Question

given: ∠1 is complementary to ∠2. ∠2 is complementary to ∠3. prove: m∠1 = m∠3
diagram of angles
statements | reasons
1 ∠1 is comp to ∠2 | 1 given
2 ∠2 is comp to ∠3 | 2 given
3 ? | 3 def of comp ∠s
4 m∠1 = 90° - m∠2 | 4 subtr equality prop
5 m∠2 + m∠3 = 90° | 5 def of comp ∠s
6 m∠3 = 90° - m∠2 | 6 subtr equality prop
7 m∠1 = m∠3 | 7 trans prop
what is the missing statement in step 3 of the proof?
○ m∠1 = m∠2
○ m∠1 + m∠2 = 90°
○ m∠2 = m∠3
○ m∠2 + m∠3 = 180°

Explanation:

Step1: Recall the definition of complementary angles

Complementary angles are two angles whose measures add up to \(90^\circ\). So if \(\angle1\) is complementary to \(\angle2\), by the definition of complementary angles, we have \(m\angle1 + m\angle2 = 90^\circ\). This should be the statement for step 3 as the reason is "def of comp \(\angle\)s" (definition of complementary angles). Let's check the other options:

• Option \(m\angle1 = m\angle2\): There's no reason to assume this from the given (they are complementary, not necessarily equal), so this is wrong.
• Option \(m\angle2 = m\angle3\): We are trying to prove \(m\angle1 = m\angle3\), and we can't assume \(m\angle2 = m\angle3\) here, this is not the correct statement for step 3.
• Option \(m\angle2 + m\angle3 = 180^\circ\): Complementary angles add to \(90^\circ\), supplementary add to \(180^\circ\), so this is incorrect as \(\angle2\) and \(\angle3\) are complementary, not supplementary.

Step2: Confirm the correct statement

From the definition of complementary angles, since \(\angle1\) is complementary to \(\angle2\), the measure of \(\angle1\) plus the measure of \(\angle2\) is \(90^\circ\), so \(m\angle1 + m\angle2 = 90^\circ\) is the correct statement for step 3.

Answer:

\(m\angle1 + m\angle2 = 90^\circ\) (the second option: \(m\angle1 + m\angle2 = 90^\circ\))