2-5 PROVING ANGLES CONGRUENT (p. 96-103)
There are a number of special angle pairs in geometry that are worth learning.
Vertical angles are a pair of angles whose sides form two pairs of opposite rays.
Example: Sketch two intersecting lines and identify two pairs of vertical angles.
Adjacent angles are a pair of coplanar angles that have a common side, a common vertex,
and no common interior points.
Example: Sketch a pair of adjacent angles that do not make up a straight angle. Then,
sketch a pair of adjacent angles that do make up a straight angle.
Complementary angles are two angles whose angle measures have a sum of 90. Each
angle is the complement of the other.
Example: Sketch a pair of complementary angles that make up a right angle. Then,
sketch a pair of complementary angles that are not adjacent.
Supplementary angles are two angles whose angle measures have a sum of 180. Each
angle is the supplement of the other.
Example: Sketch a pair of supplementary angles that make up a straight angle. Then,
sketch a pair of supplementary angles that are not adjacent.
Example:
E
H
D
3
2
1
6
A
4
C 5
B
G
In the above figure, ACH is a right angle and ACB is a straight angle.
Name a pair of vertical angles in the diagram.
Name two pairs of complementary angles in the diagram.
Name two pairs of supplementary angles in the diagram.
Do 1 a and b on p. 97.
What do you call two angles who say nice things to each other?
You can not assume many ideas from diagrams in this course. You can always conclude
that angles are adjacent, vertical, and supplementary (if they form a straight angle).
You can never conclude that angles are right angles, lines are parallel or perpendicular, or
segments and angles are congruent unless special symbols (such as tick marks) are used
in the diagram.
Do 2 on p. 97. Can you make these additional conclusions from the diagram in this
problem?
 TWQ is a right angle
 PWV and TWQ are vertical angles
 TWP and TWQ are supplementary
 PWV and QWV are congruent
The Investigation on Vertical Angles can be skipped.
How do you think vertical angles compare to each other?
We are about to prove a theorem. A theorem is simply a mathematical statement that can
be proved. When you prove something, you show that it is true beyond any doubt.
The given information represents information that you know up front. It is your starting
point for the proof. The statement after the word “Prove” represents what you must show
to be true when you are done working.
Theorem 2-1 Vertical Angles Theorem
Vertical angles are congruent.
A paragraph proof is a proof consisting of sentences within a paragraph. Writing a short
paragraph proof is like writing a short essay. When proving something, you proceed
from what you know to what you must show.
Examine Ex. 3 on p. 98. This is a paragraph proof for the Vertical Angles Theorem.
Note that the Angle Addition Postulate is used twice.
Do 3 on p. 98.
Example:
D
(2x+3) 
A
(4x-101)
B
G
E
Refer to the above figure. Set up and solve an equation to find the value of x. Then, find
the four angle measures in the diagram.
Theorem 2-2 Congruent Supplements Theorem
If two angles are supplements of the same angle (or of congruent angles), then the
two angles are congruent.
Shorter form: Supplements of the same angle are congruent. Supplements of
congruent angles are congruent.
Examine the paragraph proof for Theorem 2-2 on p. 99.
Example: Illustrate Theorem 2-2 with 3 angles and their specific measures. Do the
same for 4 angles and their specific measures.
Example: 1 and 2 are supplementary. Also, m 1 is one eighth m 2. Set up and
solve an equation to find the value of x. Then, find the measures of the two angles.
The next three theorems are presented without proof in our book. You will prove some
of these theorems in your homework problems.
Theorem 2-3 Congruent Complements Theorem
If two angles are complements of the same angle (or of congruent angles), then
the two angles are congruent.
Shorter form: Complements of the same angle are congruent. Complements of
congruent angles are congruent.
Example: Illustrate Theorem 2-3 with 3 angles and their specific measures. Do the
same for 4 angles and their specific measures.
Theorem 2-4 All right angles are congruent.
Theorem 2-5 If two angles are congruent and supplementary, then each is a right angle.
Homework p. 100-103: 1,4,5,7,8,10,13,15,18,19,26,30,33,35,37,42,43,50,62,69,70,74
37a. For example, (0,4)
b. (3,-1)