Transcript Congruent Supplements and Complements

Congruent Supplements and
Complements
Section 2.4
By: Ellen Wildner
Objective or Goal
• After studying this section you will be able
to…
– Prove angles congruent by using four new theorems that
you will learn in this section
Review
• Complementary Angles
– 2 angles are complementary if the sum of
their measures is 90 
For Ex.
1  56  therefore  2 must equal…
90  56  34 
Review Continued
• Supplementary Angles
– 2 angles are supplementary if the sum of their
measures is180 
For Ex.
1  112  therefore  2 must equal…
180   112   68 
Now we begin…Theorem 4
Theorem 4: If angles are supplementary to the same angle, then they
are congruent.
For ex. In the diagram
to the right 1 is
supplementary to A ,
and  2 is also
supplementary to  A
1
A
2
Therefore we
can prove that…
1   2
Sample Problem Thm 4.
Given: 1 is supp. to  2
3 is supp. to  4
1   4
1
4
3
Prove:  2  3
Statements
Reasons
1. 1 is supp. to  2
2. 3 is supp. to  4
1. Given
2. Given
1   4
 2  3
3.Given
3.
4.
2
4. If angles are
supplementary to 
angles they are 
Theorem 5
Theorem 5: If angles are supplementary to congruent angles, then they
are congruent.
For ex. It is given
that
 A is supp. to  B
C is supp. toD
And that B D
B

A
C
D
Therefore we
can prove that…
A
 C
Sample Problem Thm. 5
Given:  5 supp. of  6
 7 supp. of 8
 7
Conclusion:  5  8
6
Statements
5
6
1. Given
2. 7 supp. of
8
2. Given
4.
 7
 5  8
6
7
8
Reasons
1. 5 supp. of
3.
6
3. Given
4. If 2 angles are supp. To
angles they are


Theorem 6
Theorem 6: If angles are complementary to the same angle, then they
are

For ex. It is given
that  A comp  B
and  A comp C
C
A
B
Therefore we
can prove that…
B
 C
Sample Problem Thm. 6
Given: 1 comp  2
 2 comp3
Conclusion: 1
1
2
3
 3
Statements
1. 1 comp  2
2. 2 comp 3
3
3. 1

Reasons
1. Given
2. Given
3. If angles are complementary to the
same angle, then they are 
Theorem 7
Theorem 7: If angles are complementary to congruent angles, then
they are congruent.
For ex. It is given
that 1 comp  2
and3 comp  4
1
3
4
2
Therefore we
can prove that…
3

1
Sample Problem Thm. 7
Given: 1 comp to  2
3 comp to  4
1   3
Conclusion:  2
 4
Statements
1. 1 comp to  2
2.3 comp to  4
3. 1   3
4.  2   4
1
4
2
3
Reasons
1. Given
2. Given
3. Given
4. If angles are complementary to
congruent angles, then they are
congruent.
Practice Problems
Given: E is comp to G
HFG is comp to G
E
H
Conclusion:HFG  E
F
Statements
Reason
1. E is comp to HFG
1. Given
2.HFG is comp to
2. Given
3. E  HFG
G
3. If angles are complementary to the
same angle they are congruent
Practice Problems
Applying skills
Given:
is comp to 3
 4 = 131 
2
1
8
a. 3 = 49
b.  6 = 131
c. 5 = 49
d.  2 = 41
e. 1 = 139
f. 8 = 41
g.  7 = 139
2
7
4
3
6
5
Practice Problems
G
H
Given: Diagram as shown
Prove: HFE  GFJ
F
E
J
Statements
1. Diagram as shown
Reasons
1. Given
2.EFG is a straight 
2. Assumed from diagram
3.HFE is supp to HFG
3. If 2 angles form a straigt angle they are supp
4.HFJ is a straight 
4. Same as 2
5.GFJ is supp to HFG
5. Same as 3
6.HFE  GFJ
6. If angles are supplementary to the same
angle, they are congruent
Practice Problems
Given:1 is supp to 3
 2 is supp to  4
1   2
Conclusion: 3   4
2
1
3
4
Works Cited
"Complementary, Supplementary, and Vertical Angles." Algrebra Labs.
29 May 2009
<http://www.algebralab.org/lessons/lesson.aspx?file=Geometry
_AnglesComplementarySupplementaryVertical.xml>.
Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for
Enjoyment and Challenge. Boston: McDougal Littell, 1997. 7681.