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Simplifying Absolute Value Expressions - Measuring Segments and Angles

Learn how to simplify absolute value expressions and measure segments and angles using postulates and addition postulates. Find lengths of segments and determine the measures of angles.

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Simplifying Absolute Value Expressions - Measuring Segments and Angles

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  1. Bellringer • Simplify each absolute value expression • 3 • 6 • 2

  2. 1.4 Measuring Segments and Angles

  3. Postulate 1-5Ruler Postulate • The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers. • Length of =

  4. Congruent ( Segments • Two segments with the same length

  5. Comparing Segments lengths • Find AB and BC • AB = • BC = So AB = BC or A B D E C

  6. Postulate 1-6Segment Addition Postulate • If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC

  7. DS + ST = DT Using the Segment Addition Postulate • If DT = 60, find the value of x. Then find DS and ST. • Segment Addition Postulate • Substitution • Simplify • Add 20 to each side • Divide each side by 5 • DS + ST = DT • (2x – 8) + (3x – 12) = 60 • 5x – 20 = 60 • 5x = 80 • x = 16 2x - 8 3x - 12 T D S • Substitute x = 16 • DS = 2x – 8 = 2(16) – 8 = 24 • ST = 3x – 12 = 3(16) – 12 = 36

  8. Using the Segment Postulate • If DT = 100. Find the value of x. Then find DS and ST. • x = 15 • DS = 40 • ST = 60 4x - 20 2x + 30 T D S

  9. Midpoint • a point that divides a segment into two congruent segments. A midpoint, or any line, ray, or other segment through a midpoint, is said to bisectthe segment.

  10. Finding Lengths • C is the midpoint of Find AC, CB, and AB • Definition of Midpoint • Substitution • Add 4 to each sides • Subtract 2x from each side • AC = CB • 2x +1 = 3x - 4 • 2x + 5 = 3x • 5 = x 2x + 1 3x - 4 C B A • AC = 2x + 1 = 2(5) + 1 = 11 • CB = 3x – 4 = 3(5) – 4 =11 • AC + CB = AB 11 + 11 = 22

  11. Finding Lengths • Z is the midpoint , and XY = 30. Find XZ. • XZ = • XZ = 30 Z Y X

  12. Homework • Pg. 29 #’s 1 – 4, 8-15

  13. Bellringer • If XT = 12 and XZ = 21, then TZ = 9 T Z X

  14. 1.4 Continued Measuring Segments and Angles

  15. Angle() • Is formed by two rays with the same endpoint. The rays are the sides of the angle. The endpoint is the vertex of the angle.

  16. Naming Angles • Name a two other ways • 1. BAC • 2.CAB • Name two other ways • 1. • 2.DCB There are three ways to name an angle 1. Counterclockwise (CBA)2. Clockwise (ABC)3. The vertex (B)

  17. Classifying Angles • Def: Acute Angle : angle whose measure is 090. • Def: Right Angle (: angle whose measure is 90 • Def: Obtuse Angle: angle whose measure is 90. • Def: Straight Angle: angle whose measure is 180.

  18. Postulate 1-8Angle Addition Postulate • If point B is in the interior of AOC, then mAOB + mBOC = mAOC • If AOB is a straight angle, then mBOC + COA = 180

  19. Using the Angle Addition Postulate • What is mCOA if mBOC = 50 and the m.

  20. Using the Angle Addition Postulate • If mCOA = 145, find the m.

  21. Congruent Angles • Angles with the same measure. In other words, if m2, then 1 2.

  22. Homework • Pg. 30 #’s 16-28 all

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