Question

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equation explanation reason
m∠abd = m∠cbe write the equation. given
m∠abd = m∠1 + m∠2
angle addition postulate
substitution property of equality
m∠1 + m∠2 = m∠2 subtract m∠2 from each side. substitution property of equality
m∠1 = m∠3 subtraction property of equality
:: m∠cbe = m∠2 + m∠3 :: m∠1 + m∠2 = m∠2 + m∠3
:: add measures of adjacent angles. :: subtract m∠2 from each side.
:: substitute m∠1 + m∠2 for m∠abd. :: m∠abd = m∠2 + m∠3
:: subtract m∠1 from each side.
:: substitute m∠3 + m∠2 for m∠abd. :: angle addition postulate
:: substitute m∠abd for m∠cbe.
:: subtraction property of equality

Explanation:

Step1: Identify angle - addition

$m\angle ABD=m\angle1 + m\angle2$ because of the Angle Addition Postulate.

Step2: Express $m\angle CBE$ using angle - addition

$m\angle CBE=m\angle2 + m\angle3$ (Angle Addition Postulate).

Step3: Substitute based on given

Since $m\angle ABD = m\angle CBE$ (given), we substitute to get $m\angle1 + m\angle2=m\angle2 + m\angle3$ (Substitution Property of Equality).

Step4: Apply subtraction property

Subtract $m\angle2$ from both sides of the equation $m\angle1 + m\angle2=m\angle2 + m\angle3$. By the Subtraction Property of Equality, we get $m\angle1=m\angle3$.

Answer:

The steps to prove $m\angle1 = m\angle3$ are as follows: First, use the Angle Addition Postulate to express $m\angle ABD$ and $m\angle CBE$ in terms of smaller - angle measures. Then, substitute using the given equality $m\angle ABD = m\angle CBE$. Finally, apply the Subtraction Property of Equality to isolate $m\angle1$ and $m\angle3$.