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Transcript
1.1 Building Blocks of Geometry Name Definition Picture Short Rorm Point A location in space Line An infinite number of points in a straight line. TM Ray An infinite number of points in a straight line from a given point. TM Line Segment An infinite number of points in a straight line between two given points. Plane A flat surface that extends outward forever with length and width, but no thickness. The point p M T Collinear points Points on the same line Coplanar points Points on the same plane Angle Two rays sharing the same endpoint form an angle Endpoints A terminating or starting point TM T P M D Just say it Just say it BHM or MHBor H Just say it 1. Name each picture using the proper notation/symbols. If you can name the picture more than one way, then do so. P B B A Q H M T M 2. Draw RA , HB, g , and a plane containing the points JML. 3. For the following picture is the notation B a good idea? Why? P H M B 4. Use your ruler to measure the following segment in cm and then in inches and write it in the form mAB= A B 5. How many points define a line, a plane? 1.2 Pool Room Math. 1. Label each angle with a letter and measure the angles using a protractor. Write your answers in the form mA Congruence is where two shapes have the exact same size and shape. The symbol is Congruent segments have the exact same measure. Congruent angles have the exact same measure. *Numbers are EQUAL. *Shapes are CONGRUENT A B mA mB Labeling angles and segments mA 35 A B mAEB 120 mCED 80 CE 6 AB 5 EB 3 E C D Showing congruent segments and angles. A A B C D AE BE CE DE AB CD E C B D What would be wrong with using the notation E ? Mark the following on the given shape. A B C D AB AE CE BE DB CD A mC 90 mAEB 60 mCEB 90 CE 6 AB 10 EB 7 B E C D 2.3. Conditionals, Converse, Counterexample, and more definitions. Conditional- A sentence with a condition and then a result. Usually in the form of : If x 3 4 , then x 1 If it rained, then the street is wet. What do these statements all have in common? If an mA 90 , then A is a right angle. If a polygon is a square, then it is a rhombus 1. Write a conditional. Converse- The reverse statement. If x 1, then x 3 4 If street is wet, the it rained. What did I do to the previous statements to create these? If A is a right angle , then the mA 90 If a polygon is a rhombus, then it is a square 2. Write the converse to your conditional. Biconditional- A statement where the conditional and the converse are true and written using "if and only if." Examples. If x 3 4 , then x 1, If x 1, then x 3 4 x 3 4 if and only if x 1 If an mA 90 , then A is a right angle. If A is a right angle , then the mA 90 A is a right angle if and only if the mA 90 3. Can you write a biconditional using your conditional and converse? If not try writing one using another scenario. Counterexample- An example that makes something not true. It counteracts the statement. 4. Given the conditional: If it rained, then the street is wet. a) Write the converse. b) Give a counterexample to disprove the converse. 5. Given the conditional: If x 3 , then x 2 9 a) Write the converse. b) Give a counterexample to disprove the converse. Name Definition Picture (give a measurement) An angle with a measure of 90 deg. An angle with a measure of less than 90 deg. An angle with a measure of more than 90 deg. Right Angle Acute angle obtuse angles Midpoint of a segment The point equidistant from both endpoints The line segement equidistand from the segments creating the angle Angle bisector 6. Mark each figure to indicate the given information. Use the congruent slashes and symbol for a right angle. a) AB = CD, mA mD b) Point F is the midpoint of sAC, CDB is a right angle, and sAE is an angle bisector. C B A C E F D . A D B 2.4 Defining line and angle relationships. Name Definition Parallel lines Two or more lines that lie in the same plane and do not intersect Perpendicular lines Two lines that lie in the same plane and intersect at a 90 degree angle. Pair of complementary angles Two angle whose measures have a sum of 90 degrees. Pair of supplementary angles Two angle whose measures have a sum of 180 degrees. Pair of vertical angles Two nonadjacent angles formed by two intersecting lines Linear pair of angles A pair of angles that form a line. Picture (give a measurement) Short Form m lm l m p m p Draw and carefully label the following: 1. Two vertical angles 1 and 2 2. PE AR 3. PE ET 4. Supplementary angles D and F with mD 35 2.5 Defining POLYGONS (many knees) Polygon-- Sides Name Sides Name Sides 2 6 10 3 7 11 4 8 12 5 9 p Name Definition Convex polygon A polygon where no segments connecting any two vertices is outside the polygon. Concave polygon A polygon were at least one segment connecting two vertices is outside the polygon Consecutive vertices Two vertices connected by a side. Consecutive sides Two sides that share a vertice Consecutive angles Two angles that share a side Congruent polygons Polygons where all corresponding sides have equal measures and all corresponding angles have equal measures. Name Picture Name Definition Diagonal A segment connecting two nonconsecutive vertices Equilateral Polygon A polygon where all sides are equal. Equiangular polygon A polygon where all angles are equal. Regular polygon A polygon where all sides are equal and all angles are equal Picture Draw and carefully label the following: Use a ruler and protractor. 1. Hexagon HEXGON with right G and diagonal sEO 3. Draw a Hexagon ABCDEF where A D F . 2. Equiangular quadrilateral QUAD. 1.5 Triangles Picture Name Definition Right triangle A triangle with one right angle A triangle where all angles are acute. Acute triangle Obtuse triangle A triangle with one obtuse angle Scalene triangle A triangle where no sides are equal Isosceles triangle A triangle with at least two congruent sides Equilateral triangle A triangle where all sides are congruent Median of a triangle A segment connecting the midpoint of a side to the opposite vertex Altitude of a triangle A perpendicular segment from the vertex to the opposite side or the line containing the opposite side. Draw and carefully label the following: 1. An obtuse scalene triangle 2. Triangle ABC with median AE. 3. Acute triangle DEF with altitude DA 4. Obtuse triangle MOP, mO 130 , with altitude ME. 1.6 Quadrilaterals Name Definition Trapezoid A Quadrilateral with exactly on pair of parallel sides Kite A Quadrilateral with exactly two pairs of distinct congruent consecutive sides Parallelogram A Quadrilateral with two pair of parallel sides. Rhombus An equilateral Quadrilateral Rectangle An equiangular Quadrilateral Square Picture An equiangular and equilateral Quadrilateral QUADRILATERALS TRAPEZOIDS PARALLELOGRAMS RHOMBUS KITES RECTANGLE SQUARE Use your geometric tools to draw and label each figure. These must be close to exact. 9. Isosceles right triangle ABC with right angle B 10. Trapezoid ZOID with sZO // sID, mOZD = 75°, and mZOI = 45° Exercises--Group work---Review. Answer the following questions. Match each statement from 1 to 8 with a letter from the box. 1. 2. 3. 4. 5. 6. 7. 8. a. Decagon b. Isosceles c. Scalene e. Acute f. Octagon g. Hexagon i. Collinear j. Coplanar k. Protractor m. AB n. AB o. d. Obtuse h. Dodecagon l. p. _______ _______ _______ _______ _______ _______ _______ _______ BA AB AB The tool used to measure angles in degrees Three or more points on a line A triangle with all sides of unequal measure A triangle with at least two sides of equal measure A ray starting at point B and passing through point A A line segment with endpoints A and B An angle whose measure is greater than 90° A polygon with ten sides Match each statement from 1 to 8 with a letter from the box. a. Rhombus e. Angle bisector i. Parallel b. Rectangle f. Median j. Supplementary c. Trapezoid g. Altitude k. Complementary d. Parallelogram h. Perpendicular bisector 1. 2. 3. 4. 5. 6. _______ _______ _______ _______ _______ _______ 7. _______ 8. _______ Two angles whose measures add up to 180° Two lines in the same plane that do not intersect An equiangular parallelogram A quadrilateral with exactly one pair of parallel sides An equilateral quadrilateral A segment in a triangle connecting a vertex with the midpoint of the opposite side A segment in a triangle from a vertex perpendicular to the line containing the opposite side A segment in a triangle from a vertex to the opposite side dividing the angle into two parts of equal measure 9. True or False. If false, then give a counterexample or reason why it is false. a) _____An angle bisector divides an angle into two congruent angles. b) _____If two lines intersect then they form a right angle. c) _____Every square is a rectangle. d)_____Every square is a rhombus. e)_____An obtuse triangle has exactly one angle greater than 90 degrees Circles Circle- The set of points in a plane equidistant form a given point(the center of the circle). radius- A segment from the center of the circle to a point on the circle (the distance from the center to a point on the circle.) Congruent Circles- two circles with the same radius. Concentric circles - Two circles with the same center Arc of a circle- part of a circle Semicircle- An arc that is half the circle Minor Arc- less than half the circle Major Arc- more than half Chord- a segment connecting two points on the circle. Diameter- a cord containing the center of the circle. Secant line- a line that intersects two or more points of a circle or curve. Tangent line- a line that intersects one and only one point on the object. Inscribed angle- An angle created by two different cords sharing a point on the circle. Central angle- an angle with vertex at the center of the circle 1.8 Space Geometry Let's do some drawing. Cylinder Cone Prism Sphere 1. Draw each of the previous again. 2. Draw a rectangular solid 3 cm by 4 cm by 5 cm, resting on its smallest face. Pyramid Hemisphere 1.9 If you can't picture it then DRAW A PICTURE. 1. David is 3 years older than Stephen and 2 years younger than Graham. If Neil is 4 years younger than Graham, how much older than Neil is David? ____________ 2 The box on the right is wrapped with two strips of ribbon as shown. What length of ribbon was needed to decorate the box? Break the box down into it's six sides. 3. At one point in a race, Homer was 25 feet behind Bart and 28 feet ahead of Marge. Marge was trailing Lisa by 40 feet. Who is winning the race and by how much? How much is Homer losing the race by? Locus- a set of points/dots. Planes can be paper, cardboard, or any flat surface. 4. Line AB lies in a plane. Sketch the locus of points in that plane that are 3 cm from line AB. Sketch the locus of points in the space that are 3 cm from line AB. What shapes do the locus of points make for each sketch? 5. Point P lies in the plane. Sketch the locus of points in that plane that are 3 cm from the point P. What shape do the locus of points make?