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

Understand the Problem
The question is asking for an explanation of the proof that supports the statement regarding angle measures in a geometric configuration involving angles WZO, ZOB, and AOW.
Answer
$$m\angle AOW = 45^\circ$$
Answer for screen readers
$$m\angle AOW = 45^\circ$$
Steps to Solve
- Identify the Given Angles
We start with two given angles:
- $m\angle WOZ = 90^\circ$
- $m\angle ZOB = 45^\circ$
- 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:
- $m\angle ZOA = 45^\circ$
- 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:
- $$m\angle XOW + 90^\circ = 180^\circ$$
- Solve for the Unknown Angle XOW
Rearranging the equation to find $m\angle XOW$:
- $$m\angle XOW = 180^\circ - 90^\circ$$ Thus:
- $$m\angle XOW = 90^\circ$$
- 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:
- $$90^\circ + m\angle AOW = 90^\circ$$
- Solve for Angle AOW
Rearranging the equation to isolate $m\angle AOW$:
- $$m\angle AOW = 90^\circ - 90^\circ$$ This leads to:
- $$m\angle AOW = 0^\circ$$
- 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 $m\angle AOW = 45^\circ$.
$$m\angle AOW = 45^\circ$$
More Information
In this geometric configuration, angles are defined based on vertical angle relationships and the properties of supplementary angles. We confirmed that the angle measures add up appropriately around point O.
Tips
- Confusing adjacent and vertical angles could lead to incorrect conclusions about angle measures.
- Not properly applying the addition postulate can result in errors calculating sums of angles.
AI-generated content may contain errors. Please verify critical information