Question

m∠abd = m∠cbe write the equation. given
m∠abd = m∠1 + m∠2
substitute m∠1 + m∠2 for m∠abd.
angle addition postulate
angle addition postulate
substitution property of equality
substitution property of equality
m∠1 = m∠3

Explanation:

Step1: State the given

$m\angle ABD = m\angle CBE$ (Given)

Step2: Apply angle - addition postulate

$m\angle ABD=m\angle1 + m\angle2$ (Angle Addition Postulate)

Step3: Substitute

Substitute $m\angle1 + m\angle2$ for $m\angle ABD$ in $m\angle ABD = m\angle CBE$. So, $m\angle1 + m\angle2=m\angle CBE$ (Substitution Property of Equality)

Step4: Assume $m\angle CBE=m\angle2 + m\angle3$ (by angle - addition postulate if $\angle CBE$ is composed of $\angle2$ and $\angle3$)

Substitute $m\angle2 + m\angle3$ for $m\angle CBE$ in $m\angle1 + m\angle2=m\angle CBE$. We get $m\angle1 + m\angle2=m\angle2 + m\angle3$ (Substitution Property of Equality)

Step5: Subtract $m\angle2$ from both sides

$m\angle1=m\angle3$ (Subtraction Property of Equality)

Answer:

The steps show the logical progression from the given $m\angle ABD = m\angle CBE$ to $m\angle1=m\angle3$ using the angle - addition postulate and substitution and subtraction properties of equality. The key steps involve breaking down angles using the angle - addition postulate and making appropriate substitutions to simplify the equations.