Question

select the correct answer form each drop - down menu
given: ( mangle aob = 42^circ )
( mangle eof = 66^circ )
prove: ( mangle cod = 72^circ )
complete the proof.
it is given that ( mangle aob = 42^circ ) and ( mangle eof = 66^circ ). by the (\boldsymbol{\text{drop - down menu}}), ( angle eofcongangle boc ). therefore, ( mangle boc = 66^circ ). by the (\boldsymbol{\text{drop - down menu}}), ( mangle aoc = 108^circ ), and by the (\boldsymbol{\text{drop - down menu}}), ( mangle aoc + mangle cod = 180^circ ). after application of the (\boldsymbol{\text{drop - down menu}}), ( mangle cod = 72^circ ).

Explanation:

To solve this, we analyze each step using angle properties:

Step 1: Identify the first property (∠EOF ≅ ∠BOC)

Vertical angles are congruent. ∠EOF and ∠BOC are vertical angles, so we use the Vertical Angles Congruence Theorem.

Step 2: Find m∠AOC

We know \( m\angle AOB = 42^\circ \) and \( m\angle BOC = 66^\circ \). By the Angle Addition Postulate (which states \( m\angle AOC = m\angle AOB + m\angle BOC \)):
\[
m\angle AOC = 42^\circ + 66^\circ = 108^\circ
\]

Step 3: Identify \( m\angle AOC + m\angle COD = 180^\circ \)

∠AOC and ∠COD are supplementary (they form a linear pair). This is due to the Linear Pair Postulate (linear pairs of angles are supplementary).

Step 4: Solve for \( m\angle COD \)

Using the Subtraction Property of Equality (if \( a + b = c \), then \( b = c - a \)):
\[
m\angle COD = 180^\circ - m\angle AOC = 180^\circ - 108^\circ = 72^\circ
\]

Final Answers for Dropdowns:

1. First dropdown: Vertical Angles Congruence Theorem
2. Second dropdown: Angle Addition Postulate
3. Third dropdown: Linear Pair Postulate
4. Fourth dropdown: Subtraction Property of Equality

(Note: The exact wording of options may vary slightly, but the concepts are as above.)

Answer:

To solve this, we analyze each step using angle properties:

Step 1: Identify the first property (∠EOF ≅ ∠BOC)

Vertical angles are congruent. ∠EOF and ∠BOC are vertical angles, so we use the Vertical Angles Congruence Theorem.

Step 2: Find m∠AOC

We know \( m\angle AOB = 42^\circ \) and \( m\angle BOC = 66^\circ \). By the Angle Addition Postulate (which states \( m\angle AOC = m\angle AOB + m\angle BOC \)):
\[
m\angle AOC = 42^\circ + 66^\circ = 108^\circ
\]

Step 3: Identify \( m\angle AOC + m\angle COD = 180^\circ \)

∠AOC and ∠COD are supplementary (they form a linear pair). This is due to the Linear Pair Postulate (linear pairs of angles are supplementary).

Step 4: Solve for \( m\angle COD \)

Using the Subtraction Property of Equality (if \( a + b = c \), then \( b = c - a \)):
\[
m\angle COD = 180^\circ - m\angle AOC = 180^\circ - 108^\circ = 72^\circ
\]

Final Answers for Dropdowns:

1. First dropdown: Vertical Angles Congruence Theorem
2. Second dropdown: Angle Addition Postulate
3. Third dropdown: Linear Pair Postulate
4. Fourth dropdown: Subtraction Property of Equality

(Note: The exact wording of options may vary slightly, but the concepts are as above.)