Question

m∠zob = 45°
prove: m∠aow = 45°
what evidence in the proof supports the statement that 45° + m∠aow = 90°?
statements | reasons

1. m∠woz = 90° | 1. given
2. m∠zob = 45° | 2. given
3. ∠zob ≅ ∠xoa | 3. vertical angle theorem
4. m∠xoa = 45° | 4. definition of congruence
5. ∠xow + ∠woz = 180° | 5. linear pair theorem
6. m∠xow = 90° | 6. subtraction property of equality
7. m∠xoa + m∠aow = m∠xow | 7. angle addition postulate
8. 45° + m∠aow = 90° | 8. substitution property of equality
9. m∠aow = 45° | 9. subtraction property of equality

Explanation:

To find the evidence supporting $45^\circ + m\angle AOW = 90^\circ$, we trace the logical chain:

1. Step 4 establishes $m\angle XOA = 45^\circ$ via congruence.
2. Step 6 finds $m\angle XOW = 90^\circ$ using linear pair properties.
3. Step 7 (angle addition postulate) states $m\angle XOA + m\angle AOW = m\angle XOW$.
4. Step 8 substitutes the values from Steps 4 and 6 into the equation from Step 7, resulting in $45^\circ + m\angle AOW = 90^\circ$.

The supporting evidence is the combination of the angle addition postulate (Step 7) and the prior established values of $m\angle XOA = 45^\circ$ (Step 4) and $m\angle XOW = 90^\circ$ (Step 6), which are used with the substitution property of equality (Step 8).

Answer:

The statement $45^\circ + m\angle AOW = 90^\circ$ is supported by:

1. $m\angle XOA = 45^\circ$ (Statement 4, from congruent vertical angles)
2. $m\angle XOW = 90^\circ$ (Statement 6, from linear pair theorem)
3. The angle addition postulate (Statement 7: $m\angle XOA + m\angle AOW = m\angle XOW$)
4. The substitution property of equality (Statement 8, substituting the known angle measures into the angle addition equation)