2.5: Proofs about Angle Pairs and Segments
Learning Objectives
 Use theorems about special pairs of angles.
 Use theorems about right angles and midpoints.
Review Queue
Write a 2column proof
1. Given: is the angle bisector of .
is the angle bisector of .
Prove:
Know What? The game of pool relies heavily on angles. The angle at which you hit the cue ball with your cue determines if a) you hit the yellow ball and b) if you can hit it into a pocket.
The top picture on the right illustrates if you were to hit the cue ball straight on and then hit the yellow ball. The orange line shows the path that the cue ball and then the yellow ball would take. You notice that . With a little focus, you notice that it makes more sense to approach the ball from the other side of the table and bank it off of the opposite side (see lower picture with the white path). You measure and need to hit the cue ball so that it hits the side of the table at a angle (this would be ). and are called the angles of reflection. Find the measures of these angles and how they relate to and .
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 one 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 three.
This shortcut will now be used when applicable.
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 PoE 
4.  angles have = measures 
This theorem may seem redundant, but anytime right angles are mentioned, you 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.
So, if and , then . Using numbers to illustrate, we could say that if is supplementary to an angle measuring , then . is also supplementary to , so it too is . Therefore, . This example, however, does not constitute a proof.
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 PoE 
4.  Subtraction PoE 
5.  angles have = measures 
Example 1: Given that and and are right angles, show which angles are congruent.
Solution: By the Right Angle Theorem, . Also, by the Same Angles Supplements Theorem. and are a linear pair, so they add up to . and are also a linear pair and add up to . Because , we can substitute in for and then and are supplementary to the same angle, making them congruent.
This is an example of a paragraph proof. Instead of organizing the proof in two columns, you explain everything in sentences.
Same Angle Complements Theorem: If two angles are complementary to the same angle then the angles are congruent.
So, if and , then . Using numbers, we could say that if is supplementary to an angle measuring , then . is also supplementary to , so it too is . Therefore, .
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 formal proof here.
Given: Lines and intersect.
Prove: and
Statement  Reason 

1. Lines and intersect  Given 
2. and are a linear pair and are a linear pair and are a linear pair 
Definition of a Linear Pair 
3. and are supplementary and are supplementary and are supplementary 
Linear Pair Postulate 
4.  Definition of Supplementary Angles 
5.  Substitution PoE 
6.  Subtraction PoE 
7.  angles have = measures 
In this proof we combined everything. You could have done two separate proofs, one for and one for .
Example 2: In the picture and .
Each pair below is congruent. State why.
a) and
b) and
c) and
d) and
e) and
f) and
g) and
Solution:
a), c) and d) Vertical Angles Theorem
b) and g) Same Angles Complements Theorem
e) and f) 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 PoC 
Know What? Revisited If , then
. Draw a perpendicular line at the point of reflection and the laws of reflection state that the angle of incidence is equal to the angle of reflection. So, this is a case of the Same Angles Complements Theorem. because the angle of incidence and the angle of reflection are equal. We can also use this to find , which is .
Review Questions
Write a twocolumn proof for questions 110.
 Given: and Prove:
 Given: Prove:
 Given: and Prove:
 Given: is supplementary to , is supplementary to , Prove:
 Given: Prove:
 Given: and are right angles Prove:
 Given: Prove:
 Given: Prove:
 Given: Prove: and are complements
 Given: , Prove:
Use the picture for questions 1120.
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 two pairs of adjacent angles that are NOT linear pairs.
 What is the perpendicular bisector of ?
 List two bisectors of .
 List a pair of complementary angles.
 If is an angle bisector of , what two angles are congruent?
 Fill in the blanks for the proof below. Given: Picture above and Prove: is the angle bisector of
Statement  Reason 

1. is on the interior of 

2.  
3.  Angle Addition Postulate 
4.  Substitution 
5.  
6.  Division PoE 
7. 
For questions 2125, find the measure of the lettered angles in the picture below.
 (hint: is complementary to )
For questions 2635, find the measure of the lettered angles in the picture below. Hint: Recall the sum of the three angles in a triangle is .
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 