Measuring and Constructing Segments Essential Questions How do

Measuring and Constructing Segments Essential Questions How do I use length and midpoint of a segment? Holt Mc. Dougal Geometry

Measuring and Constructing Segments Vocabulary coordinate midpoint distance bisect length segment bisector construction between congruent segments Holt Mc. Dougal Geometry

Measuring and Constructing Segments 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 a Holt Mc. Dougal Geometry B b AB = |a – b| or |b - a|

Measuring and Constructing Segments 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 Mc. Dougal Geometry

Measuring and Constructing Segments 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 Mc. Dougal Geometry

Measuring and Constructing Segments Example 1: Using the Segment Addition Postulate M is between N and O. Find NO. NM + MO = NO 17 + (3 x – 5) = 5 x + 2 3 x + 12 = 5 x + 2 – 2 3 x + 10 = 5 x – 3 x 10 = 2 x 2 2 5=x Holt Mc. Dougal Geometry Seg. Add. Postulate Substitute the given values Simplify. Subtract 2 from both sides. Simplify. Subtract 3 x from both sides. Divide both sides by 2.

Measuring and Constructing Segments Example 1 Continued M is between N and O. Find NO. NO = 5 x + 2 = 5(5) + 2 Substitute 5 for x. = 27 Simplify. Holt Mc. Dougal Geometry

Measuring and Constructing Segments Check It Out! Example 2 E is between D and F. Find DF. DE + EF = DF (3 x – 1) + 13 = 6 x 3 x + 12 = 6 x – 3 x 12 = 3 x 12 3 x = 3 3 4=x Holt Mc. Dougal Geometry Seg. Add. Postulate Substitute the given values Subtract 3 x from both sides. Simplify. Divide both sides by 3.

Measuring and Constructing Segments Check It Out! Example 2 Continued E is between D and F. Find DF. DF = 6 x = 6(4) Substitute 4 for x. = 24 Simplify. Holt Mc. Dougal Geometry

Measuring and Constructing Segments 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 Mc. Dougal Geometry

Measuring and Constructing Segments Example 3: Recreation Application The map shows the route for a race. You are at X, 6000 ft from the first checkpoint C. The second checkpoint D is located at the midpoint between C and the end of the race Y. The total race is 3 miles. How far apart are the 2 checkpoints? XY = 3(5280 ft) = 15, 840 ft Holt Mc. Dougal Geometry Convert race distance to feet.

Measuring and Constructing Segments Example 3 Continued XC + CY = XY Seg. Add. Post. Substitute 6000 for XC and 15, 840 6000 + CY = 15, 840 for XY. – 6000 Subtract 6000 from both sides. Simplify. CY = 9840 D is the mdpt. of CY, so CD = CY. = 4920 ft The checkpoints are 4920 ft apart. Holt Mc. Dougal Geometry

Measuring and Constructing Segments Example 4: Using Midpoints to Find Lengths D is the midpoint of EF, ED = 4 x + 6, and DF = 7 x – 9. Find ED, DF, and EF. E 4 x + 6 Step 1 Solve for ED = DF 4 x + 6 = 7 x – 9 – 4 x D 7 x – 9 F x. D is the mdpt. of EF. Substitute 4 x + 6 for ED and 7 x – 9 for DF. Subtract 4 x from both sides. 6 = 3 x – 9 Simplify. +9 + 9 Add 9 to both sides. Simplify. 15 = 3 x Holt Mc. Dougal Geometry

Measuring and Constructing Segments Example 4 Continued D is the midpoint of EF, ED = 4 x + 6, and DF = 7 x – 9. Find ED, DF, and EF. E 4 x + 6 D 7 x – 9 F Step 2 Find ED, DF, and EF. ED = 4 x + 6 DF = 7 x – 9 = 4(5) + 6 = 7(5) – 9 = 26 Holt Mc. Dougal Geometry EF = ED + DF = 26 + 26 = 52

Measuring and Constructing Segments Check It Out! Example 5 S is the midpoint of RT, RS = – 2 x, and ST = – 3 x – 2. Find RS, ST, and RT. R – 2 x S – 3 x – 2 T Step 1 Solve for x. S is the mdpt. of RT. RS = ST – 2 x = – 3 x – 2 Substitute – 2 x for RS and – 3 x – 2 for ST. +3 x Add 3 x to both sides. x = – 2 Holt Mc. Dougal Geometry Simplify.

Measuring and Constructing Segments Check It Out! Example 5 Continued S is the midpoint of RT, RS = – 2 x, and ST = – 3 x – 2. Find RS, ST, and RT. R – 2 x S – 3 x – 2 T Step 2 Find RS, ST, and RT. RS = – 2 x = – 2(– 2) =4 Holt Mc. Dougal Geometry ST = – 3 x – 2 = – 3(– 2) – 2 =4 RT = RS + ST =4+4 =8

Measuring and Constructing 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 Mc. Dougal Geometry

Measuring and Constructing Angles Check It Out! Example 6 Write the different ways you can name the angles in the diagram. RTQ, T, STR, 1, 2 Holt Mc. Dougal Geometry

Measuring and Constructing 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 Mc. Dougal Geometry

Measuring and Constructing 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, m DOC = |d – c| or |c – d|. Holt Mc. Dougal Geometry

Measuring and Constructing Angles Holt Mc. Dougal Geometry

Measuring and Constructing Angles Check It Out! Example 7 Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. a. BOA m BOA = 40° BOA is acute. b. DOB m DOB = 125° DOB is obtuse. c. EOC m EOC = 105° EOC is obtuse. Holt Mc. Dougal Geometry

Measuring and Constructing Angles 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. Holt Mc. Dougal Geometry

Measuring and Constructing Angles Holt Mc. Dougal Geometry

Measuring and Constructing Angles Check It Out! Example 8 m XWZ = 121° and m XWY = 59°. Find m YWZ = m XWZ – m XWY Add. Post. m YWZ = 121 – 59 Substitute the given values. m YWZ = 62 Subtract. Holt Mc. Dougal Geometry

Measuring and Constructing Angles An angle bisector is a ray that divides an angle into two congruent angles. JK bisects LJM; thus LJK KJM. Holt Mc. Dougal Geometry

Measuring and Constructing Angles Example 9: Finding the Measure of an Angle KM bisects JKL, m JKM = (4 x + 6)°, and m MKL = (7 x – 12)°. Find m JKM. Holt Mc. Dougal Geometry

Measuring and Constructing Angles Example 9 Continued Step 1 Find x. m JKM = m MKL Def. of bisector (4 x + 6)° = (7 x – 12)° +12 Substitute the given values. 4 x + 18 – 4 x = 7 x – 4 x 18 = 3 x 6=x Holt Mc. Dougal Geometry Add 12 to both sides. Simplify. Subtract 4 x from both sides. Divide both sides by 3. Simplify.

Measuring and Constructing Angles Example 9 Continued Step 2 Find m JKM = 4 x + 6 = 4(6) + 6 Substitute 6 for x. = 30 Simplify. Holt Mc. Dougal Geometry

Measuring and Constructing Angles Check It Out! Example 10 Find the measure of each angle. QS bisects PQR, m PQS = (5 y – 1)°, and m PQR = (8 y + 12)°. Find m PQS. Step 1 Find y. Def. of bisector Substitute the given values. 5 y – 1 = 4 y + 6 y– 1=6 y=7 Holt Mc. Dougal Geometry Simplify. Subtract 4 y from both sides. Add 1 to both sides.

Measuring and Constructing Angles Check It Out! Example 10 Continued Step 2 Find m PQS = 5 y – 1 = 5(7) – 1 Substitute 7 for y. = 34 Simplify. Holt Mc. Dougal Geometry

Measuring and Constructing Angles Check It Out! Example 11 Find the measure of each angle. JK bisects LJM, m LJK = (-10 x + 3)°, and m KJM = (–x + 21)°. Find m LJM. Step 1 Find x. LJK = KJM (– 10 x + 3)° = (–x + 21)° +x +x – 9 x + 3 = 21 – 3 – 9 x = 18 x = – 2 Holt Mc. Dougal Geometry Def. of bisector Substitute the given values. Add x to both sides. Simplify. Subtract 3 from both sides. Divide both sides by – 9. Simplify.

Measuring and Constructing Angles Check It Out! Example 11 Continued Step 2 Find m LJM = m LJK + m KJM = (– 10 x + 3)° + (–x + 21)° = – 10(– 2) + 3 – (– 2) + 21 Substitute – 2 for x. = 20 + 3 + 21 = 46° Holt Mc. Dougal Geometry Simplify.

Measuring and Constructing Angles Assignment Segments Angles Holt Mc. Dougal Geometry page 11 -13 #12, 16 -23, 28 -29, 31 -32, 36 -40 page 18 -20 #16 -18, 27, 29 -38, 41 -44