Question

1. proof copy and complete the flowchart proof. then write a two-column proof.(see example 1.)given $angle 1 cong angle 3$prove $angle 2 cong angle 4$

Explanation:

Step1: Identify linear pairs

$\angle 1$ and $\angle 2$ are a linear pair; $\angle 3$ and $\angle 4$ are a linear pair.

Step2: State linear pair property

Linear pairs are supplementary, so:
$\angle 1 + \angle 2 = 180^\circ$
$\angle 3 + \angle 4 = 180^\circ$

Step3: Substitute given congruence

Given $\angle 1 \cong \angle 3$, so $m\angle 1 = m\angle 3$. Substitute into the first equation:
$m\angle 3 + \angle 2 = 180^\circ$

Step4: Set equations equal

Since both equal $180^\circ$:
$m\angle 3 + \angle 2 = m\angle 3 + \angle 4$

Step5: Subtract $\angle 3$ from both sides

$\angle 2 = \angle 4$, so $\angle 2 \cong \angle 4$

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Two-Column Proof:

Statements	Reasons
2. $\angle 1$ and $\angle 2$ are supplementary; $\angle 3$ and $\angle 4$ are supplementary	2. Linear Pair Postulate
3. $m\angle 1 + m\angle 2 = 180^\circ$; $m\angle 3 + m\angle 4 = 180^\circ$	3. Definition of supplementary angles
4. $m\angle 1 = m\angle 3$	4. Definition of congruent angles
5. $m\angle 3 + m\angle 2 = m\angle 3 + m\angle 4$	5. Substitution Property of Equality
6. $m\angle 2 = m\angle 4$	6. Subtraction Property of Equality
7. $\angle 2 \cong \angle 4$	7. Definition of congruent angles

Answer:

$\boldsymbol{\angle 2 \cong \angle 4}$ is proven as shown above.