1. 1 1-4 & 1-5 Angles Measures and Relationships Objectives: The student will be able to: 1.Measure and classify angles. 2.Use congruent angles and the bisector of an angle. 3.Identify and use special pairs of angles. 4.Identify perpendicular lines.

2. Classifying Angles Acute Angles 90° 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. Naming Angles:

3. Naming angles. 3

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

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

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

7. Special Angle pairs Adjacent Angles: Vertical Angles: Linear Pair:  1   4,  2   3,  5   8,  6   7 Two angles that are opposite angles. Vertical angles are congruent.  1 &  2,  2 &  4,  4 &  3,  3 &  1,  5 &  6,  6 &  8,  8 &  7,  7 &  5 Supplementary angles that form a line (sum = 180  ) 12 34 56 78 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

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

9. Complementary & Supplementary Angles Complementary Angles: Supplementary Angles: Two angles whose measures have a sum of 90°. Two angles whose measures have a sum of 180°.  A +  B = 30 + 60 = 90  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. = 180x +(x – 18) = 180 2x – 18 = 180 +18 2x = 198 x = 99 Example: Find the measures of 2 supplementary angles if the difference in their measures is 18. Are we through?NO!! If x = 99, what are the measures of the supplementary angles? 9999 -18 = 81 99 + 81 = 180 How can I check to see if that’s correct?

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

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 4.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