Step-by-step explanation:
Note:
the text that is formatted in bold and underlined fonts are the answers to the missing (blank) terms in the given problem.
Answers:
Given that [tex]\huge\sf{\overline{AB}}[/tex] is a straight line, and that ∠AOE, ∠EOF, and ∠FOB share a common vertex, at point O, whose sum equals 180°.
1. Statement: m∠AOB = 180°
Reason: Definition of a straight angle.
2. Statement: m∠AOE + m∠EOF + m∠FOB = m∠AOB
Reason: Angle Addition Postulate.
3. Statement: x + (2x + 34) + 20° = 180°
Reason: Substitution.
4. Statement: x = 42°
Reason: Algebra.
Explanation:
1) Straight angles have a measure of 180°.
2) The Angle Addition Postulate states that if O is in the interior of ∠AOB, then it means that the measure if ∠AOB is equal to the sum of the measures of ∠AOE, ∠EOF, and ∠FOB. In other words:
⇒ [tex]\huge\bf\sf{m\angle{AOB}\:=m\angle{AOE}\:+m\angle{EOF}\:+m\angle{FOB}}[/tex]
3) and 4) Since the ∠AOB is a straight angle, then it means that the sum of the measures of ∠AOE, ∠EOF, and ∠FOB equal the measure of ∠AOB = 180°. Hence, in order to solve for the value of x, simply add the given values for each angle:
x° + (2x + 34)° + 20° = 180°
x° + 2x° + 34° + 20° = 180°
Combine like terms:
3x° + 54° = 180°
Subtract 54° from both sides:
3x° + 54° - 54° = 180° - 54°
3x° = 126°
Divide both sides by 3 to solve for x:
[tex]\huge\sf{\frac{3x^\circ}{3}\:=\:\frac{126^\circ}{3}}[/tex]
x = 42°
