Question

statements
∠1 is a complement of ∠2 .
∠2 ≅ ∠3
m∠1 + m∠2 = 90°
m∠2 = m∠3
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∠1 is a complement of ∠3 .

reasons
given
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definition of congruent angles
substitution property of equality
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Explanation:

Step1: Reason for \( \angle 2 \cong \angle 3 \)

This is a given statement (assuming it's provided as given, or maybe from a diagram/previous info, but in the proof structure, if it's a given, we state "Given").

Step2: Reason for \( m\angle 1 + m\angle 2 = 90^\circ \)

By the definition of complementary angles (if two angles are complementary, their measures sum to \( 90^\circ \)).

Step3: Statement for Substitution Property

Substitute \( m\angle 2 \) with \( m\angle 3 \) in \( m\angle 1 + m\angle 2 = 90^\circ \), so we get \( m\angle 1 + m\angle 3 = 90^\circ \).

Step4: Reason for \( \angle 1 \) is a complement of \( \angle 3 \)

By the definition of complementary angles (if the sum of two angles' measures is \( 90^\circ \), they are complementary).

Filling the blanks:

• For \( \angle 2 \cong \angle 3 \): Given (or as per problem's context, if it's a given)
• For \( m\angle 1 + m\angle 2 = 90^\circ \): Definition of complementary angles
• For the substitution step: \( m\angle 1 + m\angle 3 = 90^\circ \)
• For \( \angle 1 \) is a complement of \( \angle 3 \): Definition of complementary angles

Answer:

• Reason for \( \angle 2 \cong \angle 3 \): Given
• Reason for \( m\angle 1 + m\angle 2 = 90^\circ \): Definition of complementary angles
• Statement for Substitution: \( m\angle 1 + m\angle 3 = 90^\circ \)
• Reason for \( \angle 1 \) complement of \( \angle 3 \): Definition of complementary angles