Question

given: ∠2 and ∠4 are vertical angles
prove: ∠2 ≅ ∠4
assemble the proof by dragging tiles to
the statements and reasons columns.

Explanation:

Step1: Use the linear pair postulate

If two angles form a linear pair, then they are supplementary. So, for $\angle2$ and $\angle3$ (a linear pair), $m\angle2 + m\angle3=180^{\circ}$ (by the linear - pair postulate). For $\angle3$ and $\angle4$ (a linear pair), $m\angle3 + m\angle4 = 180^{\circ}$ (by the linear - pair postulate).

Step2: Use the substitution property

Since $m\angle2 + m\angle3=180^{\circ}$ and $m\angle3 + m\angle4 = 180^{\circ}$, we can set $m\angle2 + m\angle3=m\angle3 + m\angle4$.

Step3: Use the subtraction property of equality

Subtract $m\angle3$ from both sides of the equation $m\angle2 + m\angle3=m\angle3 + m\angle4$. We get $m\angle2=m\angle4$.

Step4: Use the definition of congruent angles

If $m\angle2=m\angle4$, then $\angle2\cong\angle4$ (by the definition of congruent angles: two angles are congruent if their measures are equal).

Answer:

Statements	Reasons
$m\angle2 + m\angle3 = 180$	$\angle2$ and $\angle3$ are a linear pair (Linear - Pair Postulate)
$m\angle3 + m\angle4=180$	$\angle3$ and $\angle4$ are a linear pair (Linear - Pair Postulate)
$m\angle2 + m\angle3=m\angle3 + m\angle4$	Substitution Property of Equality
$m\angle2=m\angle4$	Subtraction Property of Equality
$\angle2\cong\angle4$	Definition of Congruent Angles