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Angles are formed by two rays, lines, or line segments with a common endpoint. The common endpoint is called the vertex. In the angle below, point A is the vertex. The angle can be denoted as BAC or CAB Note that the vertex, point A, must be in the middle. B vertex The unit of measurement of an angle is degrees. A C Right angle 90° Acute angle Less than 90° Straight angle Obtuse angle 180° Greater than 90° B C Intersection Point A E D Intersecting lines form angles at their intersection point. Angles that share a common side are called adjacent angles. The angles that are not adjacent are called vertical angles. Example EAB and BAC are adjacent angles. BAC and CAD are adjacent angles. EAD and BAC are vertical angles. (3x+15)° (4x-20)° The vertical angles property allows us to make an equation: 3x+15 = 4x -20 and solve for x. Vertical angles have equal measures of degrees. We call this being “congruent.” Two angles are supplementary angles when the sum of their measures is 180°. When two lines intersect, adjacent angles are supplementary because the sides that are not in common form a straight angle. EAD and DAC are supplementary angles. DAC = 180° Two angles are complementary angles when the sum of their measures is 90°. When two adjacent angles form a right angle with the sides that are not in common, these angles are complementary because the measures add up to 90°. B C 90° A EAD + D BAC and CAD are complementary angles. Parallel lines are lines in the same plane that never intersect (that have the same slope). If two lines, l1 and l2 are parallel, we say l1 || l2. Transversal – a line that intersects two or more lines on the same plane. A Alternate Interior Angle Property A transversal intersects parallel lines at congruent angles. Because of this and also because of the property of supplementary angles and the property of vertical angles, we can rewrite the diagram on the left as this: B C D E F G H A The angles on the inside of the parallel lines that are congruent are called “Alternate Interior Angles.” C= C and F D= D and E F are alternate interior angles. E are alternate interior angles. 180°- A 180°- A A A 180°- A 180°- A A Example Given that l1|| l2, solve for x l1 (3x+20)° l2 (3x-80)° (3x-80)° l1 (3x+20)° l2 These two angles are not equal, but we can still solve for x by using other properties. Since these two lines are parallel, the transveral intersects them at congruent angles, so the angle adjacent to (3x+20)° is (3x-80)°. (3x-80)° The supplementary angle property says that if two adjacent angles form a straight angle with uncommon sides, they are supplementary, so (3x – 80) + (3x + 20) = 180. Now that we have an equation, we can solve for x. The Sum of the Measures of the Interior Angles of a Triangle is 180° The sum of angles in a triangle is always 180 degrees. A rigorous proof takes some work, but the statement can be made plausible by the following argument. In the triangle ABC, draw a line through point C that is parallel to AB. This creates two additional angles, A' and B'. The three angles (A',C,B') add up to 180 degrees, because they are adjacent to each other and backed off against a straight line. However, by the properties of parallel lines, Angle A = Angle A' and Angle B = Angle B'. Therefore (A,C,B) also add up to 180 degrees. l1 l1 || l2 l2 Example 5: Given that Angle a = 45° and Angle x = 100°, find the measure of angles b,c, and y. x b a k Angle b = Angle c = Angle y = y c l Try this one: Given that Angle y = 55 °, and that lines m and k are perpendicular. find the measures of angles a, b, and d. m d k b a y Angle a = Angle b = Angle d = Markup and Discount $ Retail Price = $Cost + $ Markup Amount the store raises the price to make a profit Selling price for an item Amount the store paid for the item Markup = Markup Rate * $Cost $ Sale Price= $ Original Price – $ Discount $Discount = Discount Rate * $Original Price Markup rate must be converted from percent to decimal before multiplying . Original Price is the “Regular Price” of the item. Example The manager of a clothing store buys a suit for $180 and sells that suit for $252. Find the markup rate. $Markup = Markup Rate * $Cost So Markup Rate = $Markup/ $Cost We know the cost of the suit, that is $180. What is the Markup? We can use the other formula, $Retail Price = $Cost + $Markup to find Markup. $Markup = $Retail Price - $Cost = $252 - $180 = $72 Markup Rate = $Markup/ $Cost = $72/$180 = 0.4 Converting 0.4 to percent, we get 40%. Conclusion: Markup Rate is 40% You try this: The cost to a sporting goods store of a tennis racket is $120. The selling price of the racket is $180. What is the markup rate?