To prove [tex]\( x = 24 \)[/tex] given [tex]\( m \angle E D F = 120^\circ \)[/tex], [tex]\( m \angle A D B = (3x)^\circ \)[/tex], and [tex]\( m \angle B D C = (2x)^\circ \)[/tex], let's go through a step-by-step solution. We'll also find the missing reason in step 3 of the proof.
### Proof
1. Given:
[tex]\[
m \angle E D F = 120^\circ, \quad m \angle A D B = (3x)^\circ, \quad m \angle B D C = (2x)^\circ
\][/tex]
Reason: Given
2. Statement:
[tex]\[
\angle E D F \text{ and } \angle A D C \text{ are vertical angles (vert. }\angle s\text{)}
\][/tex]
Reason: Definition of vertical angles
3. Statement:
[tex]\[
\angle E D F = \angle A D C
\][/tex]
Reason: Vertical angles are congruent
The missing reason here is "Vertical angles are congruent." This tells us that vertical angles have equal measure because they are opposite each other when two lines intersect.
4. Statement:
[tex]\[
m \angle A D C = m \angle A D B + m \angle B D C
\][/tex]
Reason: Angle addition postulate
5. Statement:
[tex]\[
m \angle E D F = m \angle A D B + m \angle B D C
\][/tex]
Reason: Substitution (since [tex]\(\angle E D F = \angle A D C\)[/tex] from step 3)
6. Statement:
[tex]\[
120^\circ = (3x)^\circ + (2x)^\circ
\][/tex]
Reason: Substitution (from [tex]\(\angle E D F = 120^\circ\)[/tex])
7. Statement:
[tex]\[
120^\circ = 5x
\][/tex]
Reason: Combined like terms
8. Statement:
[tex]\[
x = \frac{120^\circ}{5}
\][/tex]
Reason: Algebraic simplification
9. Statement:
[tex]\[
x = 24
\][/tex]
Reason: Division
Thus, we have proved that [tex]\( x = 24 \)[/tex]. The missing reason in step 3 is "Vertical angles are congruent."
