Question

step statement reason 1 $overline{ab}$ and $overline{cd}$ intersect at $o$, forming four angles: $angle aoc$, $angle bod$, $angle aod$ and $angle boc$ given 2 $mangle aoc + mangle aod = 180^{circ}$ and $mangle aod + mangle bod = 180^{circ}$ linear pairs of angles are supplementary 3 substitution from step 2 4 $mangle aoc = mangle bod$ subtracting $angle aod$ from both sides of the equation from step 3 show your work here o $overline{ao}congoverline{bo}$ o $mangle aoc + mangle aod = mangle aod + mangle bod$ o $mangle aoc + mangle aod + mangle bod + mangle boc = 360^{circ}$ o $mangle aob = 180^{circ}$

Explanation:

Step1: Recall linear - pair property

From step 2, we know that \(m\angle AOC + m\angle AOD=180^{\circ}\) and \(m\angle AOD + m\angle BOD = 180^{\circ}\). Since both expressions equal \(180^{\circ}\), we can set them equal to each other.
So, \(m\angle AOC + m\angle AOD=m\angle AOD + m\angle BOD\)

Step2: Analyze other options

• Option \(\overline{AO}\cong\overline{BO}\): There is no information in the given proof about the congruence of line - segments \(AO\) and \(BO\).
• Option \(m\angle AOC + m\angle AOD + m\angle BOD + m\angle BOC=360^{\circ}\): While this is true for the sum of angles around a point, it is not the substitution step from step 2.
• Option \(m\angle AOB = 180^{\circ}\): There is no information in the given proof about \(\angle AOB\).

Answer:

\(m\angle AOC + m\angle AOD = m\angle AOD + m\angle BOD\)