Question

∠p and ∠r are complementary and ∠p and ∠q are complementary. complete the proof that ∠q ≅ ∠r.
statement | reason

1. ∠p and ∠r are complementary | given
2. ∠p and ∠q are complementary | given
3. ( mangle p + mangle r = 90^circ ) | definition of complementary angles
4. ( mangle p + mangle q = 90^circ ) | definition of complementary angles
5. ( mangle p + mangle r = mangle p + mangle q ) | transitive property of equality
6. ( mangle q = mangle r ) | properties of addition, subtraction, multiplication, and division
7. ( angle q cong angle r ) | definition of congruence

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Explanation:

Step1: Analyze Statement 4 and 5

Statement 4: \( m\angle P + m\angle Q = 90^\circ \) (from \( \angle P \) and \( \angle Q \) complementary, definition of complementary angles).
Statement 5: \( m\angle P + m\angle R = m\angle P + m\angle Q \)
We know \( m\angle P + m\angle R = 90^\circ \) (from \( \angle P \) and \( \angle R \) complementary, definition of complementary angles, Statement 3). So substituting \( m\angle P + m\angle R = 90^\circ \) into Statement 5, we get \( 90^\circ = m\angle P + m\angle Q \), which matches Statement 4. This uses the **Substitution Property of Equality** (or from the given reasons, "Properties of addition, subtraction, multiplication, and division" or more precisely, since we have two equal expressions (\( m\angle P + m\angle R = 90^\circ \) and \( m\angle P + m\angle Q = 90^\circ \)) and we set them equal, it's also related to transitive, but the reason for Statement 5 is "Transitive Property of Equality"? Wait, no—wait, Statement 5 is \( m\angle P + m\angle R = m\angle P + m\angle Q \). Let's re-express:

From Statement 3: \( m\angle P + m\angle R = 90^\circ \)
From Statement 4: \( m\angle P + m\angle Q = 90^\circ \)
So by Transitive Property of Equality (if \( a = c \) and \( b = c \), then \( a = b \)), we have \( m\angle P + m\angle R = m\angle P + m\angle Q \). Wait, but the reason for Statement 5 is given as "Transitive Property of Equality"? Wait, no—the user's table has Statement 5 with reason "Transitive Property of Equality"? Wait, the user's table:

Statement 5: \( m\angle P + m\angle R = m\angle P + m\angle Q \)
Reason: Transitive Property of Equality (option given). Wait, no—actually, the correct reason for Statement 5 is using the fact that both \( m\angle P + m\angle R \) and \( m\angle P + m\angle Q \) equal \( 90^\circ \) (from Statements 3 and 4), so by Transitive Property (if \( A = C \) and \( B = C \), then \( A = B \)), we get \( m\angle P + m\angle R = m\angle P + m\angle Q \).

But the question here is likely to fill the reason for Statement 4? Wait, no—looking at the table, Statement 4 is \( m\angle P + m\angle Q = 90^\circ \), Reason: Definition of complementary angles (since \( \angle P \) and \( \angle Q \) are complementary, their measures sum to \( 90^\circ \)).

Wait, the user's table:

1. \( \angle P \) and \( \angle R \) are complementary: Reason Given
2. \( \angle P \) and \( \angle Q \) are complementary: Reason Given
3. \( m\angle P + m\angle R = 90^\circ \): Reason Definition of complementary angles
4. \( m\angle P + m\angle Q = 90^\circ \): Reason? (The box is for Statement 4? Wait, the image shows Statement 4 with a box for the reason. Wait, Statement 3: \( m\angle P + m\angle R = 90^\circ \), Reason: Definition of complementary angles (correct, since complementary angles sum to \( 90^\circ \)).

Statement 4: \( m\angle P + m\angle Q = 90^\circ \): Reason should be "Definition of complementary angles" (since \( \angle P \) and \( \angle Q \) are complementary, so their measures sum to \( 90^\circ \)), same as Statement 3.

Then Statement 5: \( m\angle P + m\angle R = m\angle P + m\angle Q \): Reason is "Transitive Property of Equality" (since both equal \( 90^\circ \), so they equal each other).

Statement 6: \( m\angle Q = m\angle R \): Reason is "Properties of addition, subtraction, multiplication, and division" (subtract \( m\angle P \) from both sides: \( m\angle P + m\angle R - m\angle P = m\angle P + m\angle Q - m\angle P \implies m\angle R = m\angle Q \), which is subtraction property of equality).

Statement 7: \( \angle Q \cong \angle R \): Reason "Definition of congruence" (if two angles have equal measures, they are congruent).

But the specific question here—assuming the box is for Statement 4's reason:

Step1: Recall complementary angles definition

Complementary angles are two angles whose measures sum to \( 90^\circ \). Since \( \angle P \) and \( \angle Q \) are complementary (Statement 2, Given), their measures sum to \( 90^\circ \).

So the reason for Statement 4 (\( m\angle P + m\angle Q = 90^\circ \)) is "Definition of complementary angles".

Answer:

For Statement 4, the reason is "Definition of complementary angles".