Question

for items 3-5, refer to the figure shown.
the proof refers to the figure shown here. drag the word choices below to supply the missing reasons in the proof

if two angles are supplementary to the same angle, then the two angles are congruent.
the sum of the measures of a linear pair is 180°. angle addition postulate definition of perpendicular lines
definition of linear pair transitive property of equality supplementary congruent angles are right angles.

statement reason
1 ∠wzx ≅ ∠wzv given
2 ∠wzx and ∠wzv are a linear pair.
3 m∠wzx + m∠wzv = 180°
4 m∠wzx = m∠wzv = 90°
5 wy ⊥ vx

Explanation:

To solve the proof, we analyze each step:

Step 2: Reason for ∠WZX and ∠WZV being a linear pair

A linear pair is defined as two adjacent angles that form a straight line (their non - common sides are opposite rays). So the reason is the Definition of Linear Pair.

Step 3: Reason for \( m\angle WZX + m\angle WZV=180^{\circ} \)

By the definition of a linear pair, the sum of the measures of angles in a linear pair is \( 180^{\circ} \). So the reason is The sum of the measures of a linear pair is \( 180^{\circ} \).

Step 4: Reason for \( m\angle WZX = m\angle WZV = 90^{\circ} \)

We know that \( \angle WZX\cong\angle WZV \) (given), so \( m\angle WZX=m\angle WZV \). Let \( m\angle WZX = m\angle WZV=x \). From step 3, \( x + x=180^{\circ}\), \( 2x = 180^{\circ}\), \( x = 90^{\circ}\). This uses the property that supplementary congruent angles are right angles (since they are supplementary and congruent, each must be \( 90^{\circ} \)). So the reason is Supplementary congruent angles are right angles (and also substitution/transitive, but the given option here is this).

Step 5: Reason for \( \overline{WY}\perp\overline{VX} \)

Perpendicular lines are defined as two lines that intersect at right angles (angles with measure \( 90^{\circ} \)). Since \( \angle WZX \) and \( \angle WZV \) are \( 90^{\circ} \), the lines \( WY \) and \( VX \) are perpendicular. So the reason is Definition of Perpendicular Lines.

Filling in the table:

Statement	Reason
2. \( \angle WZX \) and \( \angle WZV \) are a linear pair.	Definition of Linear Pair
3. \( m\angle WZX + m\angle WZV = 180^{\circ} \)	The sum of the measures of a linear pair is \( 180^{\circ} \)
4. \( m\angle WZX = m\angle WZV = 90^{\circ} \)	Supplementary congruent angles are right angles
5. \( \overline{WY}\perp\overline{VX} \)	Definition of Perpendicular Lines

For each step:

• Step 2: Definition of Linear Pair
• Step 3: The sum of the measures of a linear pair is \( 180^{\circ} \)
• Step 4: Supplementary congruent angles are right angles
• Step 5: Definition of Perpendicular Lines

Answer:

To solve the proof, we analyze each step:

Step 2: Reason for ∠WZX and ∠WZV being a linear pair

A linear pair is defined as two adjacent angles that form a straight line (their non - common sides are opposite rays). So the reason is the Definition of Linear Pair.

Step 3: Reason for \( m\angle WZX + m\angle WZV=180^{\circ} \)

By the definition of a linear pair, the sum of the measures of angles in a linear pair is \( 180^{\circ} \). So the reason is The sum of the measures of a linear pair is \( 180^{\circ} \).

Step 4: Reason for \( m\angle WZX = m\angle WZV = 90^{\circ} \)

We know that \( \angle WZX\cong\angle WZV \) (given), so \( m\angle WZX=m\angle WZV \). Let \( m\angle WZX = m\angle WZV=x \). From step 3, \( x + x=180^{\circ}\), \( 2x = 180^{\circ}\), \( x = 90^{\circ}\). This uses the property that supplementary congruent angles are right angles (since they are supplementary and congruent, each must be \( 90^{\circ} \)). So the reason is Supplementary congruent angles are right angles (and also substitution/transitive, but the given option here is this).

Step 5: Reason for \( \overline{WY}\perp\overline{VX} \)

Perpendicular lines are defined as two lines that intersect at right angles (angles with measure \( 90^{\circ} \)). Since \( \angle WZX \) and \( \angle WZV \) are \( 90^{\circ} \), the lines \( WY \) and \( VX \) are perpendicular. So the reason is Definition of Perpendicular Lines.

Filling in the table:

Statement	Reason
2. \( \angle WZX \) and \( \angle WZV \) are a linear pair.	Definition of Linear Pair
3. \( m\angle WZX + m\angle WZV = 180^{\circ} \)	The sum of the measures of a linear pair is \( 180^{\circ} \)
4. \( m\angle WZX = m\angle WZV = 90^{\circ} \)	Supplementary congruent angles are right angles
5. \( \overline{WY}\perp\overline{VX} \)	Definition of Perpendicular Lines

For each step:

• Step 2: Definition of Linear Pair
• Step 3: The sum of the measures of a linear pair is \( 180^{\circ} \)
• Step 4: Supplementary congruent angles are right angles
• Step 5: Definition of Perpendicular Lines