What evidence in the proof supports the statement that 45° + m∠AOW = 90°?

Steps to Solve

1. Identify the Given Angles

We start with two given angles:

• $m\angle WOZ = 90^\circ$
• $m\angle ZOB = 45^\circ$

2. Use the Congruent Angles Properties

Since $m\angle ZOB = 45^\circ$ and $\angle ZOA$ is congruent to $\angle ZOB$ (by the vertical angle theorem), we can conclude:

• $m\angle ZOA = 45^\circ$

3. Apply the Relationship between Angles

We know that the sum of angles around point O (in this case, angles XOW, WOZ, and ZOB) adds up to $180^\circ$:

• $$m\angle XOW + m\angle WOZ = 180^\circ$$ Substituting the known angle values gives:
• $$m\angle XOW + 90^\circ = 180^\circ$$

4. Solve for the Unknown Angle XOW

Rearranging the equation to find $m\angle XOW$:

• $$m\angle XOW = 180^\circ - 90^\circ$$ Thus:
• $$m\angle XOW = 90^\circ$$

5. Apply the Addition of Angles

Next, we can express the relationship between angles AOW, XOW, and the given $m\angle AOW$:

• $$m\angle XOW + m\angle AOW = m\angle WOZ$$ Substituting the known values gives:
• $$90^\circ + m\angle AOW = 90^\circ$$

6. Solve for Angle AOW

Rearranging the equation to isolate $m\angle AOW$:

• $$m\angle AOW = 90^\circ - 90^\circ$$ This leads to:
• $$m\angle AOW = 0^\circ$$

7. Establish the Final Relationship

Now, review the previous findings. From earlier steps, if $m\angle ZOA$ is $45^\circ$, and hence: $$45^\circ + m\angle AOW = 90^\circ$$

Finally:

• Rearranging gives us $m\angle AOW = 45^\circ$.