Question

given that $\angle 1$ and $\angle 2$ are supplementary, $\angle 2$ and $\angle 3$ are supplementary, prove $\angle 1 \cong \angle 3$.

Explanation:

Step1: State given information

$\angle 1$ and $\angle 2$ are supplementary; $\angle 2$ and $\angle 3$ are supplementary

Step2: Apply supplementary angle definition

$m\angle 1 + m\angle 2 = 180^\circ$; $m\angle 2 + m\angle 3 = 180^\circ$

Step3: Set expressions equal (transitive property)

$m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3$

Step4: Subtract $m\angle 2$ (subtraction property)

$m\angle 1 + m\angle 2 - m\angle 2 = m\angle 2 + m\angle 3 - m\angle 2$

Step5: Simplify both sides

$m\angle 1 = m\angle 3$

Step6: Apply congruent angle definition

$\angle 1 \cong \angle 3$

Filled table (matching steps to reasons):

#	Steps	Reasons
2	$m\angle 1 + m\angle 2 = 180^\circ$; $m\angle 2 + m\angle 3 = 180^\circ$	Definition of supplementary angles
3	$m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3$	Transitive property of equality
4	$m\angle 1 + m\angle 2 - m\angle 2 = m\angle 2 + m\angle 3 - m\angle 2$	Subtraction property of equality
5	$m\angle 1 = m\angle 3$	Simplification
6	$\angle 1 \cong \angle 3$	Definition of congruent angles

Answer:

$\boldsymbol{\angle 1 \cong \angle 3}$ is proven using the transitive and subtraction properties of equality, along with definitions of supplementary and congruent angles.