Question

given: ∠aob and ∠boc are a linear pair; m∠aob = m∠boc
prove: ∠aob and ∠boc are right angles.

1. given
2. m∠aob + m∠boc = 180
3. m∠aob = m∠boc
4. m∠aob + m∠aob = 180 or 2m∠aob = 180

Explanation:

Step1: State the given information

Given \(\angle AOB\) and \(\angle BOC\) are a linear pair and \(m\angle AOB = m\angle BOC\)

Step2: Apply linear - pair property

By the definition of a linear pair, \(m\angle AOB + m\angle BOC=180^{\circ}\)

Step3: Restate given equality

We are given \(m\angle AOB = m\angle BOC\)

Step4: Substitute equal angles

Substitute \(m\angle BOC\) with \(m\angle AOB\) in \(m\angle AOB + m\angle BOC = 180^{\circ}\) to get \(2m\angle AOB=180^{\circ}\)

Step5: Solve for angle measure

Divide both sides of \(2m\angle AOB = 180^{\circ}\) by 2 to find \(m\angle AOB = 90^{\circ}\)

Step6: Find the other angle measure

Since \(m\angle AOB = m\angle BOC\), substitute \(m\angle AOB = 90^{\circ}\) to get \(m\angle BOC = 90^{\circ}\)

Step7: Conclude right - angle property

By the definition of a right - angle (an angle with measure 90°), \(\angle AOB\) and \(\angle BOC\) are right angles.

Answer:

1. Given
2. Definition of linear - pair (The sum of the measures of angles in a linear pair is 180°)
3. Given
4. Substitution (Since \(m\angle AOB=m\angle BOC\), substitute \(m\angle BOC\) with \(m\angle AOB\))
5. Solve for \(m\angle AOB\): \(m\angle AOB = 90^{\circ}\) (Divide both sides of \(2m\angle AOB=180\) by 2)
6. Since \(m\angle AOB = m\angle BOC\), \(m\angle BOC=90^{\circ}\) (Substitution)
7. Definition of right - angle (An angle with measure 90° is a right - angle, so \(\angle AOB\) and \(\angle BOC\) are right angles)