Conjectures that lead to Conjectures that lead to Theorems 2.5Theorems 2.5

DefinitionDefinition Vertical angles are the opposite Vertical angles are the opposite

angles formed by two intersecting angles formed by two intersecting lines. lines.

1 and 3 are vertical angles 2 and 4 are vertical angles

Vertical Angles TheoremVertical Angles Theorem

If two angles form a pair of If two angles form a pair of vertical angles, then they are vertical angles, then they are congruentcongruent..

1 3

2 4

Prove the vertical angle Prove the vertical angle theoremtheorem

Given: 1 and 2 are vertical angles

Prove: 1 is congruent to 2 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

3

41 2

Identify the vertical angles in the figure.

1. 1 and _____ 2. 2 and _____ 3. 3 and _____ 4. 4 and _____ 5. 5 and _____ 6. 6 and _____ _____

1. 1. 2.2.

3.3. 4. 4.

130°

x°

5x° 25°

x° 40°(x – 10)°

125 °

Find the value of x.

Find the value of x.

Find the value of x.

5y – 50

4y – 10

What type of angles

are these?

5y – 50 = 4y – 10 y = 40

Plug y back into our angle equations and we get

150

What is the measure of the angle?

1

23

4

5

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.

Example:

1 and 2

ADJACENT

1

23

4

5

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.

Example:

VERTICAL

1 and 4

1

23

4

5

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.

Example:

ADJACENT,

COMPLEMENTARY

3 and 4

1

23

4

5

Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair.

Example:

ADJACENT,

SUPPLEMENTARY,

LINEAR PAIR

1 and 5

Find x, y, and z.

Example:

51xy

z

x = 129,

y = 51,

z = 129

Find x.

Example:

x = 8

( (5 3x x - 15) = + 1) 5 15 3 1x x 2 15 1x 2 16x

(3x + 1)

L

P AT

O

(5x - 15) (20x - 5)

Find

Example:

155

m LAT(3x + 1)

L

P AT

O

(5x - 15) (20x - 5)

Since we have already found the value of x, all we need to do now is to

plug it in for LAT.

20 5 20 8 5x ( )160 5

4 17 2 9x x

13x

7 2 3 8 180x x

17x

Example:

Find the value of x.

Answer the questions for each figure

4b. Are 3 and 5 vertical angles?

3b. Are 1 and 4 vertical angles?

2b. Are 1 and 2 a linear pair?

1b. Are 1 and 5 a linear pair?

4a. Are 2 and 4 vertical angles?

3a. Are 1 and 4 a linear pair?

2a. Are 1 and 3 vertical angles?

1a. Are 1 and 2 a linear pair?

NO

YES

NO

NO

NO

NO

YES

YES

If 2 angles are supplementary to the same angle,If 2 angles are supplementary to the same angle,

then they are congruent.then they are congruent.

If If 1 & 1 & 2 are supplementary,2 are supplementary,

and and 2 & 2 & 3 are supplementary, 3 are supplementary,

then then 1 1 3.3.

1 2 3

Congruent Supplements Theorem

Congruent Complements Theorem

If 2 angles are complementary to the same If 2 angles are complementary to the same angle, angle,

then they are congruent.then they are congruent.

1 2 3

If If 1 & 1 & 2 are complementary, 2 are complementary,

and and 2 & 2 & 3 are complementary, 3 are complementary,

then then 1 1 3.3.

Right Angle Right Angle Congruence TheoremCongruence Theorem

All right angles are congruent.

90 90

DefinitionsDefinitions 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.

Find the measure of each Find the measure of each angle.angle.

12

3

456

B

G V

F

A

C

E

8

60

AssignmentAssignment

Geometry:Geometry:

2.5B and 2.5C2.5B and 2.5C

Section 10 - 20Section 10 - 20