1. 1.5 Angle Relationships

2. Objectives Identify and use special pairs of angles Identify and use special pairs of angles Identify perpendicular lines Identify perpendicular lines

3. Pairs of Angles Adjacent Angles – two angles that lie in the same plane, have a common vertex and a common side, but no common interior points Adjacent Angles – two angles that lie in the same plane, have a common vertex and a common side, but no common interior points Adjacent Angles Adjacent Angles Vertical Angles – two nonadjacent angles formed by two intersecting lines Vertical Angles – two nonadjacent angles formed by two intersecting lines Vertical Angles Vertical Angles Linear Pair – a pair of adjacent angles whose noncommon sides are opposite rays Linear Pair – a pair of adjacent angles whose noncommon sides are opposite rays Linear Pair Linear Pair

4. Name two angles that form a linear pair. A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays. Answer: The angle pairs that satisfy this definition are Example 1a:

5. Name two acute vertical angles. There are four acute angles shown. There is one pair of vertical angles. Answer: The acute vertical angles are  VZY and  XZW. Example 1b:

6. Name an angle pair that satisfies each condition. a. two acute vertical angles b.two adjacent angles whose sum is less than 90 Answer:  BAC and  CAD or  EAF and  FAN Answer:  BAC and  FAE,  CAD and  NAF, or  BAD and  NAE Your Turn:

7. Angle Relationships C C oooo mmmm pppp llll eeee mmmm eeee nnnn tttt aaaa rrrr yyyy A A A A nnnn gggg llll eeee ssss – two angles whose measures have a sum of 90º S S uuuu pppp pppp llll eeee mmmm eeee nnnn tttt aaaa rrrr yyyy A A A A nnnn gggg llll eeee ssss – two angles whose measures have a sum of 180º Remember, angle measures are real numbers, so the operations for real numbers and algebra can apply to angles.

8. ALGEBRA Find the measures of two supplementary angles if the measure of one angle is 6 less than five times the other angle. ExploreThe problem relates the measures of two supplementary angles. You know that the sum of the measures of supplementary angles is 180. PlanDraw two figures to represent the angles. Example 2:

9. Let the measure of one angle be x. Solve Given Simplify. Add 6 to each side. Divide each side by 6. Example 2:

10. Use the value of x to find each angle measure. ExamineAdd the angle measures to verify that the angles are supplementary. Answer: 31, 149 Example 2:

11. ALGEBRA Find the measures of two complementary angles if one angle measures six degrees less than five times the measure of the other. Answer: 16, 74 Your Turn:

12. Perpendicular Lines Lines that form right angles are perpendicular. Lines that form right angles are perpendicular. perpendicular We use the symbol “ ┴ ” to illustrate two lines are perpendicular. ┴ is read “ is perpendicular to.” We use the symbol “ ┴ ” to illustrate two lines are perpendicular. ┴ is read “ is perpendicular to.” AB ┴ CD AB ┴ CD

13. Perpendicular Lines The following is true for all ┴ lines: 1. ┴ lines intersect to form 4 right angles. 2. ┴ lines intersect to form congruent adjacent angles. 3. Segments and rays can be ┴ to lines or to other segments and rays. 4. The right angle symbol ( ┐ ) indicates that lines are ┴.

14. ALGEBRA Find x so that. Example 3:

15. If, then m  KJH 90. To find x, use  KJI and  IJH. Substitution Add. Subtract 6 from each side. Divide each side by 12. Answer: Sum of parts whole Example 3:

16. ALGEBRA Find x and y so that and are perpendicular. Answer: Your Turn:

17. Assumptions in Geometry As we have discussed previously, we cannot assume relationships among figures in geometry. Figures are not drawn to reflect total accuracy of the situation, merely to provide or depict it. We must be provided with given information or be able to prove a situation from the given information before we can state truths about it. As we have discussed previously, we cannot assume relationships among figures in geometry. Figures are not drawn to reflect total accuracy of the situation, merely to provide or depict it. We must be provided with given information or be able to prove a situation from the given information before we can state truths about it.

18. The diagram is marked to show that From the definition of perpendicular, perpendicular lines intersect to form congruent adjacent angles. Answer: Yes; and are perpendicular. Determine whether the following statement can be assumed from the figure below. Explain. m  VYT 90 Example 4a:

19. Determine whether the following statement can be assumed from the figure below. Explain.  TYW and  TYU are supplementary. Answer: Yes; they form a linear pair of angles. Example 4b:

20. Determine whether the following statement can be assumed from the figure below. Explain.  VYW and  TYS are adjacent angles. Answer: No; they do not share a common side. Example 4c:

21. Determine whether each statement can be assumed from the figure below. Explain. a. b.  TAU and  UAY are complementary. c.  UAX and  UXA are adjacent. Answer: Yes; lines TY and SX are perpendicular. Answer: No; they do not share a common side. Answer: No; the sum of the two angles is 180, not 90. Your Turn:

22. Assignment Honors Geometry Pgs. 37 – 40 #1 – 30, 37, 40 Honors Geometry Pgs. 37 – 40 #1 – 30, 37, 40 Foundations Geometry Pgs. 40 – 41 #1 – 13, 22-28, 35 Foundations Geometry Pgs. 40 – 41 #1 – 13, 22-28, 35