Geometry Worksheet Angles by oza12622

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									Honors Geometry


          Lesson 2.6
      Deductive Reasoning
What You Should Learn
Why You Should Learn It
 Goal 1: How to identify a special
 angle relationships
 Goal 2: How to use deductive
 reasoning to verify angle relationships
 Deductive reasoning allows you to
 build a mathematical system and to
 solve real-life problems
Identifying Special Pairs of Angles

 Vertical Angles
   Two angles are vertical angles if
   their sides form two pairs of
   opposite rays

                    1

                   2
Identifying Special Pairs of Angles

 Linear pair
   Two adjacent angles are a linear
   pair if their noncommon sides are
   on opposite rays


                  1   2
Identifying Special Pairs of Angles

 Complementary Angles
   Two angles are complementary if
   the sum of their measures is 90°.
   Each angle is the complement of
   the other.



          60°
                30°
Identifying Special Pairs of Angles

 Supplementary Angles
   Two angles are supplementary if the
   sum of their measures is 180°. Each
   angle is the supplement of the other.




                   80°   100°
Example 1

 Use the terms vertical angles, linear
 pair, complementary angles &
 supplementary angles to describe the
 relationships between the labeled
 angles in the figure

                    3
                6       4
                    5
Example 1 Solution
 Vertical Angles: 3 and 5, 4 and 6
 Linear Pairs: 3 and 4, 4 and 5,
                      5 and 6, 6 and 3

 Complementary Angles: None
 Supplementary Angles: 3 and 4,            4 and 5,
                                 5 and 6, 6 and 3


              3
          6       4
              5
Postulate 11: Linear Pair Postulate

 If two angles form a linear
 pair, then they are
 supplementary, i.e., the sum
 of their measures is 180°
Deductive Reasoning
 To deduce means to reason from
 known facts
 When you prove a theorem, you are
 using deductive reasoning
   using the existing structure to deduce new
   parts of the structure
 In geometry, as in construction, new
 parts of the structure are built upon
 existing parts
Theorem 2.1:
Congruent Supplements Theorem
 If two angles are
 supplementary
 to the same
 angle or to
 congruent
 angles, then they
 are congruent
Proof of Theorem 2.1:
Congruent Supplements Theorem
Given: 1 is the supplement of 2
      3 is the supplement of 4
       2  4
Prove: 1  3
m1 + m2 = 180 Def. of Supplementary Angles
m3 + m4 = 180 Def. of Supplementary Angles
m1 + m2 = m3 + m4 Transitive Prop. of =
2  4 Given
m2 = m4      Def. of Congruence
m1 + m2 = m3 + m2 Subsitution Prop. of =
m1 = m3     Subtraction Property of Equality
1  3   Definition of Congruence
Theorem 2.2
Congruent Complements Theorem
 If two angles are
 complementary
 to the same
 angle or to
 congruent
 angles, then they
 are congruent
Proof of Theorem 2.2
Congruent Complements Theorem
Fill in the missing steps and missing reasons
Proof of Theorem 2.2 (Solution)
Congruent Complements Theorem
Theorem 2.3:
Vertical Angles Theorem
             If two angles are
             vertical angles, then
             they are congruent.

								
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