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1-2&3 Measuring and Constructing Segments and Angles Warm Up Solve. 1. 2 x 9 x 18 0 2 3 x or x 6 2 2. 6 x 54x 84 0 2 x 7 or x 2 Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Objectives Use length and midpoint of a segment. Construct midpoints and congruent segments. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Vocabulary coordinate midpoint distance bisect length segment bisector construction between congruent segments Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles 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. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles 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 Holt Geometry b 1-2&3 Measuring and Constructing Segments and Angles 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. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles 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. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Check It Out! Example 2 Continued Sketch, draw, and construct a segment congruent to JK. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles 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. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Example 3A: Using the Segment Addition Postulate G is between F and H, FG = 6, and FH = 11. Find GH. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Example 3B: Using the Segment Addition Postulate M is between N and O. Find NO. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles The midpoint M 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. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Example 5: Using Midpoints to Find Lengths D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. F E 4x + 6 Holt Geometry D 7x – 9 1-2&3 Measuring and Constructing Segments and Angles Objectives Name and classify angles. Measure and construct angles and angle bisectors. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Vocabulary angle vertex interior of an angle exterior of an angle measure degree acute angle Holt Geometry right angle obtuse angle straight angle congruent angles angle bisector 1-2&3 Measuring and Constructing Segments and Angles 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 angle is 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. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles The set of all points between the sides of the angle is the interior of an angle. The exterior of an angle is 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. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles The measure of an angle is usually given in degrees. Since there are 360° in a circle, one degree is of a circle. When you use a protractor to measure angles, you are applying the following postulate. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles 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. If OC corresponds with c and OD corresponds with d, mDOC = |d – c| or |c – d|. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Example 2: Measuring and Classifying Angles Find the measure of each angle. Then classify each as acute, right, or obtuse. A. WXV mWXV = 30° WXV is acute. B. ZXW mZXW = |130° - 30°| = 100° ZXW = is obtuse. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Congruent angles are angles that have the same measure. In the diagram, mABC = mDEF, 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. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles An angle bisector is a ray that divides an angle into two congruent angles. JK bisects LJM; thus LJK KJM. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Example 4: Finding the Measure of an Angle KM bisects JKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Check It Out! Example 4a Find the measure of each angle. QS bisects PQR, mPQS = (5y – 1)°, and mPQR = (8y + 12)°. Find mPQS. Holt Geometry 1-2&3 Measuring and Constructing Segments and Angles Homework: Pg 18 #21, 23, 27, 32, 40-44 Pg 25 #18, 29-32, 37, 38, 46-50 Holt Geometry