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Given: \(m \angle ADE = 60^{\circ}\) and \(m \angle CDF = (3x + 15)^{\circ}\)

Prove: \(x = 15\)

What is the missing statement and the missing reason in step 5?

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
& Statements & Reasons \\
\hline
1 & \(m \angle ADE = 60^{\circ}\) \newline \(m \angle CDF = (3x + 15)^{\circ}\) & Given \\
\hline
2 & \(\angle ADE\) and \(\angle CDF\) are vertical angles & Definition of vertical angles \\
\hline
3 & \(\angle ADE = \angle CDF\) & Vertical angles are equal \\
\hline
4 & \(m \angle ADE = m \angle CDF\) & Definition of angle equality \\
\hline
5 & \(60 = 3x + 15\) & Substitution \\
\hline
6 & \(45 = 3x\) & Subtraction property of equality \\
\hline
7 & \(x = 15\) & Division property of equality \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnswer
To solve the problem and to find the missing statement and reason in step 5, let's go through the detailed, step-by-step solution:

1. Given Information:
- \( m \angle ADE = 60^\circ \)
- \( m \angle CDF = (3x + 15)^\circ \)

Reason 1: Given.

2. Identify Vertical Angles:
- \( \angle ADE \) and \( \angle CDF \) are vertical angles.

Reason 2: Definition of vertical angles.

3. Vertical Angles are Equal:
- \( \angle ADE = \angle CDF \)

Reason 3: Vertical angles have equal measures.

4. Equal Measures:
- \( m \angle ADE = m \angle CDF \)

Reason 4: Definition of congruent angles.

5. Substitute the Given Values:
- Substitute the given values into the equation.
- \( 60 = 3x + 15 \)

Reason: Substitution.

6. Solve for \( x \):
- Subtract 15 from both sides:
[tex]\[ 60 - 15 = 3x \][/tex]
[tex]\[ 45 = 3x \][/tex]

Missing Statement and Reason:
- Statement: \( 45 = 3x \),
- Reason: Subtraction property of equality.

7. Solve for \( x \):
- Divide both sides by 3:
[tex]\[ \frac{45}{3} = x \][/tex]
[tex]\[ x = 15 \][/tex]

Reason: Division property of equality.

So, the missing statement and reason in step 5 are:

Statement: \( 45 = 3x \)

Reason: Subtraction property of equality.
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