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Warm Up Solve. 1. 2.

Warm Up Solve. 1. 2. Objectives. Use length and midpoint of a segment. Construct midpoints and congruent segments. Vocabulary. coordinate midpoint distance bisect length segment bisector construction between congruent segments.

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Warm Up Solve. 1. 2.

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  1. Warm Up Solve. 1. 2.

  2. Objectives Use length and midpoint of a segment. Construct midpoints and congruent segments.

  3. Vocabulary coordinate midpoint distance bisect length segment bisector construction between congruent segments

  4. A ruler can be used to measure the distance between two points. A point corresponds to one and only one number on a ruler. The number is called a coordinate. The following postulate summarizes this concept.

  5. The distance between any two points is the absolute value of the difference of the coordinates. If the coordinates of points A and B are a and b, then the distance between A and B is |a – b| or |b – a|. The distance between A and B is also called the length of AB, or AB. A B AB = |a – b| or |b - a| a b

  6. Congruent segments are segments that have the same length. In the diagram, PQ = RS, so you can write PQRS. This is read as “segment PQ is congruent to segment RS.” Tick marks are used in a figure to show congruent segments.

  7. You can make a sketch or measure and draw a segment. These may not be exact. A construction is a way of creating a figure that is more precise. One way to make a geometric construction is to use a compass and straightedge.

  8. Sketch, draw, and construct a segment congruent to JK. Check It Out! Example 2 Continued

  9. In order for you to say that a point B is between two points A and C, all three points must lie on the same line, and AB + BC = AC.

  10. Example 3A: Using the Segment Addition Postulate G is between F and H, FG = 6, and FH = 11. Find GH.

  11. Example 3B: Using the Segment Addition Postulate M is between N and O. Find NO.

  12. The midpointM of AB is the point that bisects, or divides, the segment into two congruent segments. If M is the midpoint of AB, then AM = MB. So if AB = 6, then AM = 3 and MB = 3.

  13. D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. Example 5: Using Midpoints to Find Lengths F E 4x + 6 7x – 9 D

  14. Objectives Name and classify angles. Measure and construct angles and angle bisectors.

  15. Vocabulary angle right angle vertex obtuse angle interior of an angle straight angle exterior of an angle congruent angles measure angle bisector degree acute angle

  16. A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points. An angleis a figure formed by two rays, or sides, with a common endpoint called the vertex(plural: vertices). You can name an angle several ways: by its vertex, by a point on each ray and the vertex, or by a number.

  17. The set of all points between the sides of the angle is the interior of an angle. The exterior of an angleis the set of all points outside the angle. Angle Name R, SRT, TRS, or 1 You cannot name an angle just by its vertex if the point is the vertex of more than one angle. In this case, you must use all three points to name the angle, and the middle point is always the vertex.

  18. The measureof an angle is usually given in degrees. Since there are 360° in a circle, one degreeis of a circle. When you use a protractor to measure angles, you are applying the following postulate.

  19. If OC corresponds with c and OD corresponds with d, mDOC = |d– c| or |c– d|. You can use the Protractor Postulate to help you classify angles by their measure. The measure of an angle is the absolute value of the difference of the real numbers that the rays correspond with on a protractor.

  20. Example 2: Measuring and Classifying Angles Find the measure of each angle. Then classify each as acute, right, or obtuse. A. WXV mWXV = 30° WXV is acute. B. ZXW mZXW = |130° - 30°| = 100° ZXW = is obtuse.

  21. Congruent angles are angles that have the same measure. In the diagram, mABC = mDEF, so you can write ABC  DEF. This is read as “angle ABC is congruent to angle DEF.” Arc marks are used to show that the two angles are congruent. The Angle Addition Postulate is very similar to the Segment Addition Postulate that you learned in the previous lesson.

  22. An angle bisector is a ray that divides an angle into two congruent angles. JK bisects LJM; thus LJKKJM.

  23. KM bisects JKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM. Example 4: Finding the Measure of an Angle

  24. QS bisects PQR, mPQS = (5y – 1)°, and mPQR = (8y + 12)°. Find mPQS. Check It Out! Example 4a Find the measure of each angle.

  25. Homework: Pg 18 #21, 23, 27, 32, 40-44 Pg 25 #18, 29-32, 37, 38, 46-50

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