conjectures that lead to theorems 2.5. definition vertical angles are the opposite angles formed by...
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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