Question

given: \\(\angle 2\\) and \\(\angle 4\\) are vertical angles.
prove: \\(\angle 2 \cong \angle 4\\)

assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Identify given information

We are given that \(\angle 2\) and \(\angle 4\) are vertical angles. We need to prove that \(\angle 2 \cong \angle 4\).

Analyze the available tiles

The available tiles for Statements and Reasons are:

• Statements:
• \(m\angle 2 + m\angle 3 = 180\)
• \(m\angle 3 + m\angle 4 = 180\)
• \(\angle 2\) and \(\angle 4\) are vert. angles
• \(\angle 2\) and \(\angle 3\) are a linear pair
• \(\angle 3\) and \(\angle 4\) are a linear pair
• \(m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4\)

Construct the proof sequence

Using Intersecting Lines and Vertical Angles concepts, we build the logical flow:

1. Statement: \(\angle 2\) and \(\angle 4\) are vert. angles

Reason: Given

1. Statement: \(\angle 2\) and \(\angle 3\) are a linear pair

Reason: Definition of linear pair (from diagram)

1. Statement: \(m\angle 2 + m\angle 3 = 180\)

Reason: Linear Pair Postulate

1. Statement: \(\angle 3\) and \(\angle 4\) are a linear pair

Reason: Definition of linear pair (from diagram)

1. Statement: \(m\angle 3 + m\angle 4 = 180\)

Reason: Linear Pair Postulate

1. Statement: \(m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4\)

Reason: Transitive Property of Equality (or Substitution)

1. Statement: \(m\angle 2 = m\angle 4\)

Reason: Subtraction Property of Equality

1. Statement: \(\angle 2 \cong \angle 4\)

Reason: Definition of congruent angles

Match the target blank tiles

The image shows a specific step to be filled in the Statements and Reasons table.
Looking at the layout, the first row of the proof is typically the Given statement:

• Statement: \(\angle 2\) and \(\angle 4\) are vert. angles
• Reason: Given

Let's look at the next logical step in a standard vertical angles theorem proof:

• Statement: \(\angle 2\) and \(\angle 3\) are a linear pair
• Reason: Definition of linear pair (or from diagram)

Alternatively, the tiles shown in the drag-and-drop area are:

• Statements:
• \(m\angle 2 + m\angle 3 = 180\)
• \(m\angle 3 + m\angle 4 = 180\)
• \(\angle 2\) and \(\angle 4\) are vert. angles
• \(\angle 2\) and \(\angle 3\) are a linear pair
• \(\angle 3\) and \(\angle 4\) are a linear pair
• \(m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4\)

Since the prompt asks to assemble the proof, we will provide the complete step-by-step table mapping these statements to their corresponding reasons.

Answer:

The complete step-by-step proof is assembled as follows:

Statements	Reasons
2. \(\angle 2\) and \(\angle 3\) are a linear pair	2. Definition of linear pair
3. \(m\angle 2 + m\angle 3 = 180\)	3. Linear Pair Postulate
4. \(\angle 3\) and \(\angle 4\) are a linear pair	4. Definition of linear pair
5. \(m\angle 3 + m\angle 4 = 180\)	5. Linear Pair Postulate
6. \(m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4\)	6. Transitive Property of Equality
7. \(m\angle 2 = m\angle 4\)	7. Subtraction Property of Equality
8. \(\angle 2 \cong \angle 4\)	8. Definition of congruent angles