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Lesson 1-4

Lesson 1-4. Measuring Segments and Angles. The Ruler Postulate. The Ruler Postulate: Points on a line can be paired with the real numbers in such a way that: Any two chosen points can be paired with 0 and 1.

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Lesson 1-4

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  1. Lesson 1-4 Measuring Segments and Angles

  2. The Ruler Postulate • The Ruler Postulate:Points on a line can be paired with the real numbers in such a way that: • Any two chosen points can be paired with 0 and 1. • The distance between any two points on a number line is the absolute value of the difference of the real numbers corresponding to the points. Formula: Take the absolute value of the difference of the two coordinates a and b: │a – b │ Lesson 1-2: Segments and Rays

  3. Ruler Postulate : Example Find the distance between P and K. Note: The coordinates are the numbers on the ruler or number line! The capital letters are the names of the points. Therefore, the coordinates of points P and K are 3 and -2 respectively. Substituting the coordinates in the formula │a – b │ PK = | 3 --2 | = 5 Remember : Distance is always positive Lesson 1-2: Segments and Rays

  4. Between Definition: X is between A and B if AX + XB = AB. AX + XB = AB AX + XB > AB Lesson 1-2: Segments and Rays

  5. 12 AC + CB = AB x + 2x = 12 3x = 12 x = 4 The Segment Addition Postulate Postulate: If C is between A and B, then AC + CB = AB. If AC = x , CB = 2x and AB = 12, then, find x, AC and CB. Example: 2x x Step 1: Draw a figure Step 2: Label fig. with given info. Step 3: Write an equation x = 4 AC = 4 CB = 8 Step 4: Solve and find all the answers Lesson 1-2: Segments and Rays

  6. If numbers are equal the objects are congruent. AB: the segment AB ( an object ) AB: the distance from A to B ( a number ) Congruent Segments Definition: Segments with equal lengths. (congruent symbol: ) Congruent segments can be marked with dashes. Correct notation: Incorrect notation: Lesson 1-2: Segments and Rays

  7. Midpoint Definition: A point that divides a segment into two congruent segments Formulas: On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is . In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and is . Lesson 1-2: Segments and Rays

  8. Angle and Points • An Angle is a figure formed by two rays with a common endpoint, called the vertex. ray vertex ray • Angles can have points in the interior, in the exterior or on the angle. A E D B C Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex. Lesson 1-4: Angles

  9. Naming an angle:(1) Using 3 points (2) Using 1 point (3) Using a number – next slide Using 3 points: vertex must be the middle letter This angle can be named as Using 1 point: using only vertex letter *Use this method is permitted when the vertex point is the vertex of one and only one angle. Since B is the vertex of only this angle, this can also be called . A C B Lesson 1-4: Angles

  10. Example • K is the vertex of more than one angle. Therefore, there is NO in this diagram. There is Lesson 1-4: Angles

  11. Naming an Angle - continued Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named as . A B 2 C * The “1 letter” name is unacceptable when … more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present. Lesson 1-4: Angles

  12. 4 Types of Angles Acute Angle: an angle whose measure is less than 90. Right Angle: an angle whose measure is exactly 90 . Obtuse Angle: an angle whose measure is between 90 and 180. Straight Angle: an angle that is exactly 180 . Lesson 1-4: Angles

  13. Measuring Angles • Just as we can measure segments, we can also measure angles. • We use units called degrees to measure angles. • A circle measures _____ • A (semi) half-circle measures _____ • A quarter-circle measures _____ • One degree is the angle measure of 1/360th of a circle. 360º ? 180º ? ? 90º Lesson 1-4: Angles

  14. Adding Angles • When you want to add angles, use the notation m1, meaning the measure of 1. • If you add m1 + m2, what is your result? m1 + m2 = 58. m1 + m2 = mADC also. Therefore, mADC = 58. Lesson 1-4: Angles

  15. Angle Addition Postulate Postulate: The sum of the two smaller angles will always equal the measure of the larger angle. Complete: m  ____ + m ____ = m  _____ MRK KRW MRW Lesson 1-4: Angles

  16. Example: Angle Addition K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK. First, draw it! 3x + x + 6 = 90 4x + 6 = 90 – 6 = –6 4x = 84 x = 21 3x x+6 Are we done? mMRK = 3x = 3•21 = 63º Lesson 1-4: Angles

  17. Congruent Angles Definition: If two angles have the same measure, then they are congruent. Congruent angles are marked with the same number of “arcs”. The symbol for congruence is  3 5 Example: 3   5. Lesson 1-4: Angles

  18. Example • Draw your own diagram and answer this question: • If is the angle bisector of PMY and mPML = 87, then find: • mPMY = _______ • mLMY = _______ Lesson 1-4: Angles

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