Question

$\angle x$ and $\angle v$ are complementary and $\angle w$ and $\angle v$ are complementary. complete the proof that $\angle w \cong \angle x$.

	statement	reason
1	$\angle x$ and $\angle v$ are complementary
2	$\angle w$ and $\angle v$ are complementary
3	$m\angle x + m\angle v = 90^\circ$
4	$m\angle w + m\angle v = 90^\circ$
5	$m\angle x + m\angle v = m\angle w + m\angle v$
6	$m\angle w = m\angle x$
7	$\angle w \cong \angle x$

Explanation:

Step1: Identify Given Info (Statement 1)

The first statement is given in the problem: \( \angle X \) and \( \angle V \) are complementary. Reason: Given.

Step2: Identify Given Info (Statement 2)

The second statement is also given: \( \angle W \) and \( \angle V \) are complementary. Reason: Given.

Step3: Definition of Complementary Angles (Statement 3)

If two angles are complementary, their measures sum to \( 90^\circ \). So \( m\angle X + m\angle V = 90^\circ \). Reason: Definition of complementary angles.

Step4: Definition of Complementary Angles (Statement 4)

Similarly, for \( \angle W \) and \( \angle V \), \( m\angle W + m\angle V = 90^\circ \). Reason: Definition of complementary angles.

Step5: Transitive Property of Equality (Statement 5)

Since both \( m\angle X + m\angle V \) and \( m\angle W + m\angle V \) equal \( 90^\circ \), they equal each other. Reason: Transitive property of equality (\( a = c \), \( b = c \) implies \( a = b \)).

Step6: Subtraction Property of Equality (Statement 6)

Subtract \( m\angle V \) from both sides of \( m\angle X + m\angle V = m\angle W + m\angle V \). So \( m\angle X = m\angle W \) (or \( m\angle W = m\angle X \)). Reason: Subtraction property of equality (if \( a + c = b + c \), then \( a = b \)).

Step7: Definition of Congruent Angles (Statement 7)

If two angles have equal measures, they are congruent. So \( \angle W \cong \angle X \). Reason: Definition of congruent angles (angles with equal measures are congruent).

Answer:

1. Reason for Statement 1: Given
2. Reason for Statement 2: Given
3. Reason for Statement 3: Definition of complementary angles
4. Reason for Statement 4: Definition of complementary angles
5. Reason for Statement 5: Transitive property of equality
6. Reason for Statement 6: Subtraction property of equality
7. Reason for Statement 7: Definition of congruent angles

(For each statement, the corresponding reason is as above. If filling the table, each row's "Reason" column is filled with the respective reason.)