1.5 Angle Relationships

Objectives Identify and use special pairs of angles Identify perpendicular lines

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 Vertical Angles – two nonadjacent angles formed by two intersecting lines Linear Pair – a pair of adjacent angles whose noncommon sides are opposite rays  

Example 1a: 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 1b: 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.

Your Turn: 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 FAE, CAD and NAF, or BAD and NAE Answer: BAC and CAD or EAF and FAN

Angle Relationships Complementary Angles – two angles whose measures have a sum of 90º Supplementary Angles – 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.

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

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

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

Your Turn: 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

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

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 ┴.

Example 3: ALGEBRA Find x so that .

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

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

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.

Example 4a: Determine whether the following statement can be assumed from the figure below. Explain. mVYT 90 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.

Example 4b: 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 4c: 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.

Your Turn: 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; the sum of the two angles is 180, not 90. Answer: No; they do not share a common side.

Assignment Geometry Pg. 42 #11 – 35, omit #29 and #30 Pre-AP Geometry Pg. 42 #11 - 35