Ch 1 Note Sheet L2 Key.doc 1.1 Building Blocks of Geometry

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1 1.1 uilding locks of Geometry Read page 28. It s all about vocabulary and notation! To name something, trace the figure as you say the name, if you trace the figure you were trying to describe you re correct! Points, lines, and planes are the building blocks of geometry. They are the undefined terms of geometry. Term Size? Is a How to write/name? How to model? Point None location capital letter. point P or just P dot with a capital letter. P Line has infinite length no width straight, continuous arrangement of infinitely many points. Extends forever in two directions. ny two points on the line with a line symbol drawn on top:,,,, or or a lower case letter: line j names the entire line. Straight line with arrows at the ends. j Plane length and width no thickness flat surface that extends infinitely along its length and width. script capital letter plane P or just P. or any three points in the plane: plane or plane names the entire plane. Many possible names here! 4-sided figure, usually tilted for perspective P definition is a statement that explains the meaning of a word or phrase. Using these three undefined terms, you can define all other geometric figures and terms. S. Stirling Spring 2014 Page 1 of 18

2 lassifying Points Read bottom page 29. Definition Illustration Example/nonexample collinear points on the same line coplanar points on the same plane F P E T P D S N F, E, and D are collinear. F, D and P are NOT collinear. D and P are collinear, since they can be contained by one line. (The line is not drawn.) Through any two points there is exactly one line. P, N, and T are coplanar. P, N, T and S are NOT coplanar. P, T and S are coplanar, since they can be contained by one plane. (The plane is not drawn.) Through any three points there is exactly one plane. Parts of Lines Read page 30 to the middle of page 32. Definition How to model? How to write/name? Ray part of a line that starts at one point, called the endpoint, and extends infinitely through another point. Line segment (or just segment) part of a line that consists of two points called endpoints and all of the collinear points between them. Length [of a segment] 2 in Y R Note: the arrow at one end. The endpoint 1st and then any other point that the ray goes through. Put the ray symbol above it. R or RY NOT R The ray symbol does NOT indicate the direction of the ray in the drawing!!! The endpoints with the segment symbol above it. and are the same segment. Two ways to write: 1. The distance from to is two inches. = 2 in refers to numbers Note: the measure is written somewhere between the two endpoints near the segment. Midpoint [of a segment] point on a segment that is the same distance from both endpoints. point that divides the segment into two congruent segments. 2. The measure of the segment is two inches, m is measure of. 2 in m = refers to the figures M Note: the tic marks. M is the midpoint of. so mm mm or M = M S. Stirling Spring 2014 Page 2 of 18

3 Two segments with equal measures, or lengths, are said to be congruent. The symbol for congruence is. The equals symbol, =, is used between equal numbers or measures. The congruence symbol,, is used between congruent figures. In geometric drawings, congruent segments are marked with identical symbols. ongruent segments, when the exact measures are not known, are marked by identical tic marks. Never assume that two segments are congruent!! They must be marked. isect means to cut into two equal parts. fter studying the EXMPLE on page 31, complete the example below: Study the diagrams. a. Name each midpoint and the segment it bisects. b. Name all the congruent segments. Use the congruence symbol and/or the equal symbol to write your answers. P is the midpoint of and D. P bisects and D. mp mp P = PD Q is the midpoint of GH. Q bisects GH. mgq mqh No midpoints or bisectors! or m md = D Space is the collection of all points. (3-D geometry K the real world!) You may start the homework Exercises. heck your syllabus for the exercise numbers. S. Stirling Spring 2014 Page 3 of 18

4 1.2 Poolroom Math (aka ngles) Read pages 38 through definition of congruent on page 40. Study both Example and Example! ngles Definition How to model? How to write/name? ngle n angle is a figure formed by two rays with a common endpoint, provided the two rays do not lie on the same line. The two rays are the sides of the angle, and the common endpoint is the vertex. N 2 Note: the arrow at both ends. Indicates that the sides of the angles are rays (there is no length). L G Use the angle symbol followed by the vertex (if there is no confusion) or by a point on one side, followed by the vertex, followed by a point on the other side. Or by a number written inside the angle near the vertex. N, NG, GN, LN, NL or 2 NOT NG or LN vertex N; sides N, NL Measure [of an angle] is the smallest amount of rotation about the vertex from one ray to the other. ngles are usually measured in degrees. You use a protractor to measure an angle. The reflex measure of an angle is the largest amount of rotation less than 360 between the two rays. Note: the degree measure is written inside the angle near the vertex. Sometimes with an arc. 316 N N 44 Note: The arc is needed to indicate the reflex measure. G G Use m is measure of. m N = 44 The reflex measure of NG is 316º. Since = 316º Using a protractor: 1. Place the center on the vertex of the angle. 2. Line up the 0-mark with one side. 3. Read the measure on the protractor. (Make sure you are going from 0º!) S. Stirling Spring 2014 Page 4 of 18

5 Read pages 40 through Example on page 41. Two angles are congruent if and only if they have equal measures. gain The equals symbol, =, is used between equal numbers or measures, m = m. The congruence symbol,, is used between congruent figures,. In geometric drawings, congruent angles are marked with identical symbols. ongruent angles, when the exact measures are not known, are marked by arcs with identical tic marks through them. Never assume that two angles are congruent!! Remember that bisect means to cut into two equal parts, so Definition How to model? How to write/name? ngle bisector ray that contains the FH bisects vertex of the angle and divides the angle into two congruent angles. : OM does not bisect fter studying the EXMPLE on page 41. Note: the arcs with the identical tic marks. LON because m LOM m MON EFG so EFH HFG or m EFH = m HFG dditional EXMPLE Look for angle bisectors and congruent angles in the diagrams. a. Name each angle bisector and the angle it bisects. b. Name all the congruent angles. Use the congruence symbol and/or the equal symbol to write your answers. Use the drawings above. OM does not bisect LON because m LOM m MON UR bisects QUS because m QUR = m RUS US bisects RUT because TUS RUS Definition Example: Non-examples: djacent angles LOM is adjacent to QUR Two angles that share a vertex and a side, and they MON. SUT share no interior points. adjacent kinda means next to RUS is adjacent to SUT. QUS is adjacent to SUT. is NOT adjacent to (no shared side) RUS is NOT adjacent to RUT (they overlap) S. Stirling Spring 2014 Page 5 of 18

6 Sum of the Parts equals the Whole properties. Segment ddition If points, and are collinear and is between and, then + =. ngle ddition If point D is in the interior of, then m D + m D = m. D + = + because is not between and. Optional Investigation Virtual Pool pages Know the property on page 41. Summary: The incoming angle equals the outgoing angle. S. Stirling Spring 2014 Page 6 of 18

7 1.3 What s a Widget? (aka how to start to write good geometric definitions) Read pages 47 and 48. Make sure to study both Example and Example! Writing Good Definitions 1. lassify your term. What is it? 2. Differentiate your term. How does it differ from others in that class? 3. Test your definition. Look for a counterexample (an example to show that your definition doesn t work). Types of Lines (two or more lines) Definition How to model? How to write/name? Parallel lines Lines in the same plane that never intersect. Perpendicular lines Lines that intersect at a 90º angle. Skew lines Lines that are not coplanar and never intersect. n Note: identical arrow marks on the lines indicate that the lines are parallel to eachother. q m Note: the little square is used to indicate 90º angles and for perpendicular lines. t D R O v Note: must indicate that they are not in the same plane. p T n m or D read is parallel to lso parts D p q or OR TO read is perpendicular to lso parts RO OT line t and line v are skew. (no symbol) Do Investigation: Defining ngles pages 49 and 50. Make sure to study Example page 51. lassifying Individual ngles Definition cute angle n angle that measures less than 90º (more than 0º). Right angle n angle that measures exactly 90º. Obtuse angle n angle that measures more than 90º and less than 180º. Straight angle n angle that measures exactly 180º. Example: N G N N G N G S. Stirling Spring 2014 Page 7 of 18

8 lassifying Pairs of ngles (two angles) Definition Example: Non-example(s): Nonadjacent angles 1 and 2 are djacent angles complements. are not complements. omplementary angles ny two angles that add to 90º. and D They do not need to be adjacent! 2 1 D Supplementary ngles ny two angles that add to 180º. They do not need to be adjacent! m 1+ m 2 = 90 djacent angles 3 and 4 are supplements. 3 4 m + m D = 60 djacent angles and D are not supplementary. D Vertical ngles Two non-adjacent angles formed by two intersecting lines. Linear Pair of ngles Two adjacent angles formed by a line and a ray. m 3 + m 4 = 180 Vertical angles E and ED. m E = m ED 5 and 6 are a linear pair. 5 E 6 D m + m D = 150 Non-vertical angles 3 and djacent angles 7 and 8 are not a linear pair. 7 8 m 5 + m 6 = 180 m 7 + m 8 = 160 S. Stirling Spring 2014 Page 8 of 18

9 1.4 Polygons Read page 54 and 55. Make sure to study the diagrams and the Examples! onsecutive means one right after the other, in order. orresponding means the figures match up. Definition How to classify? How to write/name? Polygon is a closed figure in a plane, formed by connecting line segments endpoint to endpoint with each segment intersecting exactly 2 others. Each segment is a side. Each endpoint is a vertex. Diagonal [of a polygon] line segment that connects two nonconsecutive vertices. Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Undecagon 12 Dodecagon n n-gon Only 2 diagonals: QS, PR Name by the type followed by consecutive vertices. Quadrilateral RSPQ, RQPS, or PQRS NOT PQSR (trace it!) sides: PQ, QR, RS, SP vertices: P, Q, R, S onvex [polygon] polygon in which no diagonal is outside the polygon. oncave [polygon] polygon that has at least one diagonal outside the polygon. Definition How to model? How to write/name? ongruent polygons have the same shape (the corresponding angles are congruent) and have the same size (the corresponding sides are congruent). Note: ll 6 corresponding parts are marked as equal. orresponding vertices must match: EFG or GEF Need all 6 parts congruent to get congruent triangles: orresponding angles congruent: E, F, G orresponding sides congruent: EF, FG, GE S. Stirling Spring 2014 Page 9 of 18

10 EXMPLE Which polygon is congruent to TUVW? D or D The perimeter of a polygon equals the sum of the lengths of its sides. The perimeter = = 28 cm Do Investigation: Special Polygons page 56. Special Polygons Definition Example: Non-example(s): n equilateral quadrilateral (that is not equiangular). n equilateral polygon has all if its sides equal in length. n equiangular quadrilateral (that is not equilateral). n equiangular polygon has all if its angles equal in measure. n equiangular hexagon (that is not equilateral). n equilateral hexagon (that is not equiangular). regular polygon is both equilateral and equiangular. Regular pentagon. Not regular pentagon. S. Stirling Spring 2014 Page 10 of 18

11 1.5 Triangles Read page 59 to top of 60. Make sure to study the things you may assume section and the Example! Do Investigation: Triangles pages 60 and 61. lassifying Triangles (types of triangles) lassifying Triangles by ngles right triangle is a triangle with one right angle. n acute triangle is a triangle with three acute angles. n obtuse triangle is a triangle with only one obtuse angle. n equiangular triangle is a triangle with all angles congruent. lassifying Triangles by Sides scalene triangle is a triangle with no congruent sides. n isosceles triangle is a triangle with at least two sides congruent. n equilateral triangle is a triangle with all sides congruent. Triangles can also be classified by both angles and sides. dditional EXMPLE Tell whether the triangle described is possible or not possible. If the triangle is possible, sketch an example on your paper. 1. Obtuse isosceles triangle 2. cute right triangle NP 3. Obtuse equiangular triangle NP 4. Isosceles right triangle 5. cute scalene triangle 6. cute equilateral triangle S. Stirling Spring 2014 Page 11 of 18

12 Terms used to describe parts of Polygons. opposite means across from is opposite TR R is opposite T adjacent means next to is adjacent to T is adjacent to R R is adjacent to T between parts sandwiched between two other parts is between T and R, R is between and R R Special Vocabulary for Special Triangles Right triangle triangle with one right angle. Hypotenuse The side opposite the right angle. The longest side of the right triangle. R leg leg G hyp otenuse hypotenuse: GT legs: GR and RT Leg(s) of a right triangle The sides that form the right angle. T Isosceles triangle triangle with at least two sides congruent. Leg(s) of an isosceles triangle The two sides of equal length. Vertex ngle The angle between the legs. ase The side opposite the vertex angle. ase ngles The two angles opposite the two sides of equal length. The angles adjacent to the base. S leg base O leg I legs: SO and OI base: SI vertex angle: O base angles: S and I S. Stirling Spring 2014 Page 12 of 18

13 1.6 Special Quadrilaterals Read page 64. Do Investigation: Special Quadrilaterals pages 64 and 65. Quadrilateral four-sided polygon. Kite quad. with exactly two distinct pairs of consecutive congruent sides. Parallelogram quad with [both pairs] opposite sides parallel. Trapezoid quad with exactly one pair of parallel sides. Rhombus n equilateral parallelogram. 4 sides congruent. Rectangle n equiangular parallelogram. 4 angles congruent. Square n equiangular rhombus or an equilateral rectangle. S. Stirling Spring 2014 Page 13 of 18

14 dditional EXMPLEs 1. Look carefully at the quadrilaterals. lassify each figure as a trapezoid, kite, rhombus, rectangle, or square. Explain your thinking. Quadrilateral D is a kite because two pairs of adjacent sides are congruent. Quadrilateral EFGH is a trapezoid because it has only one pair of parallel sides. Quadrilateral IJKL is a rhombus because it is a parallelogram and all of its sides have equal length. Quadrilateral MNOP is a rectangle because it is a parallelogram and all of its angles have equal measures. In Questions 2 8, complete the sentence with always, sometimes, or never. Explain. 2. parallelogram is a square. sometimes 3. rectangle is a rhombus. sometimes 4. square is a rhombus. always 5. rectangle is a parallelogram. always 6. parallelogram that is not a rectangle is a square. never 7. square is a rectangle. always 8. This is a concept map showing the relationships among the different types of parallelograms. This type of concept map is known as a Venn diagram. Fill in the missing names. S. Stirling Spring 2014 Page 14 of 18

15 1.7 ircles Read page 69. Do Investigation: Defining ircle Terms page 70 and 71 or notes. ircles and Segments Term ircle is a set of points, a given distance from a given point, called the center. How to write? Name it using the circle symbol with the center point. P R Radius [of a circle] is segment that goes from the center to any point on the circle. the distance from the center to any point on the circle. Name all radii: P, RP and P s a measure: r = 6 cm or P = 6 cm D P E Term hord is a segment connecting any two points on the circle. How to write? Name all chords: and D Diameter [of a circle] is chord that goes through the center of a circle. the length of the diameter. d = 2r or ½ d = r. Name all diameters: If PR = 5 cm, = 10 cm. If d = 15 in., r = 7.5 in. Tangent [to a circle] is a line that intersects a circle in only one point. The point of intersection is called the point of tangency. Name all tangents: E with point of tangency,. Note: the word tangent can also be used in reference to circles. Two circles can be tangent to each other if they intersect in only one point. S. Stirling Spring 2014 Page 15 of 18

16 Parts of ircles Term rc [of a circle] is formed by two points on a circle and a continuous part of the circle between them. The two points are called endpoints. How to write? R rc R with endpoints and R. R P Types of rcs Semicircle is an arc whose endpoints are the endpoints of the diameter. Minor arc is an arc that is smaller than a semicircle. Major arc is an arc that is larger than a semicircle. Name all semicircles: R, D Must use 3 letters! Endpoints at ends. Name all minor arcs:, R, R, RD Must use 2 letters! Name all major arcs: D, RD, R Must use 3 letters! D Measuring rcs entral ngle is an whose vertex is the center of a circle. Measure of an arc is equal to the measure of its central angle. m RP = 60, so m = 180 m PR = 120, so If mr = 60 mr = 120 md = 110, then m PD = 70 ongruent circles are two or more circles with the same radius measure. oncentric circles are two or more circles with the same center point. S. Stirling Spring 2014 Page 16 of 18

17 1.8 Space Geometry Read page 75 to middle of 77. Draw each of the following 3-D figures. Prism Sketch a pentagonal prism. Pyramid Sketch a hexagonal pyramid. Sphere ylinder prism with circular bases. one pyramid with circular bases. Hemisphere Do Investigation: Space Geometry page 70. See hints below. Pay close attention to the phrase exactly one!! Investigation: Space Geometry In this investigation you need to decide whether statements about geometric objects are true or false. You can make sketches or use physical objects to help you visualize each statement. For example, you might use a sheet of paper to represent a plane and a pencil to represent a line. In each case, try to find a counterexample to the statement. If you find one, the statement must be false. If a statement is false, draw a picture and explain why it is false. elow are some suggestions for visualizing the situations described in the statements. Try to determine whether each statement is true or false on your own before you read the suggestion. 1. For any two points, there is exactly one line that can be drawn through them. True Draw two points on a sheet of paper and draw a line through them. Is there a way to draw another straight line through the points? Remember that you are not limited to the surface of the paper. S. Stirling Spring 2014 Page 17 of 18

18 2. For any line and a point not on the line, there is exactly one plane that contains them. True Draw a dot on a sheet of paper to represent the point, and use a pencil to represent the line. Hold the pencil above the paper and imagine a plane passing through both the point and the line. Without moving the point or the line, try to imagine a different plane passing through them. an you do it? hange the position of the pencil and the paper so that they represent a different point and line. an you imagine more than one plane passing through them? Experiment until you think you know whether the statement is true or false. 3. For any two lines, there is exactly one plane that contains them. False Skew lines. There are three situations that you must consider: intersecting lines, parallel lines, and skew lines. First, look at the intersecting lines. They are drawn on a sheet of paper, which can represent a plane containing the lines. Try to imagine a different plane that also contains both lines. an you do it? Next, study the parallel lines contained in the plane of the sheet of paper. an a different plane contain both parallel lines? Finally, look at the third pair of lines, which are skew lines, or lines that are not parallel and do not intersect. an you imagine a sheet of paper that will contain these lines? 4. If two coplanar lines are both perpendicular to a third line in the same plane, then the two lines are parallel. True Notice that all the lines mentioned in this statement are in the same plane. You can use a sheet of paper to represent the plane. On the paper, draw a line and then draw two lines that are each perpendicular to the line. re the two lines parallel? Make more sketches if you need to. 5. If two planes do not intersect, then they are parallel. True Use two sheets of paper or cardboard to represent the planes. You ll need to picture the sheets extending forever. an you arrange the planes so that they will never intersect but so they are not parallel? 6. If two lines do not intersect, then they are parallel. False Skew lines. You know that if lines in the same plane do not intersect, then they must be parallel. ut what if the lines are in different planes? You can use two pencils to represent two lines. See if you can position the lines so that they will not intersect and are not parallel. 7. If a line is perpendicular to two lines in a plane, but the line is not contained in the plane, then the line is perpendicular to the plane. True You can use a sheet of paper to represent the plane. Draw two lines on the paper to represent the two lines in the plane. The third line is not contained in the plane. Represent this line with a pencil. Hold the pencil so that it is perpendicular to both of the lines in the plane. (Note: In order for you to do this, the lines in the plane must intersect.) Is the pencil perpendicular to the plane? Experiment until you are convinced you know whether the statement is true or false. S. Stirling Spring 2014 Page 18 of 18