Question

given: ∠1 is complementary to ∠2. ∠2 is complementary to ∠3. prove: m∠1 = m∠3 what is the missing statement in step 3 of the proof?
o m∠1 = m∠2
o m∠1 + m∠2 = 90°
o m∠2 = m∠3
o m∠2 + m∠3 = 180°
statements | reasons

1. ∠1 is comp. to ∠2 | 1. given
2. ∠2 is comp. to ∠3 | 2. given

3.? | 3. def. of comp. ∠s

1. m∠1 = 90° - m∠2 | 4. subtr. equality prop.
2. m∠2 + m∠3 = 90° | 5. def. of comp. ∠s
3. m∠3 = 90° - m∠2 | 6. subtr. equality prop.
4. m∠1 = m∠3 | 7. trans. prop.

Explanation:

To prove \( m\angle1 = m\angle3 \), we start with the definition of complementary angles (comp. \( \angle s \)). If two angles are complementary, their measures add up to \( 90^\circ \).

• For \( \angle1 \) and \( \angle2 \) (since \( \angle1 \) is comp. to \( \angle2 \)), by the definition of complementary angles, we have \( m\angle1 + m\angle2 = 90^\circ \).
• Let's analyze the options:
• \( m\angle1 = m\angle2 \): There's no reason to assume this from the given info (complementary angles don't have to be equal).
• \( m\angle2 = m\angle3 \): We can't conclude this directly from the given complementary relationships without further steps.
• \( m\angle2 + m\angle3 = 180^\circ \): This is the definition of supplementary angles, not complementary. Complementary angles add to \( 90^\circ \), so this is incorrect.
• \( m\angle1 + m\angle2 = 90^\circ \): This matches the definition of complementary angles for \( \angle1 \) and \( \angle2 \), which is the correct statement for step 3 (using the "def. of comp. \( \angle s \)" as the reason).

Answer:

\( m\angle1 + m\angle2 = 90^\circ \)