical, if possible Guided Practice: Division of Radicals Tips: This is NOT the only way to simplify these problems Simplify: Simplify each radical, if possible Divide numbers outside the radical 4 Divide numbers under the radical no radical remaining Simplify radical, if possible Geometry Summer Review Packet Page 6

7 The following set of notation and definitions will be used throughout the entire course. Notation Meaning Diagram or Line or Line or Line or Can t switch order!! or Has one dimension. Through any two points, there is eactly one line. Segment Consists of two endpoints and and all of the points on between and Ray Consists of one endpoint and all the points on that are on the same side as The length of segment (has no segment bar on top) Equal to * Geometric Notation and Definitions m. m. C in. D E in. F Equal CD EF Congruent CD EF Congruent (has the same measure) * Equal m m Congruent or ngle has a verte of The verte should be the middle letter or The measure of angle C C m C Degree(s), a unit measure for angles Perpendicular Two lines that intersect to form a right angle. Parallel Two coplanar lines that never intersect. They have the same slope. Triangle C Geometry Summer Review Packet Page 7

8 Other Definitions Point Point point has no dimension. It is represented by a dot. Plane Plane plane has two dimensions. It is represented by a shape that looks like a floor or a wall and etends without end. C M Plane Through any three points not on the same line, there is eactly one plane. Opposite Rays Collinear Points Coplanar Points djacent ngles You can use three points, not all on the same line, to name it. Sometimes, you can use a capital letter without a point, if it is provided. Two rays with a common end point that go in opposite directions. The first letter is the endpoint. Points that are on the same line. Points that are on the same plane. ngles that share a common verte and a common side that do not overlap each other. ngles must be coplanar. Plane C or Plane M C and C are opposite rays See diagram above for OPPOSITE RYS. are collinear points. See diagram for PLNE above. are coplanar points and angles. and angles. are adjacent are adjacent and are NOT adjacent angles. Geometry Summer Review Packet Page 8

9 Segments Congruent Segments - Line segments that have the same length. tick mark is used on each segment to show that they are congruent. C D Lengths are equal = CD Segments are congruent CD Segment ddition Postulate - If point is between and C, then + C = C. lso, if + C = C, then point is between point and C. C C + C = C = C 7 8 C 15 = C Midpoint - the point that divides a segment into two congruent segments. M M is the midpoint of. M M and M = M ngles Classifying angles: 1. cute: an angle with measure between and 2. Right: an angle with measure equal to 3. Obtuse: an angle with measure between and 4. Straight: an angle with measure equal to Naming angles: 1. Use 3 letters (middle letter is always the verte) 2. Use the verte letter (only when one angle is present) 3. Sometimes an angle can be named with a number This angle can be named as: C or C or or C Geometry Summer Review Packet Page 9

ngle ddition Postulate: The sum of two adjacent angle measures is equal to the measure of the larger angle. m C m D m DC m C m C 7 ngle isector: a ray that divides an angle into two congruent angles.

10 ngle ddition Postulate: The sum of two adjacent angle measures is equal to the measure of the larger angle. m C m D m DC m C m C 7 ngle isector: a ray that divides an angle into two congruent angles. (Their measures are equal) Given: FR bisects CFT therefore CFR RFT Complementary ngles: Two angles whose sum is. 1 2 djacent Nonadjacent Supplementary ngles: Two angles whose sum is. 1 2 djacent Nonadjacent Geometry Summer Review Packet Page 10

11 Linear Pair: two adjacent angles whose non-common sides are opposite rays. Linear pairs are supplementary. and are linear pairs m m Vertical ngles: two angles that are opposite each other when two lines intersect. Vertical angles are congruent. 3 and and are vertical angles and are vertical angles and PRLLEL LINES CUT Y TRNSVERSL Line m is parallel to Line n. Line m and Line n are cut by a transversal, Line t. line that intersects two or more lines is called a transversal. Corresponding angles are congruent. These angles lie on the same side of the transversal in corresponding positions. 1 & 5, 4 & 8, 2 & 6, 3 & 7 ngles on opposite sides of the transversal are called alternating. lternate Interior angles are congruent: 3 & 5, 4 & 6 lternate Eterior angles are congruent: 1 & 7, 2 & 8 Same Side Interior ngles are supplementary: 3 & 6, 4 & 5 The following angles are all congruent: 1, 3, 5, and 7 The following angles are all congruent: 2, 4, 6, and 8 Geometry Summer Review Packet Page 11

12 Polygons and their Measures Polygon- closed plane figure with the following properties: 1- It is formed by 3 or more line segments, called sides. (no curves) 2- Each side intersects eactly two sides, one at each verte, so that no two sides with a common endpoint are collinear. Naming a Polygon: name in a clockwise or counterclockwise fashion Conve all vertices point out Concave at least one verte points inward * polygon is concave if it is not conve. * Equilateral Polygon- polygon where all sides are equal or congruent. Equiangular Polygon- polygon where all interior angles are equal or congruent Regular Polygon- conve polygon that is both equilateral and equiangular. Diagonal (of a polygon)- segment that joins two non-adjacent vertices. Classifying Triangles by Sides Scalene Triangle Isosceles Triangle Equilateral Triangle 0 Congruent Sides t least 2 Congruent Sides 3 Congruent Sides Geometry Summer Review Packet Page 12

13 Classifying Triangles by ngles cute Triangle Right Triangle Obtuse Triangle Equiangular Triangle ll angles are less than. Eactly one angle is. One angle is greater than. ll angles are. Interior and Eterior ngles of a Polygon: Conve Polygon # Sides Sum of the Measures of Interior ngles Each Interior ngle of a Regular Polygon Sum of the Measures of Eterior ngles Each Eterior ngle of a Regular Polygon n-gon n Geometry Summer Review Packet Page 13

rea and Perimeter Terms to know: rea: how much is contained within a

contained shape o for a circle, called circumference ase and Height: one must be

they must be perpendicular to each other rea of a square (b h), rectangle (b h),

Circumference of a circle (2πr) Pythagorean Theorem: the relationship between

14 rea and Perimeter Terms to know: rea: how much is contained within a 2-dimensional, contained shape Perimeter: the distance around a 2-dimensional, contained shape o for a circle, called circumference ase and Height: one must be a side length. they must be perpendicular to each other rea of a square (b h), rectangle (b h), triangle( ½b h), circle (πr 2 ) Perimeter of polygons (sum of all the sides) Circumference of a circle (2πr) Pythagorean Theorem: the relationship between the sides of a right triangle o a 2 + b 2 = c 2, where c is the hypotenuse 3-D Figures Surface rea and Volume Formulas OR OR Geometry Summer Review Packet Page 14

15 Practice Problems Solving Quadratics For 1-4 solve by factoring or the quadratic formula. 1. = System of Equations For 5-6 solve the following system of equations. 5. { 7 6. { Geometry Summer Review Packet Page 15

Radicals For 7-12 simplify. Epress your answer in simplified radical form. 7. 8. 9. 10. 11. 12. Points, Lines and Planes 13.. C D are collinear True False. and F are collinear..true False C.

16 Radicals For 7-12 simplify. Epress your answer in simplified radical form Points, Lines and Planes 13.. C D are collinear True False. and F are collinear..true False C., C, and F are coplanar. True False D. I can form eactly one plane with points C D and E True False E. Plane N and Plane CF intersect at line l..true False F. Give another name for G. Give another name for line l H. Give another name for I. Give another name for Plane J. Given that is the midpoint of, what can you conclude? K. Name a pair of opposite rays. Geometry Summer Review Packet Page 16

17 Segments 14. Solve for : 15. Solve for : 27 F G 9 H F G H Point M is the midpoint of. Solve for V M W ngles For #17-20, use the diagram to answer the following questions: 17. Name 3 different angles. D 18. The, find, then classify the angle. C 19. If, find and if. 20. If, find if and. Geometry Summer Review Packet Page 17

18 For #21-28, label each angle pair as vertical angles, linear pairs, or neither and and and and and and and and 8 For solve for Geometry Summer Review Packet Page 18

19 34. Find the measure of two supplementary angles if the measure of the larger angle is more than the measure of the smaller angle. 35. Find the measure of two complementary angles if the measure of one angle is 3 more than the measure of its complement. PRLLEL LINES CUT Y TRNSVERSL For #36-37, solve for the missing variables y y 62 Geometry Summer Review Packet Page 19

20 For 38-40, classify the triangle based on its sides and angles Polygons and their Measures Tell whether or not the figure is a polygon and whether it is conve or concave. a) b) c) d) 42. Name the polygon according to the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. a) c) b) d) Geometry Summer Review Packet Page 20

21 43. Given a conve regular 18-gon, find: a.) the sum of the measures of the interior angles. b.) each interior angle. c.) the sum of the measures of the eterior angles. d.) each eterior angle. 44. Given a conve regular 30-gon, find: a.) the sum of the measures of the interior angles. b.) each interior angle. c.) the sum of the measures of the eterior angles. d.) each eterior angle. 45. Find the value of in the diagram a.) b.) D C The sum of the measures of the interior angles of a conve polygon is. Determine the number of sides of the polygon and name it. Geometry Summer Review Packet Page 21

22 47. Each interior angle of a regular polygon is. Determine the number of sides of the polygon and name it. 48. How many sides will a regular polygon have if each of its eterior angles is 12? 49. Find the value of. Geometry Summer Review Packet Page 22

23 Perimeter and rea For find the area ND perimeter/circumference of each shape: 23m in 16m 8in cm 18mi 12 cm 54. square with side length 17km. 55. circle with radius 3ft 56. Right triangle with legs 2m and 6m. Geometry Summer Review Packet Page 23

Identify the number of faces, edges, and vertices in the rectangular prism. 58.

24 Volume and Surface rea For use the rectangular prism given below. 57. Identify the number of faces, edges, and vertices in the rectangular prism. 58. Find the surface area of the rectangular prism. 59. Find the volume of the cylinder. Write the eact answer with and the estimated answer rounded to the nearest tenth. Geometry Summer Review Packet Page 24

Moore Catholic High School Math Department

Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during