Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Golden ratio wikipedia , lookup
Perspective (graphical) wikipedia , lookup
Technical drawing wikipedia , lookup
Perceived visual angle wikipedia , lookup
Multilateration wikipedia , lookup
Line (geometry) wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Rational trigonometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euler angles wikipedia , lookup
Pythagorean theorem wikipedia , lookup
IMA 101 Basic Mathematics 1 LECTURE 22 IMA101 2/2010 – Lecture 22 February 3, 2011 Geometry 2 Angles Lines Planes Polygons Triangle properties Circles IMA101 2/2010 – Lecture 22 February 3, 2011 Definitions 3 SECTIONS 9.1 -9.3 T&G IMA101 2/2010 – Lecture 22 February 3, 2011 Basic Definitions 4 A point is an exact location in space A point has no dimension. A line is a collection of points along a straight path. A line extends forever in both directions. A line segment is a part of a line having two endpoints. A line segment has only one dimension-its length. IMA101 2/2010 – Lecture 22 A (read “point A”) C B (read “line CB”) A B (read “line segment AB”) AB February 3, 2011 Basic Definitions 5 A ray is a part of a line having only one endpoint. C D (read “ray CD”) An angle consists of two rays that have a common endpoint called the vertex of the angle. C B Vertex A ABC (read “angle ABC”) A plane is a flat surface that extends endlessly in two dimentions. IMA101 2/2010 – Lecture 22 February 3, 2011 Angles 6 Angles are measured in Degrees, where a full revolution is 360°. We can use a protractor to measure degrees. B A C D V The measure of AVC is m(AVC ) IMA101 2/2010 – Lecture 22 February 3, 2011 NOTE: Angles are measured in degrees (°). 7 ABC is a straight angle if it A measures 180°. B C ABC is an obtuse angle if it measures less than 180° and more than 90°. A C B ABC is an right angle if it measures 90°. A C B ABC is an acute angle if it measures less than 90°. IMA101 2/2010 – Lecture 22 A B C February 3, 2011 Basic Definitions 8 Complementary angles are two angles whose measures sum to 90°. Supplementary angles are two angles whose measures sum to 180°. IMA101 2/2010 – Lecture 22 65° 25° H J 40° K 140° L February 3, 2011 Exercises 9 Find x. Is x acute, obtuse, right, or straight? 100° 90° x° x° 60° J 35° K 135° 20° x° L IMA101 2/2010 – Lecture 22 14° x° 45° L February 3, 2011 Basic Definitions 10 Vertical angles are two angles formed by two intersecting lines. Vertical angles have the same measurement 120° 60° 60° 120° Two angles are congruent when their measurement is the same. A is congruent to B when IMA101 2/2010 – Lecture 22 A B February 3, 2011 Exercises 11 Find x. Is x acute, obtuse, right, or straight? x° 100° (2x)° (x+30)° (4x+15)° (7x-60)° IMA101 2/2010 – Lecture 22 February 3, 2011 Basic Definitions 12 Intersecting lines are two lines that meet. l2 l1 Parallel lines are two lines in the same plane that do not intersect. l2 l1 || l2 l2 || l1 Perpendicular lines are two lines that intersect to form right angles. l1 l2 l2 l1 IMA101 2/2010 – Lecture 22 l1 l2 l1 February 3, 2011 Transversal 13 A line that intersects 2 or more co-planar lines is called a transversal. l3 is a transversal intersecting l1 and l2 l3 l1 l2 IMA101 2/2010 – Lecture 22 February 3, 2011 Interior Angles 14 Angles 3,4, 5, and 6 are interior angles Specifically the pairs 3 and 6 ; as well as 4 and 5 are both pairs of alternate interior angles l3 8 7 5 3 1 IMA101 2/2010 – Lecture 22 4 l1 6 l2 2 February 3, 2011 Parallel lines 15 If two parallel lines are cut by a transversal then their alternate interior angles are congruent. if l1 || l2 then 4 5 and 3 6 l3 8 7 5 3 1 IMA101 2/2010 – Lecture 22 4 l1 6 l2 2 February 3, 2011 Corresponding Angles 16 The following pairs of angles are corresponding angles 1 and 5 2 and 6 7 and 3 8 and 4 l3 5 3 1 IMA101 2/2010 – Lecture 22 8 7 4 l1 6 l2 2 February 3, 2011 Parallel lines- alternate interior angle property 17 If two parallel lines are cut by a transversal then their alternate interior angles are congruent. if l1 || l2 then 4 5 and 3 6 l3 8 7 5 3 1 IMA101 2/2010 – Lecture 22 4 l1 6 l2 2 February 3, 2011 Parallel lines- corresponding angle property 18 If two parallel lines are cut by a transversal then their corresponding angles are congruent. if l1 || l2 then 1 5 and 2 6 3 7 and 4 8 l3 5 3 1 IMA101 2/2010 – Lecture 22 8 7 4 l1 6 l2 2 February 3, 2011 Parallel lines- supplementary angle property 19 If two parallel lines are cut by a transversal then the interior angles on the same side of the transversal are supplementary. l3 if l1 || l2 then 3 and 5 are supplements 4 and 6 are supplements 5 3 1 IMA101 2/2010 – Lecture 22 8 7 4 l1 6 l2 2 February 3, 2011 Parallel lines- perpendicular transversal property 20 If a transversal is perpendicular to on of two parallel lines, then it is also perpendicular to the other line || l2 and l3 l1 then l3 l2 if l1 l3 8 7 5 3 1 IMA101 2/2010 – Lecture 22 4 l1 6 l2 2 February 3, 2011 Exercise 21 If l1 || l2 and m(3) 120 find the measures of the other angles. l3 8 7 5 3 1 IMA101 2/2010 – Lecture 22 4 l1 6 l2 2 February 3, 2011 Exercise 22 Find x if AB || CD B C (7x-2)° (2x+33)° A IMA101 2/2010 – Lecture 22 D February 3, 2011 Exercise 23 Find x if AB || DE E (9x-38)° B C D (6x-2)° A IMA101 2/2010 – Lecture 22 February 3, 2011 Polygons 24 IMA101 2/2010 – Lecture 22 February 3, 2011 Polygon 25 Is a closed geometric figure that has line segments for its sides. Is a closed figure, which means that if you start at any point on the figure, you can trace completely around it and return to the starting point. IMA101 2/2010 – Lecture 22 February 3, 2011 Triangle 26 Is a polygon with three sides. It also has 3 vertices, and is called by its vertices IMA101 2/2010 – Lecture 22 February 3, 2011 Types of Triangles 27 An equilateral triangle is a triangle with three sides equal in length. All angles are equal An isosceles triangle is a triangle with two sides equal in length. 2 base angles are equal An scalene triangle is a triangle with no sides equal in length. IMA101 2/2010 – Lecture 22 February 3, 2011 Types of Triangles 28 An acute triangle is a triangle with three acute angle. An right triangle is a triangle with one right angle. An obtuse triangle is a triangle with one obtuse angle. In all triangles, the sum of the measures of all three angles is 180° The sum of any two sides is greater than the third IMA101 2/2010 – Lecture 22 February 3, 2011 Exercises 29 If the one base angle is 70° then how large is the vertex angle? What is x? x° 40° IMA101 2/2010 – Lecture 22 February 3, 2011 Exercise 30 Find ACB if AB || DE E (9x-36)° B C D (6x-6)° A IMA101 2/2010 – Lecture 22 February 3, 2011 Quadrilateral 31 Is a polygon with four sides. IMA101 2/2010 – Lecture 22 February 3, 2011 Rectangle: properties 32 All angles are right angles Opposite sides are parallel Opposite sides are of equal length Diagonals are equal length IMA101 2/2010 – Lecture 22 February 3, 2011 Exercise 33 Find ADB if AB || CD B C (7x-2)° (3x+22)° A IMA101 2/2010 – Lecture 22 D February 3, 2011 Pentagon 34 Is a polygon with five sides. IMA101 2/2010 – Lecture 22 February 3, 2011 Hexagon 35 Is a polygon with six sides, and so forth. IMA101 2/2010 – Lecture 22 February 3, 2011 Sum of the angles of a polygon 36 the sum of the measures of the angles of a polygon with n sides is given by S = (n – 2) 180° IMA101 2/2010 – Lecture 22 February 3, 2011 Triangle Properties 37 IMA101 2/2010 – Lecture 22 February 3, 2011 Congruent triangles 38 Triangles with the same area and the same shape are congruent Same side lengths and the same angle measures But we don’t need to know all these pieces of information to know whether two triangles are congruent IMA101 2/2010 – Lecture 22 February 3, 2011 SSS 39 Side-Side-Side Property If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent 4 3 5 IMA101 2/2010 – Lecture 22 4 3 5 February 3, 2011 SAS 40 Side-Angle-Side Property If two sides and the angle between them in one triangle are congruent respectively to two sides and the angle between them in a second triangle, the triangles are congruent 5 80° IMA101 2/2010 – Lecture 22 80° 3 3 5 February 3, 2011 ASA 41 Angle-Side-Angle Property If two angles and the side between them in one triangle are congruent, respectively, to two angles and the side between them in a second triangle, the triangles are congruent 80° 5 80° 10° IMA101 2/2010 – Lecture 22 5 10° February 3, 2011 THERE IS NO SSA 42 Even if two sides and the angle beside them in one triangle are congruent, respectively, to two sides and the angle beside them in a second triangle, the triangles may NOT be congruent 2 5 5 2 10° IMA101 2/2010 – Lecture 22 10° February 3, 2011 Are the congruent triangles? 43 Our properties are : SSS SAS ASA 5 6 6 5 IMA101 2/2010 – Lecture 22 February 3, 2011 Are the congruent triangles? 44 Our properties are : SSS SAS ASA 5 5 6 6 IMA101 2/2010 – Lecture 22 February 3, 2011 Are the congruent triangles? 45 Our properties are : SSS SAS ASA 6 5 5 IMA101 2/2010 – Lecture 22 6 February 3, 2011 Which of the following are congruent triangles? 46 Our properties are : SSS SAS ASA 4 50° 50° 4 4 2 2 4 IMA101 2/2010 – Lecture 22 February 3, 2011 Which of the following are congruent triangles? 47 Our properties are : 50° IMA101 2/2010 – Lecture 22 SSS SAS ASA 50° February 3, 2011 Similar Triangles 48 If two angles of one triangle are congruent to two angles of a second triangle, the triangles will have the same shape. (similar triangles) If two triangles are similar then all pairs of corresponding sides are proportional IMA101 2/2010 – Lecture 22 February 3, 2011 Similar Triangles 49 H 1m 4m IMA101 2/2010 – Lecture 22 0.5 m February 3, 2011 The Pythagorean Theorem (right triangles) 50 For right triangles: Hypotenuse is the side opposite the right angle c2 = a2 + b2 This is only true for right triangles! a c b IMA101 2/2010 – Lecture 22 February 3, 2011