Lesson 1-5 Angle Relationships

5-Minute Check on Lesson 1-4 Transparency 1-5 5-Minute Check on Lesson 1-4 G Refer to the figure for questions 1 through 5. Name the vertex of 3. Name a point in the interior of ACB. Name the sides of BAC Name an acute angle with vertex B If BD bisects ABC, m ABD = 2x + 5 and m DBC = 3x – 16, find m ABD. If P is in the interior of MON and m MOP = ½ m MOP, what can you conclude? B A D 3 C Standardized Test Practice: PON  NOM MON is an acute angle A B m MOP > m PON OP is the angle bisector of MON C D Click the mouse button or press the Space Bar to display the answers.

5-Minute Check on Lesson 1-4 Transparency 1-5 5-Minute Check on Lesson 1-4 G Refer to the figure for questions 1 through 5. Name the vertex of 3. Name a point in the interior of ACB. Name the sides of BAC Name an acute angle with vertex B If BD bisects ABC, m ABD = 2x + 5 and m DBC = 3x – 16, find m ABD. If P is in the interior of MON and m MOP = ½ m MOP, what can you conclude? B C A G D 3 BA, AC C ABD or DBC 2x + 5 = 3x – 16 x = 21 m ABD = 47 Standardized Test Practice: PON  NOM MON is an acute angle A B m MOP > m PON OP is the angle bisector of MON C D Click the mouse button or press the Space Bar to display the answers.

Objectives Identify and use special pairs of angles Identify perpendicular lines

Vocabulary Adjacent angles – two coplanar angles that have a common vertex, a common side, but no common interior points Linear pair – a pair of adjacent angles whose noncommon sides are opposite rays (always supplementary) Vertical angles – two non adjacent angles formed by two intersecting lines Vertical angles are congruent (measures are equal)!! Complementary Angles – two angles whose measures sum to 90° Supplementary Angles – two angles whose measures sum to 180° Perpendicular – two lines or rays are perpendicular if the angle (s) formed measure 90°

Interior of angle AVB or V Angles 360º A Circle Exterior of angle Ray VA Interior of angle AVB or V Vertex (point V) V Ray VB B Angles measured in degrees A degree is 1/360th around a circle Acute Right Obtuse A A A mA < 90º mA = 90º 90º < mA < 180º Names of angles: Angles have 3 letter names (letter on one side, letter of the vertex, letter on the other side) like AVB or if there is no confusion, like in most triangles, then an angle can be called by its vertex, V

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 5-1a

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 5-1b

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 Example 5-1c

Plan Draw two figures to represent the angles. 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 5-2a

Let the measure of one angle be x. Solve Given Simplify. Add 6 to each side. Divide each side by 6. Example 5-2b

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 Example 5-2c

ALGEBRA Find x and y so that and are perpendicular. AD  CE  4 right angles sum of parts = whole 4x° + 5x° = 90° 9x° = 90° x = 10° (7y – 15)° = 90° 7y° = 105° y = 15° Answer: x = 10° y = 15° Example 5-3c

Summary & Homework Summary: Homework: There are many special pairs of angles such as adjacent angles, vertical angles, complementary angles, supplementary angles, and linear pairs. Homework: pg 41-43; 11-16, 21, 32-33, 44-49