Question

according to the vertical angle theorem, vertical angles are congruent. which of the following statements can be used to prove that ∠aob≅∠cod? draw show your work here o m∠aob = 180° o m∠aoc + m∠aod + m∠bod + m∠boc = 360° o m∠aob + m∠boc = 180° and m∠cod + m∠boc = 180° o m∠aod = m∠boc

Explanation:

Step1: Recall vertical - angle property

Vertical angles are formed when two lines intersect. $\angle AOB$ and $\angle COD$ are vertical angles. If we can show that they are supplementary to the same angle, it proves their congruence.

Step2: Analyze each option

• Option 1: $m\angle AOB = 180^{\circ}$ only gives the measure of $\angle AOB$ alone, not related to $\angle COD$.
• Option 2: $m\angle AOC + m\angle AOD + m\angle BOD + m\angle BOC=360^{\circ}$ is about the sum of all angles around the intersection point, not directly related to proving $\angle AOB\cong\angle COD$.
• Option 3: If $m\angle AOB + m\angle BOC = 180^{\circ}$ and $m\angle COD + m\angle BOC = 180^{\circ}$, then by the subtraction property of equality, we can subtract $m\angle BOC$ from both equations. So $m\angle AOB=m\angle COD$, which means $\angle AOB\cong\angle COD$.
• Option 4: $m\angle AOD = m\angle BOC$ is about another pair of vertical angles, not helpful for proving $\angle AOB\cong\angle COD$.

Answer:

$m\angle AOB + m\angle BOC = 180^{\circ}$ and $m\angle COD + m\angle BOC = 180^{\circ}$