Remember from Lesson 1 that a ray has one fixed end and extends indefinitely in one direction. For example YV!!!"
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1 Lesson 3 Lesson 3, page 1 of 1 Glencoe Geometry Chapter 1.6 & 1.7 Angles: Exploration & Relationships By the end of this lesson, you should be able to 1. Identify angles and classify angles. 2. Use the Angle Addition Postulate to find the measure of angles. 3. Identify and use congruent angles and the bisector of an angle. 4. Identify and use special pairs of angles. 5. Identify your favorite Math television program host. Remember from Lesson 1 that a ray has one fixed end and extends indefinitely in one direction. For example YV!!!" in the figure at right. Since direction matters, YV!!!"!!" and YZ are called opposite rays, but they share a common endpoint. Opposite rays are always collinear. An angle is usually formed by two non-collinear rays with a common endpoint. The common endpoint is called the vertex. Give some names for the angle at right: Q, RQS, SQR, or 3, But NOT RS!!
2 Lesson 3, page 2 of 2 Notice in the last diagram, there was only one angle. You must be more careful when naming different angles that share a common vertex. In the diagram below, you CANNOT name either of the angle as just B!!! What are some names? ABC, CBA CBD, DBC DBE, EBD!!!" A!! ng le ABE or EBA is called a straight angle, since BA and BE!" are opposites. An angle separates a plane into three distinct parts: 1. The interior of the angle. A 2. The exterior of the angle. B 3. and the angle itself. X We typically measure angles in degrees using a protractor. *All angles this year will be in degrees. The degree symbol is sometimes used, but without it, we infer that the measure is still in degrees: # 85 = 85 Using the inner scale, we can say that the degree measure of ABC is 60, or equivalently, m ABC = 60
3 By the Angle Addition Postulate, in the figure below, m HIJ + m JIK = m HIK Duhhhh!! So, what is m HIJ if m JIK = 45 # and # m HIK =100? m HIJ + m JIK = m HIK m HIJ + 45 = 100 # # m HIJ = = 55 # # # We can also classify individual angles by their measures: Lesson 3, page 3 of 3 right angle = 90 acute angle < 90 obtuse angle > 90 straight angle = Congruent angles have the same measure. Which of the angles above are congruent to all others in the same class? right angles and straight angles Two angles that add to 180 are said to be supplementary angles. Two angles that add to 90 are said to be complementary angles.
4 An angle bisector is a ray that divides and angle into two congruent angles. Lesson 3, page 4 of 4!!!" If GD bisects CGE, which angle is congruent to CGD? EGD What other angle is congruent to CGE? FGE When two lines intersect, they form four angles. When they intersect to form four right angles, we say the lines are perpendicular, and denoted by the symbol Not all lines are perpendicular to each other, though. hics/perpendicular.jpg When two lines intersect, it is useful to classify angles by their relationship to other angles. Adjacent Angles have a common vertex and a common side with no common interior points Ex) 1& 2, 2 & 3, 3& 4, 4 & 1 Vertical Angles non-adjacent angles across from each other. Vertical angles are congruent!!! Ex) 1& 3, 2& 4 Linear Pair adjacent angles formed by opposite rays. Linear pairs will always be supplements of each other. Which angle above are linear pairs? All the adjacent angles
5 If m MYZ = 160, what is m MYW? MYW + WYZ = MYZ ( x) ( x ) = 160 6x + 10= 160 6x = x = = 25 6 Name two angles that are adjacent to WTV. A. 1 and 2 B. 2 and 3 Lesson 3, page 5 of 5 C. WTV and 3 D. 1 and 3 If m 1 = 2x and m 2 = 4x. Find the value of x if 1 and are complementary. m + 1 m 2= 2x+ 4x= 90 6x = m ( ) m ( ) 1= 2 15 = 30 2 = 4 15 = 60 2 x 90 = = 15 6
6 Find the value of x. 2x + 40 These are vertical angles so they are congruent. 2x+ 40= 4x+ 8 2x 4x= x = 32 x = 16 4x + 8 Lesson 3, page 6 of 6 If m ABC = 180 and x = 12, what is m GBH? m ABG + m GBH + m HBC = 180 ( x ) m GBH ( x ) = m GBH + 50 = m GBH = 180 m GBH = = 30
7 Circle the right Answer: Say What??!! Lesson 3, page 7 of 7 1. Angles are measured in units called (sides) or (degrees). 2. In Figure 1, 2and 3 are (complementary) or (supplementary) angles. 3. A (compass) or (protractor) is used to find the measure of an angle. 4. In Figure 2, the two angles shown are (supplementary) or (congruent) angles 5. In Figure 3, 5 and 6 are (vertical) or (adjacent) angles. 6. Perpendicular lines intersect to form (obtuse) or (right) angles. 7. In Figure 3, A is called (a side) or (the vertex) of In Figure 1,!!! 1 and 4 form a (linear pair) or (right angle). 9. In Figure 4, KM!" is the (vertex) or (bisector) of JKL. Figure 1 Figure 2 Figure 3 Figure 4 *unless otherwise noted, all images are created using TI-Interactive Software or from
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