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Transcript
3-7-1 Geometry Vocabulary It is important to know the vocabulary to be able to communicate and understand the questions you come across. Geometry is the study of the size, shape and positions of object in space. Plane geometry studies these objects in flat surfaces. Terms A point is a position in space represented by a dot and usually named with a capital letter. A point can’t be measured because it has no length and no height. Diagram A C B A line, represented on a diagram by a straight line with arrows on both ends, has no width and extends infinitely in both directions. It take two points to define a line, but an infinite number of points are on the line. It is named with two of the points on the line or with a lowercase A C k B D italicized letter. AB , DA and line k are the same in the diagram. A ray is part of a line. It starts at one point and goes infinitely in one direction. BD starts at point B and goes through point D and continues forever in that direction. DA is not the same ray. A segment is between two points. AB is the segment between point A and point B. Notice the part of ray BD that is the same as part of ray DA is segment BD . An intersection is where geometric objects share space. Two lines intersect at a point. Line k and line m intersect at point B. BD is the A C intersection of DA and BD . m k B Two rays with a common starting point form an angle. The common point is called the vertex. An angle may be named by it's vertex, B when the vertex alone is not ambiguous. Three points can also designate an angle with the vertex as the middle point, ABC . Angles might also be numbered, 1 . D A B 1 C 103 Perpendicular lines intersect at 90 degree A angles. CE BD is read line CE is perpendicular to line BD. C m E Parallel lines don't intersect, even if they are extended much farther than shown in the diagram. Line k is parallel to line m but is not parallel to line n. k B D n An acute angle is less than 90 . DBE is acute. Think "a cute little angle" to remember the term. E An obtuse angle is greater than 90 . CBD is obtuse. Think "obtuse angles are obese" to remember. D A B A right angle is 90 . EBC and ABE are right angles. The little square at the vertex indicates the angle is 90 . C Complementary angles add to 90 . ABD and DBE are complementary angles. Supplementary angles add to 180 ABD and DBC are supplementary angles. Vertical angles are across the intersection of two lines from each other. 8 and 5 are vertical angles. Vertical angles are equal. There are two pairs of vertical angles in one intersection. 2 A transversal crosses a set of parallel lines. CD is a transversal to the parallel lines m and k. This grouping makes two sets of four equal angles. m2 m3 m8 m5 and m1 m4 m7 m6 The "m" stand for "the measure of". C m 1 4 D 8 3 6 5 7 k Alternate interior angles are equal and between the parallel lines and across the transversal from one another. 8 and 3 are one set of alternate interior angles. Alternate exterior angles are equal and outside the parallel lines and across the transversal from one another. 4 and 6 are one set of alternate interior angles. Corresponding angles are on the same side of the transversal and either both above each parallel line or both below. 4 and 7 are one set of corresponding angles. 104 3-7-2 Geometry Vocabulary Practice T Practice: a) Color point E green. Points have no width, but we represent them with a dot. b) Highlight line l yellow. It takes two points to make a line. Lines go on forever in both directions. c) Highlight AD brown. Lines also have no thickness. The symbol is read “line AD.” d) Color ray IQ red. A ray starts at one point and goes forever in one direction. e) Color the intersection BI of ED and purple. The intersection of two lines is where they cross. f) Highlight two lines that never intersect orange. Lines that don’t intersect are parallel. g) Find and color black a line that crosses the parallel lines. This line is called a transversal. h) Color ADE blue. An angle is two rays that start at the same point. i) Color the vertex of ADE yellow. The vertex is the point on an angle. j) Fill in an acute angle with vertex at point C red. An acute angle is less than 90 . k) Fill in an obtuse angle with vertex C blue. An obtuse angle is greater than 90 . l) Fill in a right angle purple. A right angle is 90 . m) Measure each angle with vertex I. n) Color a set of vertical angles orange. Vertical angles are equal. Vertical angles are the angles opposite each other when two lines intersect. 0) Color a set of supplementary angles green. Supplementary angles sum to 180 p) Color a set of complementary angles brown. Complimentary angles sum to 90 . q) Find a set of alternate interior angles. They are equal. Alternate angles are on opposite sides of a transversal. Interior angles are inside parallel lines. r) Find a set of alternate exterior angles. The are equal. Exterior angles are outside the parallel lines. s) Find a set of corresponding angles. Angles formed with parallel lines and a transversal and located in the same position compared to the transversal are corresponding angles They are equal. 105 3-7-3 Geometry Vocabulary Puzzle 1 2 3 4 6 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Across 1 Units angles are measured in 3 A location in space; has no dimension 4 Angles that are formed when two lines intersect 8 Angles that have a measure equal to 90 degrees 9 Angles on opposite sides of a transversal 11 angles formed by the transversal and parallel lines on the inside of the parallel lines 106 12 Angle greater than 90 degrees 14 Angle measuring less than 90 degrees 16 Two angles that add up to 90 degrees 17 Extending endlessly in two directions in space. Takes two points to Define 19 A tool used to measure angles 21 A line that intersects two parallel lines 22 Two lines in the same plane that never meet 23 Angles on the outside of parallel lines Down 2 When the sum of the measures of two angles is 180 degrees, the angles are________. 5 Equal angles located in the same position compared to the transversal 6 When two lines cross 7 Part of a line between two points 10 two rays starting at a common point 13 The study of flat surfaces 15 two lines that intersect form 90 degree angles 18 The study of size, shape, positions, of objects in space 20 A part of a line that starts at one point and extends endlessly in the other direction 3-7-4 Geometry with Algebra The following will provide practice with both geometry vocabulary and algebra. Think about what the picture or vocabulary means and make the equation accordingly. 3x-7 2x-3 Examples: Find x. The two angles are complementary and must add to 90 degrees. 2x-3 + 3x-7 = 90 When this is worked out x is 20. The angles could be found by plugging the 20 into the expression for each angle. E D A B C The angles in a triangle add to 180 degrees. Cut out any triangle. Tear off the angles and arrange them next to each other as shown. This works with any triangle. Example: Find the measure of the angles in an equilateral triangle. First draw a picture. An equilateral triangle has all sides the same length and all angles the same size. x+x+x=180 3x = 180 x= 60 The angles in an equilateral triangle are each 60 degrees. x x x Practice: 2x-5 a) Find x. Remember the complementary angles add to ninety degrees. E D x A B C b) One angle in a set of supplementary angles is 5 more than twice the other. Find the measurement of the angles. c) Solve for x in each of the following diagrams. Assume lines that appear parallel are parallel. 3x+85 2x-8 3(x-8) 5(2x-15) 3(2x -5) x+15 107 8x+22 2(3x -5) 3(x+15) 5(x-2) 3(4x -12) 3(x-4) d) The angles on any triangle add up to 180 degrees. Find the measure of each angle if the triangle is isosceles with one 40-degree angle. e) What is the measure of the angles in an equilateral triangle? (all sides and all angles equal) f) One angle of a triangle is three less than twice another. The other is 4 times the sum of the small angle and 2. What are the angle measurements? g) One vertical angle is a number plus 12. The other 40 less than is three times the same number. What is the number? h) What is the measure of the other two angles in a right isosceles triangle? (Angles in any triangle add to 180.) 2x 2(5(x-3)+14) i) 3x+15 5(x-3)+14 2x Find x. Use a protractor to measure angles. A protractor has a notch or hole to mark where the vertex of the angle must be. One ray of an angle is along the line marked 0 degrees. Most times this is not along the bottom of the protractor, but in line with the hole for the vertex. The size of the angle is read from the numbers on the curved part of the protractor. 108 Students sometimes have a hard time knowing which number to read of the two that the angle touches. There is an easy way to decide. If the angle is acute it is the measurement that is less than 90 degrees. If the angle is obtuse the measurement is more than 90 degrees. The angle is positioned with the vertex on the hole and one ray along the zero degree line. The numbers the other side of the angle touches are 60 degrees and 120 degrees. Which measurement is correct? Sometimes you must use a straight edge to extend the angel farther than is presented in the original problem to see exactly where the line hits the protractor. Practice: Measure the following angles. a) b) c) d) Draw a 65 angle. Draw a 32 angle. Draw a 15 angle. e) Draw a 120 angle. Draw a 155 angle. Draw a 175 angle. 109