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1-4 & 1-5 Angles Measures and Relationships

1-4 & 1-5 Angles Measures and Relationships. Objectives: The student will be able to: Measure and classify angles. Use congruent angles and the bisector of an angle. Identify and use special pairs of angles. Identify perpendicular lines. Classifying Angles.

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1-4 & 1-5 Angles Measures and Relationships

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  1. 1-4 & 1-5 Angles Measures and Relationships Objectives: The student will be able to: Measure and classify angles. Use congruent angles and the bisector of an angle. Identify and use special pairs of angles. Identify perpendicular lines.

  2. Classifying Angles Acute Angles < 90° Right Angles = 90° Obtuse Angles > 90° Naming Angles: When naming angles using 3 letters, the vertex must be the second of the 3 letters. You can name an angle using a single letter only when there is exactly one angle located at the vertex.

  3. Naming angles.

  4. Congruent Angles & Angle Bisector: A ray that divides an angle into two congruent angles. PQS ≅TQS The bisector of PQT is QS . In the figure, QS is the angle bisector of . Point S lies in the interior of and . If and , find the value of x. 50 = 4x + 14 -14-14 36 = 4x 9 = x

  5. Example: In the figure, QS is the angle bisector of . Point S lies in the interior of and . If and , find the value of . 6x - 2 = 3x + 13 -3x + 2 3x = 15 x = 5 NO! Did we answer the question?

  6. If and , find the value of . 6(5) – 2 + 3(5) + 13 = PQT 30 – 2 + 15 + 13 = PQT 56° = PQT

  7. 1 2 Special Angle pairs 3 4 5 6 7 8 Adjacent Angles: Vertical Angles: Linear Pair: Two angles that lie in the same plane and have a common vertex and a common side, but no common interior points. 1 & 2,1 & 3, 2&4, 3&4, 5 &6, 5 &7, 6 &8, 7 & 8 Two angles that are opposite angles. Vertical angles are congruent. 1  4, 2  3, 5  8, 6  7 Supplementary angles that form a line (sum = 180) 1 & 2 ,2 & 4 , 4 &3, 3 & 1, 5 & 6,6 & 8, 8 & 7, 7 & 5

  8. Special Angle pairs E Congruent Angles: Perpendicular angles: Two or more angles that have the same measure. •  AEB &  BEC,CED &DEA, • AEB &DEC, BEC &AED Lines, segments, and rays that form right angles (90 degrees).  AEC &  BED

  9. Complementary & Supplementary Angles Complementary Angles: Supplementary Angles: Two angles whose measures have a sum of 90°. A + B = 30 + 60 = 90 Two angles whose measures have a sum of 180°. F + G = 120 + 60 = 180

  10. Identify: Two Obtuse vertical angles: Two acute adjacent angles: An angle supplementary to TNU:

  11. Find x so that . If the two angles are perpendicular they MUST = 90° . (9x + 5) + (3x + 1) = 90 12x + 6 = 90 - 6-6 12x = 84 x = 7

  12. Example: Find the measures of 2 supplementary angles if the difference in their measures is 18. Are we through? NO!! = 180 x + (x – 18) = 180 If x = 99, what are the measures of the supplementary angles? 2x – 18 = 180 +18 +18 2x = 198 x = 99 99 99 -18 = 81 How can I check to see if that’s correct? 99 + 81 = 180

  13. Find x and y so that KOand HM are perpendicular. 1. Find x by setting the two angles equal to 90. 2. Vertical angels tell us if , then . 3. Find y by setting . 90 90 (3x + 6) + (9x) = 90 (3y + 6) = - 6-6 12x + 6 = 90 3y = 84 - 6-6 y = 28 12x = 84 x = 7

  14. 1. Are the angles congruent? • Yes – set the expressions equal to each other. • A = B • 2. Do the angles add up to 90°? • Yes – add the expressions and set them equal to 90°. • A + B = 90 • 3. Do the angles add up to 180°? • Yes – add the expressions and set them equal to 180°. • A + B = 180 • Do the angles add up to some other value given in the problem? • Yes – add the expressions and set them equal to the value. • A + B = other value

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