Question

examine the figure below, prove that ∠aob ≅ ∠coe. proof: 2 - column given: ∠aoc ≅ ∠boe prove: ∠aob ≅ ∠coe statement reason ∠aoc ≅ ∠boe given select definition of congruent m∠aoc = m∠boa + m∠boc angle addition postulate m∠boe = m∠boc + m∠coe select m∠boa + m∠boc = m∠boc + m∠coe substitution m∠aob = m∠coe subtraction of equality ∠aob ≅ ∠coe definition of congruent

Explanation:

Step1: Recall congruent - angle property

If $\angle AOC\cong\angle BOE$, then $m\angle AOC = m\angle BOE$ by the Definition of Congruent.

Step2: Apply angle - addition postulate

We know that $m\angle AOC=m\angle BOA + m\angle BOC$ (Angle Addition Postulate) and $m\angle BOE=m\angle BOC + m\angle COE$ (also by the Angle Addition Postulate).

Step3: Substitute equal angles

Since $m\angle AOC = m\angle BOE$, we substitute to get $m\angle BOA + m\angle BOC=m\angle BOC + m\angle COE$.

Step4: Use subtraction property of equality

Subtract $m\angle BOC$ from both sides of the equation $m\angle BOA + m\angle BOC=m\angle BOC + m\angle COE$. We get $m\angle AOB=m\angle COE$.

Step5: Use definition of congruent

Since $m\angle AOB=m\angle COE$, by the Definition of Congruent, $\angle AOB\cong\angle COE$.

Answer:

The first blank should be filled with "If $\angle AOC\cong\angle BOE$, then $m\angle AOC = m\angle BOE$" and the second blank should be filled with "Angle Addition Postulate".