Chapter+Six

**6.1** **6.2**  · Congruent triangles are triangles that are exactly the same shape and size, though they might be rotated differently or flipped over in the opposite direction. · Similar triangles have the same shape, but not the same size. They are mathematically proportional, meaning that if one triangle is twice as big as the other, each side is exactly twice as long as the corresponding side on the other triangle. · Finding missing sides of similar triangles o Similar triangles are used to find the length of missing sides in the same way that proportions were used to find the missing value of rates and ratios. To use it, you just build a proportion comparing the sides of three know lengths, using X for the unknown length. However, you must make sure that you are comparing the same corresponding sides. (hypotenuse to hypotenuse, short leg to short leg, etc.)
 * Chapter 6 **
 * A ratio is a __fraction__ that compares same units
 * A ratio could compare part to whole, such as 4 boys out of 9 students, or it could compare part to part, such as 4 boys to 5 girls. You really have to watch the words in the sentence to see what is being compared, or the units in the ratio.
 * Having the same units is kind of misleading. If you are comparing the number of boys who swim to the number of total boys in the room, boys are the units. But, you could compare boys who swim to girls who swim and still have a ratio, with person as the unit. When we talk about rate you will see the difference between comparing like units (ratios) and unlike units (rates).
 * Always reduce ratios to lowest terms
 * If there are decimals in ratio, convert to whole numbers. So, 4.5 clowns to 10 circus people needs to become 9 clowns to 20 circus people.
 * If there are mixed numbers in ratios, convert to whole numbers. So, 2⅓ to 1 becomes 7 to 3.
 * Ratios must be in same units, such as lengths. So, if you are comparing inches to feet, you need to put the numerator and denominator into the same units (inches or feet).
 * Rates are like ratios except that they compare different units
 * For example, 18 miles for each gallon of gas or $650 for working 30 hours.
 * You should still reduce rates to lowest terms, unless they ask for a unit rate.
 * Unit rate is when the denominator is 1
 * This is often describe as “per”, meaning “per one unit”. So, miles per gallon means how many miles per one gallon, meaning that 1 gallon will be the denominator.
 * You do not have to have a whole number for the numerator in unit rates. Usually it will be a decimal, such as $12.55 per hour.
 * 6.3**
 * A proportion is equivalent rates or ratios, written as a fraction, followed by an equal sign, followed by the other fraction. The fractions are equivalent, but don’t look the same.
 * Proportions are used to find a missing number or amount. An example would be when you know that it costs $3 to buy 4 bags of peanuts and you want to know how many bags you could get for $15.
 * You use “cross products” are used to find unknown numbers. The numerator of the first fraction (rate or ratio) times the denominator of the second one, equals the denominator of the first fraction times the numerator of the second. After you do this, you just solve for X.
 * It doesn’t matter where the X is in the proportion, but it does matter that the fractions are equivalent. So if the X represents dogs in one fraction and is the denominator, dogs must be the denominator in the other fraction as well.
 * 6.4**
 * Proportions and application problems. Most proportions are story problems anyway. They tell you the two units, like donuts and police officers, and give you a relationship between the two (each officer eats 3 donuts becomes 1 to 3, or 3 to 1). Then they ask you to find the missing value for one of the variables: If there are 6 officers, how many donuts, or if 500 donuts are ordered, how many officers will that cover. Just build a proportion and solve for X.
 * 6.5**
 * ** A line is a line running through 3-dimensional space that passes through at least two named points.
 * A line is written as the letters that represent the points with a double headed arrow above them. See the diagram.
 * A line segment is the line starting and ending at the two named points.
 * Line segments are written as the letters representing the points with a line with no arrows above them. See the diagram.
 * A ray is a line that starts at the first named point and goes through the second point, but goes on forever in that direction.
 * A ray is written as the two points (starting point first) and a line with one arrow above them. See diagram.
 * Parallel lines are two lines that are running in exactly the same direction, some space apart. The distance between the lines will always be the same. The symbol for parallel lines is usually //
 * Intersecting lines are lines that cross each other at some point.
 * Perpendicular lines are lines that cross each other so that it creates four 90 degree angles. Think of the lines as making a plus sign, but the sign could be rotated so that the lines aren’t exactly up and down and straight across. Perpendicular lines are shown as __|__
 * If there is only a single angle, or only one that could be meant, the angle is named for the point that is its vertex or the point where the two rays intersect. So, in the diagram it could be __/__ A.
 * If there are multiple angles using A as the vertex, use a point on one side, then the vertex, then a point on the other side. So, in the diagram, __/__ BAC and __/__ CAB are the same angle and they are the same as __/__ A.
 * Acute, obtuse, right, straight angles
 * Acute angles measure less than 90 degrees.
 * Obtuse angles measure more than 90 degrees.
 * Right angles measure exactly 90 degrees.
 * Straight angles measure exactly 180 degrees. (straight line)
 * Complementary, supplementary angles
 * Complementary angles are two angles that together add up to 90 degrees.
 * Supplementary angles are two angels that together add up to 180 degrees.
 * Vertical angles are angles that are formed when two lines intersect. This will create four angles, two with one size and two with another (unless all are exactly 90 degrees). Any two angles that are directly opposite from each other will have the same measure and those are called vertical angles.
 * This is used when you have two parallel lines being intersected by another line. Using the concepts of supplementary angles and vertical angles, you can find the measure of all of the angles that are created if you know only a single measure. In the diagram, all of the angles labeled A will have the same measure and all of the angles labeled B will have the same measure. Also A + B = 180. So if A is 130 degrees, all of the angles marked A will be 130 degrees and all of the angles marked B will be 50 degrees.
 * 6.6**

o The flagpole, or a tree, etc., acts as one leg of a triangle, making a right angle with the ground. So if you were sitting some distance from the flagpole and made an imaginary triangle with you, the base of the flagpole and the top of the flagpole as the three points, you could measure a similar triangle (such as using a ruler) and use a proportion to figure out the missing info. See the diagram.