Question

proving statements about segments and angles > section exercises 2.5 > exercise 4
statements
∠1 is a complement of ∠2
∠2 ≅ ∠3
m∠1 + m∠2 = 90°
m∠2 = m∠3
m∠1 + m∠3 = 90°
∠1 is a complement of ∠3
reasons
given
blank
blank
definition of congruent angles
substitution property of equality
blank
properties: addition property of equality, definition of complement, definition of complementary angles, given, reflexive property of equality, symmetric property of equality

Answer:

To solve the proof, we analyze each step:

Step 1: Reason for \( \angle 2 \cong \angle 3 \)

We need to determine the reason. Since no specific info is given here, but looking at the next step, we know \( m\angle 2 = m\angle 3 \) comes from the definition of congruent angles (if \( \angle A \cong \angle B \), then \( m\angle A = m\angle B \)). But for \( \angle 2 \cong \angle 3 \), if it's given or from a prior step, but in the proof structure, the reason for \( \angle 2 \cong \angle 3 \) (if it's a given or from a property, but here, maybe it's a given? Wait, no, the first given is \( \angle 1 \) is complement of \( \angle 2 \). Wait, maybe the reason for \( \angle 2 \cong \angle 3 \) is "Given" or another property, but actually, the reason for \( m\angle 2 = m\angle 3 \) is "Definition of congruent angles" (since congruent angles have equal measures). Then, for \( m\angle 1 + m\angle 2 = 90^\circ \), the reason is "Definition of complement" (since if \( \angle 1 \) is complement of \( \angle 2 \), then \( m\angle 1 + m\angle 2 = 90^\circ \)). Then, the step \( m\angle 1 + m\angle 3 = 90^\circ \) uses "Substitution Property of Equality" (substituting \( m\angle 2 \) with \( m\angle 3 \) because \( m\angle 2 = m\angle 3 \)). Finally, the reason for \( \angle 1 \) is complement of \( \angle 3 \) is "Definition of complement" (since \( m\angle 1 + m\angle 3 = 90^\circ \) means they are complementary).

Let's fill in the reasons:

1. \( \angle 1 \) is a complement of \( \angle 2 \): Given (already filled)
2. \( \angle 2 \cong \angle 3 \): Let's assume it's a given or from a diagram, but in the proof, the reason for \( \angle 2 \cong \angle 3 \) (if it's a statement) – maybe "Given" or another property, but actually, the next step \( m\angle 2 = m\angle 3 \) is "Definition of congruent angles" (since \( \angle 2 \cong \angle 3 \) implies \( m\angle 2 = m\angle 3 \))
3. \( m\angle 1 + m\angle 2 = 90^\circ \): Definition of complement (because \( \angle 1 \) is complement of \( \angle 2 \))
4. \( m\angle 2 = m\angle 3 \): Definition of congruent angles (given \( \angle 2 \cong \angle 3 \))
5. \( m\angle 1 + m\angle 3 = 90^\circ \): Substitution Property of Equality (substitute \( m\angle 2 \) with \( m\angle 3 \) in \( m\angle 1 + m\angle 2 = 90^\circ \))
6. \( \angle 1 \) is a complement of \( \angle 3 \): Definition of complement (because \( m\angle 1 + m\angle 3 = 90^\circ \))

Filling the Reasons:

• For \( \angle 2 \cong \angle 3 \): If it's a given, then "Given", but if it's from a diagram, maybe "Given". But in the proof, the reason for \( \angle 2 \cong \angle 3 \) (the statement) – let's check the options. The options are: Addition Property of Equality, Reflexive Property of Equality, Symmetric Property of Equality, Definition of complement, Definition of complementary angles, Given.

Wait, the "Reasons" column has boxes:

• For \( \angle 2 \cong \angle 3 \): Maybe "Given" (but the first given is \( \angle 1 \) is complement of \( \angle 2 \)). Wait, maybe \( \angle 2 \cong \angle 3 \) is a given, so reason is "Given".
• For \( m\angle 1 + m\angle 2 = 90^\circ \): Reason is "Definition of complement" (since \( \angle 1 \) is complement of \( \angle 2 \), so their measures add to 90°)
• For \( m\angle 2 = m\angle 3 \): Reason is "Definition of congruent angles" (but that's not in the options? Wait, the options are: Addition Property of Equality, Reflexive Property of Equality, Symmetric Property of Equality, Definition of complement, Definition of complementary angles, Given. Wait, maybe "Definition of congruent angles" is implied, but maybe the problem has a typo. Wait, the "Definition of congruent angles" is that if two angles are congruent, their measures are equal. So that's the reason.
• For \( m\angle 1 + m\angle 3 = 90^\circ \): Reason is "Substitution Property of Equality" (already filled)
• For \( \angle 1 \) is a complement of \( \angle 3 \): Reason is "Definition of complement" (since their measures add to 90°)

Let's correct:

1. \( \angle 1 \) is a complement of \( \angle 2 \): Given (filled)
2. \( \angle 2 \cong \angle 3 \): Let's say "Given" (if it's a given statement)
3. \( m\angle 1 + m\angle 2 = 90^\circ \): Definition of complement (because \( \angle 1 \) and \( \angle 2 \) are complementary)
4. \( m\angle 2 = m\angle 3 \): Definition of congruent angles (since \( \angle 2 \cong \angle 3 \))
5. \( m\angle 1 + m\angle 3 = 90^\circ \): Substitution Property of Equality (substitute \( m\angle 2 \) with \( m\angle 3 \))
6. \( \angle 1 \) is a complement of \( \angle 3 \): Definition of complement (because \( m\angle 1 + m\angle 3 = 90^\circ \))

Final Proof Table:

Statements	Reasons
\( \angle 2 \cong \angle 3 \)	Given (or from diagram)
\( m\angle 1 + m\angle 2 = 90^\circ \)	Definition of complement
\( m\angle 2 = m\angle 3 \)	Definition of congruent angles
\( m\angle 1 + m\angle 3 = 90^\circ \)	Substitution Property of Equality
\( \angle 1 \) is a complement of \( \angle 3 \)	Definition of complement

Key Reasons:

• For \( m\angle 1 + m\angle 2 = 90^\circ \): Definition of complement (since complementary angles sum to 90°)
• For \( m\angle 2 = m\angle 3 \): Definition of congruent angles (congruent angles have equal measures)
• For \( m\angle 1 + m\angle 3 = 90^\circ \): Substitution Property of Equality (replace \( m\angle 2 \) with \( m\angle 3 \))
• For \( \angle 1 \) is a complement of \( \angle 3 \): Definition of complement (since their measures sum to 90°)

If we need to fill the dashed boxes:

• For \( \angle 2 \cong \angle 3 \): Reason is "Given" (if it's a given) or another property, but likely "Given".
• For \( m\angle 1 + m\angle 2 = 90^\circ \): Reason is "Definition of complement".
• For \( \angle 1 \) is a complement of \( \angle 3 \): Reason is "Definition of complement".

Final Answer (Filling the Reasons):

• Reason for \( \angle 2 \cong \angle 3 \): Given (if it's a given statement)
• Reason for \( m\angle 1 + m\angle 2 = 90^\circ \): Definition of complement
• Reason for \( \angle 1 \) is a complement of \( \angle 3 \): Definition of complement

(Note: The exact reasons depend on the problem's context, but the above is the logical flow.)