2.5: Proofs about Angle Pairs and Segments
Learning Objectives
 Use theorems about pairs of angles, right angles and midpoints.
Review Queue
Fill in the 2column proof.
1. Given: is the angle bisector of .
is the angle bisector of .
Prove:
Statement  Reason 

1.  Given 
2.  Definition of an Angle Bisector 
3.  
4. 
Know What? The game of pool relies heavily on angles. The angle at which you hit the cue ball with your cue determines if you hit the yellow ball and if you can hit it into the pocket.
The best path to get the yellow ball into the corner pocket is to use the path in the picture to the right. You measure and need to hit the cue ball so that it hits the side of the table at a angle (this would be ). Find and how it relates to .
If you would like to play with the angles of pool, click the link for an interactive game. http://www.coolmathgames.com/0poolgeometry/index.html
Naming Angles
As we learned in Chapter 1, angles can be addressed by numbers and three letters, where the letter in the middle is the vertex. We can shorten this label to only the middle letter if there is only one angle with that vertex.
All of the angles in this parallelogram can be labeled by one letter, the vertex, instead of all three.
Right Angle Theorem: If two angles are right angles, then the angles are congruent.
Proof of the Right Angle Theorem
Given: and are right angles
Prove:
Statement  Reason 

1. and are right angles  Given 
2. and  Definition of right angles 
3.  Transitive 
4.  angles have = measures 
Anytime right angles are mentioned in a proof, you will need to use this theorem to say the angles are congruent.
Same Angle Supplements Theorem: If two angles are supplementary to the same angle then the angles are congruent.
Proof of the Same Angles Supplements Theorem
Given: and are supplementary angles. and are supplementary angles.
Prove:
Statement  Reason 

1. and are supplementary and are supplementary 
Given 
2.

Definition of supplementary angles 
3.  Substitution 
4.  Subtraction 
5.  angles have = measures 
Example 1: and and are right angles.
Which angles are congruent and why?
Solution: By the Right Angle Theorem, . Also, by the Same Angles Supplements Theorem because and they are linear pairs with these congruent angles.
Same Angle Complements Theorem: If two angles are complementary to the same angle then the angles are congruent.
The proof of the Same Angles Complements Theorem is in the Review Questions. Use the proof of the Same Angles Supplements Theorem to help you.
Vertical Angles Theorem Recall the Vertical Angles Theorem from Chapter 1. We will do a proof here.
Given: Lines and intersect.
Prove:
Statement  Reason 

1. Lines and intersect  Given 
2. and are a linear pair and are a linear pair 
Definition of a Linear Pair 
3. and are supplementary and are supplementary 
Linear Pair Postulate 
4.

Definition of Supplementary Angles 
5.  Substitution 
6.  Subtraction 
7.  angles have = measures 
You can also do a proof for , which would be exactly the same.
Example 2: In the picture and .
Each pair below is congruent. State why.
a) and
b) and
c) and
d) and
Solution:
a) and
b) Same Angles Complements Theorem
c) Vertical Angles Theorem
d) Vertical Angles Theorem followed by the Transitive Property
Example 3: Write a twocolumn proof.
Given: and
Prove:
Solution:
Statement  Reason 

1. and  Given 
2.  Vertical Angles Theorem 
3.  Transitive 
Know What? Revisited If , then .
Draw a perpendicular line at the point of reflection. The laws of reflection state that the angle of incidence is equal to the angle of reflection (see picture). This is an example of the Same Angles Complements Theorem.
Review Questions
Fill in the blanks in the proofs below.
 Given: and Prove:
Statement  Reason 

1.  
2.  
3.  lines create right angles 
4.


5.


6.  Substitution 
7.  
8.  Substitution 
9.  Subtraction 
10. 
 Given: Prove:
Statement  Reason 

1.  
2.  angles have = measures 
3.  Angle Addition Postulate 
4.  Substitution 
5.  
6.  angles have = measures 
 Given: and Prove:
Statement  Reason 

1.  
2.  lines create right angles 
3.


4.  Angle Addition Postulate 
5.  Substitution 
6.  
7.  Subtraction 
8.  angles have = measures 
 Given: is supplementary to is supplementary to Prove:
Statement  Reason 

1.  
2.  
3.  Definition of supplementary angles 
4.  Substitution 
5.  Substitution 
6.  Subtraction 
7. 
 Given: Prove:
Statement  Reason 

1.  
2.  
3.  Definition of a Linear Pair 
4. and are supplementary and are supplementary 

5.  Definition of supplementary angles 
6.  
7.  
8.  
9. 
 Given: and are right angles Prove:
Statement  Reason 

1.  
2.  
3.  
4. 
 Given: Prove:
Statement  Reason 

1.  
2. and are right angles  
3. 
 Given: Prove:
Statement  Reason 

1.  
2. and are a linear pair  
3.  Linear Pair Postulate 
4.  Definition of supplementary angles 
5.  Substitution 
6. 
 Given: Prove: and are complements
Statement  Reason 

1.  
2.  lines create right angles 
3.  
4. and are complementary 
 Given: and Prove:
Statement  Reason 

1.  
2.  
3.  
4.  
5. 
Use the picture for questions 1121.
Given: is the midpoint of and
is the midpoint of
is the midpoint of
 List two pairs of vertical angles.
 List all the pairs of congruent segments.
 List two linear pairs that do not have as the vertex.
 List a right angle.
 List a pair of adjacent angles that are NOT a linear pair (do not add up to ).
 What line segment is the perpendicular bisector of ?
 Name a bisector of .
 List a pair of complementary angles.
 If is an angle bisector of , what two angles are congruent?
 Find .
For questions 2125, find the measure of the lettered angles in the picture below.
 a
 b
 c
 d
 e (hint: is complementary with )
Review Queue Answers
1.
Statement  Reason 

1. is an bisector of is an bisector of 
Given 
2.

Definition of an angle bisector 
3.  Transitive Property 