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Name: Geometry Unit 1: Essentials of Geometry Extra Practice Worksheets (with answer keys) are available at popkoskimath.pbworks.com These are optional and are not counted as homework. You are encouraged to complete these on your own and/or after school to practice your skills and prepare for tests. 1-2: Points, Lines, and Planes 1-3: Measuring Segments TERM Point DIAGRAM DEFINITION Indicates a location. Named w/ capital letter. Line A set of continuous points that extends endlessly in both directions. Named with a lower case letter or by two points. Plane A flat surface with no thickness. Extends indefinitely in all directions. Formed by three noncollinear points. Could also be named with a capital letter. Line Segment Part of a line consisting of two points. Named with its two endpoints. Ray Part of a line consisting of an endpoint and the set of points on other side. Named by endpoint then another point on line. Opposite Rays Rays that have the same endpoint and lie on the same line. Collinear Points Points that are on the same line. Coplanar Points Points that are on the same plane. Angle Union of two rays that have the same endpoint (vertex). Named with three points, the vertex, or label interior with a number. Acute Angle An angle whose measure is greater than 0 and less than 90 Right Angle An angle whose measure equal to 90 1 Obtuse Angle An angle whose measure is greater than 90 and less than 180 Reflex Angle An angle whose measure is greater than 180 and less than 360 Straight Angle An angle whose measure is equal to 180 Parallel Lines Lines that do not intersect and have equal slopes Skew Lines Lines that are in different planes that do not intersect, but are NOT parallel. Perpendicular Lines Lines that intersect forming right angles and have negative reciprocals for slopes Linear Pair When the exterior sides of adjacent angles lie on opposite rays. Midpoint Divides a segment into two congruent segments. Bisector (Angles/Segments) Divides an angle or segment into two congruent angles or segments. Segment/Angle Addition Postulate If point X is between points A and B, then AX+XB = AB Congruent -Shapes: same size and shape -Segments: same length -Angle: same measure -A figure with no dimension cannot be measured (no length) -A one-dimensional figure has a length. -A two-dimensional figure has a length and a width -A three-dimensional figure has a length, width, and height Dimension If OX is between the sides of <AOB, then mAOX+XOB = mAOB 2 Intersection of two lines 1 point Intersection of two planes 1 line Intersection of line and plane 1 point Practice: Using the figure at right, answer the following questions. For #1 – 8, write TRUE or FALSE 1. ____________ Points A, D, and E are coplanar. 2. ____________ Points E, H, and B are coplanar. 3. ____________ Points B, C, F, and E are coplanar. 4. ____________ The intersection of plane ADB and plane CHG is CG . 5. ____________ AE&BF are coplanar. 6. ____________ EF&DC are coplanar. 7. ____________ AB& GF are coplanar. 8. ____________ GH & point B are coplanar. 9. Shade the plane that contains points B, C, and H. 10. Darken the intersection of planes ABF and ADC. Practice: Practice: If EG = 59, what are EF and FG? Q is the midpoint of PR . What are PQ, QR and PR? 3 Postulate: A statement of fact. Important Postulates and Theorems To Remember! Use the diagram below to answer the following. Give two other names for line PQ: Give two other names plane R: Name three points that are collinear: Name four points that are coplanar: Give two other names for line ST: Name a point that is not coplanar with points Q, S, and T: Draw a sketch of the given question then solve. 1. Line RS bisects PQ at point R. Find RQ if PQ = 14 cm. 2. Point T is a midpoint of UV. Find UV if UT = 4 ½ yards. In the diagram, M is the midpoint of the segment. Find the indicated length. 3. Find LN 4. Find MR 5. Name two planes that intersect in the given line: AB HW 1-2: p. 16 #15, 18, 25, 32 – 36, 40 – 45 HW 1-3: p. 24 #20, 40, 43 4 1-4: Measuring Angles 1-5: Exploring Angle Pairs Angles: Name the angle at the right. ______________ ______________ ______________ Note: You may also see this referred to as A, but we will NOT be using this label in proofs in this class! Angle Measures: 1. Classify each of the following angles according to their measures 2. Write an equation/inequality to express the possible measures ________________________ ________________________ ________________________ ________________________ How To Mark Congruent Angles Given: 1 3 and 2 4 Practice: If mRQT = 155, what are mRQS and mTQS? 5 Angle Pairs Definition Example Adjacent Angles: Vertical Angles: Complimentary Angles: Supplementary Angles: Practice: Practice: KPL and JPL are a linear pair. What conclusions can be made from the diagram? What are the measures of KPL and JPL? 6 Important Postulates and Theorems To Remember! VERTICAL ANGLES ARE CONGRUENT 1. DEF is a straight angle. What is the mDEC and mCEF? 2. Are BFD and CFD adjacent angles? Explain. 3. Are AFB and EFD are vertical angles? Explain. Additional Examples 1) Name the three angles in the diagram. Why can’t you just call this angle B??????? _______ , or _______ _______ , or _______ _______ , or _______ 7 2) Name the three angles in the diagram. Why can’t you just call this angle G??????? _______ , or _______ _______ , or _______ _______ , or _______ 3) What type of angles do the x and y axis form in the coordinate plane? _________________ 4) Given that m GFJ = 155o, find a) m GFH __________ b) m HFJ __________ 5) Given that VRS is a right angle, find a) m VRT __________ b) m TRS __________ 6) Identify all pairs of congruent angles in the diagram. If m P = 120o, what is m N? _______ _______ _______ _______ m N = ______o 7) WY bisects XWZ, and m XWY = 29o. Find m XWZ _________o 8 8) Identify all pairs of congruent angles in the diagram. If m B = 135o, what is m A? _______ _______ _______ _______ m A = ______o 9) KM bisects LKN and m LKM = 78o. Find m LKN _________o 10) Two angles form a linear pair. The measure of one angle is 4 times the measure of the other. Find the measure of each angle. 11) The exterior sides of two adjacent angles are opposite rays. The measure of one angle is 3 times the measure of the other. Find the measure of each angle. 12) In the diagram: a) All points shown are _________________. b) Points A, B, and C are ________________. c) DBE and EBC are ______________ angles. d) ABC is a ____________ angle. 9 13) Find m<EGT and m<TGC if EG is perpendicular to CG, and m<EGT = 7x + 2 and m<TGC = 4x. 14) Find the value of each angle. HW 1-4: p. 31 – 32 #18 – 23, 30 HW 1-5: p. 38 – 39 #25 – 28, 32 – 36 Extra Practice: Angle Pair Relationship Worksheet 10 1-6 (Day 1): Basic Constructions Refer to my website popkoskimath.pbworks.com for videos and printable instructions! Constructions: _________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ 1. Congruent Segments 11 2. Congruent Angles Proof: This construction works by creating two congruent triangles. The angle to be copied has the same measure in both triangles (Side-Side-Side Triangle Congruence Theorem). 12 3. Perpendicular Bisector Proof: This construction works by effectively building 4 congruent triangles that result in right angles being formed at the midpoint of the line segment. 13 4. Angle Bisector Proof: This construction works by effectively building two congruent triangles. The angle has the same measure in both triangles (Side-Side-Side Triangle Congruence Theorem). 14 1-6 (Day 2): Applied Constructions 1. a) Construct an equilateral triangle whose sides are all the same length as XY . b) Using this equilateral triangle, construct a 30 angle. 2. Construct AB so that AB = MN + OP 15 3. Construct KL so that KL = OP MN. 4. Construct A so that mA = m1 + m2 16 5. Construct B so that mB = m1 - m2 6. Construct C so that mC = 2m2. 17 7. Construct a segment whose length is 14 AB. HW 1-6: Finish 1-6 Class Notes Extra Practice: Constructions WS (Angle Bisector) Constructions WS (Perpendicular Bisector) 18 1-7 (Day 1): Midpoint and Distance in the Coordinate Plane Midpoint of a Line Segment: Midpoint Formula: Find the coordinates of the midpoint of AB . Practice: Practice: EF has endpoints E(7, 5) and F(2, -4). What are the coordinates of the midpoint of EF ? The midpoint of CD is (-2, 1). One endpoint is C(-5, 7). What are the coordinates of D? 19 Length of a Line Segment (Distance): Distance Formula: Find the length of AB . Practice: What is the distance between U(-5, 4) and V(3, 2)? Express your answer in simplest radical form. Practice: In circle O, a diameter has an endpoints at (-5, 4) and (3, -6). What is the length of the diameter? 20 Find the coordinates of the midpoint of the segment with the given endpoints. 1. L(4, 2) and P(0, 2) 2. G(-2, -8) and H( -3, -12) Using the given point R and the midpoint M, find the other endpoint. 3. R(6, 0), M(0, 2) 4. R( 3, 4), M(3, -2) Horizontal & Vertical Lines Find the length of line segment GF: _________ Find the length of line segment HJ: _________ So, GF _____ HJ Find the length of the line segment. Round to the nearest tenth of a unit. If there are two segments compare the lengths. 5. 6. RS: R(5, 4), S(0, 4) TU: T(-4, -3), U(-1, 1) HW 1-7 (Day 1): p. 54 – 55 #10, 13, 16, 19, 22, 25, 28, 51 Extra Practice: Midpoint WS 1, Midpoint WS 2, Midpoint WS 3 Distance WS 1, Distance WS 2 21 1-7 (Days 2&3): Partitioning a Line Segment 1. AB is a directed line segment from A(1,3) to B(8,3). What are coordinates of point P that partitions the segment in the ratio 1:2? 2. AB is a directed line segment from A(7,9) to B(1,3). What are coordinates of point P that partitions the segment in the ratio 1:3? 22 3. AB is a directed line segment from A(0, 1) to B(8,3). What are coordinates of point P that partitions the segment in the ratio 1:3? 4. The endpoints of XY are X (2, -6) and Y (-6, 2). What are the coordinates of point P on XY such that XY is ¾ of the distance from X to Y? 23 *5. AB is a directed line segment from A(2, 2) to B(4,7). What are coordinates of point P that partitions the segment in the ratio 1:2? Partition Point Formula: The section formula is just a fancier version of the midpoint formula. If a line segment has endpoints (x1, y1) and (x2, y2), and a partition point P will separate the line segment into a ratio of m:n, then students should plug the numbers into the section formula to find the coordinates of P. Watch for starting and ending points! x1 and y1 are where you begin…x2 and y2 are where you end! m= n= x1 = y1 = x2 = y2 = 24 Partition Point Formula Practice: 1. Find L on JK such that JL: LK = 4:1, when J = (-3, 5) and K = (1, -10). m= n= x1 = y1 = x2 = y2 = 2. Find R on MP such that MR:RP = 5:2, when M = (-10, 8) and P = (-2, 11). m= n= x1 = y1 = x2 = y2 = 3. Point C lies on directed line segment from A(5, 16) to B (-1, 2) and partitions the segment into a ratio of 1 to 2. What are the coordinates of C? m= n= x1 = y1 = x2 = y2 = 25 More Practice: 1. Points A(-2, -3) and B(8, 2) are the endpoints of AB. What are the coordinates of point C on AB such that AC is 2/5 the length of AB? 2. LM is the directed line segment from L (-4, 1) to M (5, -5). What are the coordinates of the point that partitions the segment in the ratio of 2 to 3? 26 3. Points X (6, 4) and Y(-4, -16) are the endpoints of XY. What are the coordinates of point Z on XY such that XZ is 4/5 the length of XY? m= n= x1 = y1 = x2 = y2 = *4. Points X (6, 4) and Y(-4, -16) are the endpoints of YX. What are the coordinates of point Z on YX such that YZ is 4/5 the length of YX? m= n= x1 = y1 = x2 = y2 = HW 1-7 (Days 2&3): Midpoint Practice WS/Finish 1-7 Class Notes Extra Practice: Directed Line Segment WS 27 1-8: Perimeter, Circumference and Area (With Review of Polygons) Classifying Polygons Polygon __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ Convex Polygon ____________________________________________________________________________________ Concave Polygon ____________________________________________________________________________________ n-gon _____________________________________________________________________________________ Equilateral Polygon _________________________________________________________________________ Equiangular Polygon ________________________________________________________________________ Regular Polygon____________________________________________________________________________ Name Sides Names of Polygons Name Sides 3 11 4 12 5 n 6 7 8 9 10 28 1. Tell whether the figure is a polygon and whether it is convex or concave. a) b) c) 2. The head of a bolt is shaped like a regular hexagon. The expressions shown represent side lengths of the hexagonal bolt. Find the length of a side. 3. The expressions (4x + 8)o and (5x – 5)o represent the measures of two congruent angles . Find the measure of an angle. 29 1. Find the dimensions of the garden, path. 2. What is the circumference and area of the including the circle in terms of ? What is the circumference and area of the circle to the nearest tenth? 3. What is the area and perimeter of EFG? 4. Graph quadrilateral JKLM with vertices J(-3, -3), K(1, -3), L(1, 4) and M(-3,1). What is the perimeter of JKLM? 5. You are designing a poster that will be 3 yds wide and 8 ft high. How much paper do you need to make the poster? Leave your answer in terms of feet. 30 6. What is the area of the figure at the right? All angles are right angles. 7. Triangle JKL has vertices J(1, 6), K(6, 6) and L(3, 2). Find the approximate perimeter and area of triangle JKL. Round all answers to the nearest tenth. 8. The base of a triangle is 24 feet. Its area is 216 square feet. Find the height of the triangle. HW 1-8: p. 64-65 #14, 22, 32-33, 41-42 31 Unit 1 Common Core Test Questions 1. 2. 32 3. 4. 5. 33 Unit 1 Common Core Test Questions Answer Key 1. 2. 3. 4. 5. 34