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Given: [tex]\( m \angle EDF = 120^\circ \)[/tex], [tex]\( m \angle ADB = (3x)^\circ \)[/tex], [tex]\( m \angle BDC = (2x)^\circ \)[/tex]

Prove: [tex]\( x = 24 \)[/tex]

What is the missing reason in step 3?

[tex]\[
\begin{array}{|c|c|c|}
\hline
& \text{Statements} & \text{Reasons} \\
\hline
1. & m \angle EDF = 120^\circ, m \angle ADB = (3x)^\circ, m \angle BDC = (2x)^\circ & \text{Given} \\
\hline
2. & \angle EDF \text{ and } \angle ADC \text{ are vertical angles} & \text{Definition of vertical angles} \\
\hline
3. & \angle EDF = \angle ADC & \text{Vertical angles are congruent} \\
\hline
4. & m \angle ADC = m \angle ADB + m \angle BDC & \text{Angle addition postulate} \\
\hline
5. & m \angle EDF = m \angle ADB + m \angle BDC & \text{Substitution} \\
\hline
6. & 120 = 3x + 2x & \text{Substitution} \\
\hline
7. & 120 = 5x & \text{Addition} \\
\hline
8. & x = 24 & \text{Division} \\
\hline
\end{array}
\][/tex]


Sagot :

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."
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