Answer:
1. Vertical Angles Theorem
2. Angle Addition Postulate
3. Linear Pair Theorem
4. Subtraction Property of Equality
Step-by-step explanation:
The given diagram shows three lines, AD, CF and BE that intersect at point O.
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Vertical Angles Theorem
The vertical angles theorem states that when two straight lines intersect, the opposite vertical angles are congruent. Therefore, as ∠EOF is vertically opposite ∠BOC, then:
∠EOF ≅ ∠BOC
Given that m∠EOF = 66°, then it follows that m∠BOC = 66°.
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Angle Addition Postulate
The Angle Addition Postulate states that the total measure of an angle formed by two adjacent angles is equal to the sum of their individual measures.
Given that m∠AOB = 42° and m∠BOC = 66°, then by the angle addition postulate:
m∠AOC = m∠AOB + m∠BOC
m∠AOC = 42° + 66°
m∠AOC = 108°
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Linear Pair Theorem
A linear pair consists of two adjacent angles that sum to 180°.
Since the sum of ∠AOC and ∠COD forms a straight line, they constitute a linear pair, so:
m∠AOC + m∠COD = 180°
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Subtraction Property of Equality
According to the subtraction property of equality, if we subtract the same quantity from both sides of an equation, the two sides remain equal.
Substitute m∠AOC = 108° into the sum equation of ∠AOC and ∠COD:
m∠AOC + m∠COD = 180°
108° + m∠COD = 180°
Now, subtract 108° from both sides of the equation to find m∠COD:
108° + m∠COD - 108° = 180° - 108°
m∠COD = 72°
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Completed Proof
It is given that m∠AOC = 42° and m∠EOF = 66°. By the Vertical Angles Theorem, ∠EOF ≅ ∠BOC. Therefore, m∠BOC = 66°. By the Angle Addition Postulate, m∠AOC = 108°, and by the Linear Pair Theorem, m∠AOC + m∠COD = 180°. After application of the Subtraction Property of Equality, m∠COD = 72°.
