Questions: Given: triangle XOB congruent to angle AOX Prove: m angle XOB=90 degrees Statements Reasons 1. angle XOB congruent to angle AOX 1. given 2. angle XOB and angle AOX are supplementary 2. linear pair theorem 3. m angle XOB+m angle AOX=180 degrees 3. definition of supplementary angles 4. m angle XOB=m angle AOX 4. definition of congruence 5. 2 m angle XOB=180 degrees 5. substitution property of equality m angle XOB=90 degrees 6. division property of equality Since line segment AOB forms a line segment, angle XOB and angle AOX are supplementary by the linear pair theorem. Using the definition of supplementary angles, m angle XOB+m angle AOX=180 degrees. Since it is given that angle XOB congruent to angle AOX, then m angle XOB=m angle AOX.

Solution Steps

To prove that \( \mathrm{m} \angle \mathrm{XOB} = 90^{\circ} \), we start by using the given information that \( \angle \mathrm{XOB} \cong \angle \mathrm{AOX} \). This means the measures of these angles are equal. Since \( \angle \mathrm{XOB} \) and \( \angle \mathrm{AOX} \) are supplementary (they form a linear pair), their measures add up to \( 180^{\circ} \). By substituting the congruence into the equation for supplementary angles, we can solve for the measure of \( \angle \mathrm{XOB} \).

We are given that \(\angle \mathrm{XOB} \cong \angle \mathrm{AOX}\). This means that the two angles are congruent, or in other words, they have the same measure.

Since \(\overline{\mathrm{AOB}}\) forms a line segment, \(\angle \mathrm{XOB}\) and \(\angle \mathrm{AOX}\) are supplementary. According to the linear pair theorem, two angles that form a linear pair are supplementary.

By the definition of supplementary angles, the sum of the measures of \(\angle \mathrm{XOB}\) and \(\angle \mathrm{AOX}\) is \(180^\circ\). Therefore, we have: \[ \mathrm{m} \angle \mathrm{XOB} + \mathrm{m} \angle \mathrm{AOX} = 180^\circ \]

Since \(\angle \mathrm{XOB} \cong \angle \mathrm{AOX}\), it follows that: \[ \mathrm{m} \angle \mathrm{XOB} = \mathrm{m} \angle \mathrm{AOX} \]

Substituting the congruence from Step 4 into the equation from Step 3, we get: \[ 2 \mathrm{~m} \angle \mathrm{XOB} = 180^\circ \]

Dividing both sides of the equation by 2, we find: \[ \mathrm{m} \angle \mathrm{XOB} = 90^\circ \]

Final Answer