Question

try it given: ∠2 and ∠4 are vertical angles. prove: ∠2 ≅ ∠4

Explanation:

Step1: Identify Linear Pairs

∠2 and ∠3 are a linear pair (given by the statements). By the definition of a linear pair, their measures sum to 180°, so \( m\angle2 + m\angle3 = 180^\circ \). Similarly, ∠3 and ∠4 are a linear pair, so \( m\angle3 + m\angle4 = 180^\circ \).

Step2: Set Equations Equal

Since \( m\angle2 + m\angle3 = 180^\circ \) and \( m\angle3 + m\angle4 = 180^\circ \), we can set them equal: \( m\angle2 + m\angle3 = m\angle3 + m\angle4 \).

Step3: Subtract \( m\angle3 \)

Subtract \( m\angle3 \) from both sides of the equation \( m\angle2 + m\angle3 = m\angle3 + m\angle4 \). This gives \( m\angle2 = m\angle4 \).

Step4: Conclude Congruence

If the measures of two angles are equal, the angles are congruent. So, \( \angle2 \cong \angle4 \).

Now, filling the two - column proof:

Statements	Reasons
2. \( m\angle2 + m\angle3 = 180^\circ \); \( m\angle3 + m\angle4 = 180^\circ \)	2. Definition of a linear pair (linear pairs are supplementary)
3. \( m\angle2 + m\angle3 = m\angle3 + m\angle4 \)	3. Substitution property (both equal to \( 180^\circ \))
4. \( m\angle2 = m\angle4 \)	4. Subtraction property of equality (subtract \( m\angle3 \) from both sides)
5. \( \angle2 \cong \angle4 \)	5. Definition of congruent angles (if \( m\angle A=m\angle B \), then \( \angle A\cong\angle B \))

Answer:

The two - column proof is filled as above, and we have proven that \( \angle2 \cong \angle4 \) using the properties of linear pairs and equality of angle measures.