1. Conjectures that lead to Theorems 2.5

2. Definition
Vertical angles are the opposite angles formed by two intersecting lines.
1 and 3 are vertical angles
2 and 4 are vertical angles

3. Vertical Angles Theorem
If two angles form a pair of vertical angles, then they are congruent.

4. Prove the vertical angle theorem3 1 2
Given: 1 and 2 are vertical angles
Prove: 1 is congruent to 2 4
Statement Reason
1. 1 and 2 are vertical angles
1. Given
2. 1 + 3 = 180° ,
2 + 3 = 180°
2. Linear Pair Property
3. 1 + 3 = 2 + 3
3. Substitution Property of Equality
4. 1 = 2
4. Subtraction Property of Equality
5. 1 is congruent to 2
5. Definition of Congruent

5. Identify the vertical angles in the figure.
1. 1 and _____ 2 and _____
3. 3 and _____ 4 and _____
5. 5 and _____ 6 and _____

6. Find the value of x. 1. 2. 3. 4. 130° 5x° 25° x° 125 ° x° 40°
130°
5x°
25° x°
125 ° x°
40°
(x – 10)°

7. Find the value of x.

8. Find the value of x.

9. What is the measure of the angle?
5y – 50
What type of angles are these?
4y – 10
5y – 50 = 4y – 10
y = 40
Plug y back into our angle equations and we get

10. Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.
Example: 3 2 4 1
ADJACENT 5

11. Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.
Example: 3 2 4 1
VERTICAL 5

12. Example: ADJACENT, COMPLEMENTARY
Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.
Example: 3 2 4 1
ADJACENT,
COMPLEMENTARY 5

13. Example: ADJACENT, SUPPLEMENTARY, LINEAR PAIR
Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.
Example: 3 2 4
ADJACENT,
SUPPLEMENTARY,
LINEAR PAIR 1 5

14. Find x, y, and z.
Example:
x = 129,
y = 51,
z = 129

15. L P A T O
Example:
Find x.
x = 8

16. L P A T O
Example:
Find
Since we have already found the value of x, all we need to do now is to plug it in for LAT.
155

17. Example:
Find the value of x.

18. Answer the questions for each figureNO
1b. Are 1 and 5 a linear pair?
YES
1a. Are 1 and 2 a linear pair?
2b. Are 1 and 2 a linear pair? NO
2a. Are 1 and 3 vertical angles? NO
3b. Are 1 and 4 vertical angles?
YES
3a. Are 1 and 4 a linear pair?
YES NO
4b. Are 3 and 5 vertical angles? NO
4a. Are 2 and 4 vertical angles?

19. Congruent Supplements Theorem
If 2 angles are supplementary to the same angle,
then they are congruent.
If 1 & 2 are supplementary,
and 2 & 3 are supplementary,
then 1  3.

20. Congruent Complements Theorem
If 2 angles are complementary to the same angle,
then they are congruent.
If 1 & 2 are complementary,
and 2 & 3 are complementary,
then 1  3.

21. Right Angle Congruence Theorem
All right angles are congruent.
90 

22. Definitions
Inductive Reasoning: The process of forming conjectures based on observations or experiences.
Deductive Reasoning: The process of drawing conclusions by using logical reasoning in an argument.

23. Find the measure of each angle.1 2 3 4 5 6 B G V F A C E 8
60

24. Assignment
Geometry:
2.5B and 2.5C
Section