Question

1. given: ∠1 and ∠4 form a linear pair; ∠1 and ∠2 are supplementary prove: ∠3 ≅ ∠4

statements | reasons

1. ∠1 and ∠4 form a linear pair | 1.
2. | 2. supplement theorem
3. | 3. given
4. | 4. congruent supplements theorem
5. ∠2 ≅ ∠3 | 5.
6. | 6. transitive property

Explanation:

Step1: Reason for Statement 1

The first statement is given, so the reason is "Given".

Step2: Statement for Reason 2

By the Supplement Theorem, if two angles form a linear pair, they are supplementary. So the statement is "$\angle 1$ and $\angle 4$ are supplementary".

Step3: Statement for Reason 3

The given information is that $\angle 1$ and $\angle 2$ are supplementary, so the statement is "$\angle 1$ and $\angle 2$ are supplementary".

Step4: Statement for Reason 4

By the Congruent Supplements Theorem, if two angles are supplementary to the same angle, they are congruent to each other. So $\angle 2\cong\angle 4$ (wait, no, let's correct. Wait, we have $\angle 1$ supplementary to $\angle 4$ (from step 2) and $\angle 1$ supplementary to $\angle 2$ (from step 3). So by Congruent Supplements Theorem, $\angle 2\cong\angle 4$? Wait, no, the final goal is $\angle 3\cong\angle 4$. Wait, maybe I made a mistake. Wait, $\angle 2$ and $\angle 3$: are they vertical angles? Wait, looking at the diagram, $\angle 2$ and $\angle 3$: if the two lines are parallel? Wait, no, the diagram has two parallel lines cut by a transversal? Wait, the top line and bottom line are parallel, cut by a transversal. So $\angle 2$ and $\angle 3$: $\angle 2$ and $\angle 3$ are vertical angles? Wait, no, $\angle 2$ and $\angle 3$: maybe $\angle 2$ and $\angle 3$ are vertical angles? Wait, no, let's re-examine. Wait, the bottom line: the angle labeled 2 and angle 3: if the transversal crosses the bottom line, then $\angle 2$ and $\angle 3$ are vertical angles? Wait, no, vertical angles are opposite each other when two lines intersect. So if the transversal intersects the bottom line, then the angle adjacent to 2 (vertical) would be 3? Wait, maybe $\angle 2\cong\angle 3$ by Vertical Angles Theorem. Wait, step 5 is $\angle 2\cong\angle 3$, so the reason for step 5 is Vertical Angles Theorem. Then step 6: using Transitive Property, since $\angle 2\cong\angle 4$ (from step 4) and $\angle 2\cong\angle 3$ (from step 5), then $\angle 3\cong\angle 4$. Let's correct the steps:

Wait, let's redo the proof properly:

1. $\angle 1$ and $\angle 4$ form a linear pair - Given
2. $\angle 1$ and $\angle 4$ are supplementary - Linear Pair Postulate (which is the Supplement Theorem for linear pairs)
3. $\angle 1$ and $\angle 2$ are supplementary - Given
4. $\angle 2\cong\angle 4$ - Congruent Supplements Theorem (since both are supplementary to $\angle 1$)
5. $\angle 2\cong\angle 3$ - Vertical Angles Theorem (because $\angle 2$ and $\angle 3$ are vertical angles)
6. $\angle 3\cong\angle 4$ - Transitive Property (since $\angle 2\cong\angle 4$ and $\angle 2\cong\angle 3$, then $\angle 3\cong\angle 4$)

So let's fill in the table:

1. Reason: Given
2. Statement: $\angle 1$ and $\angle 4$ are supplementary; Reason: Linear Pair Postulate (or Supplement Theorem for linear pairs)
3. Statement: $\angle 1$ and $\angle 2$ are supplementary; Reason: Given
4. Statement: $\angle 2\cong\angle 4$; Reason: Congruent Supplements Theorem
5. Reason: Vertical Angles Theorem
6. Statement: $\angle 3\cong\angle 4$; Reason: Transitive Property

So going back to the original table:

1. Reason: Given
2. Statement: $\angle 1$ and $\angle 4$ are supplementary
3. Statement: $\angle 1$ and $\angle 2$ are supplementary
4. Statement: $\angle 2\cong\angle 4$ (Wait, but the final step is $\angle 3\cong\angle 4$, so maybe I messed up. Wait, no, maybe $\angle 2$ and $\angle 3$ are equal because they are alternate interior angles? Wait, no, the lines are parallel? Wait, the top and bottom lines are parallel, so $\angle 1$ and $\angle 3$ are corresponding angles? Wait, no, the diagram: top line, bottom line, transversal. $\angle 1$ is top left, $\angle 4$ is top right (linear pair). $\angle 2$ is bottom left, $\angle 3$ is bottom right. So $\angle 2$ and $\angle 3$: if the lines are parallel, $\angle 2$ and $\angle 3$: no, $\angle 2$ and $\angle 3$: $\angle 2$ and $\angle 3$ are vertical angles? Wait, no, when the transversal intersects the bottom line, the angle at the bottom left is $\angle 2$, and the angle at the bottom right (opposite to $\angle 2$) would be $\angle 3$? Wait, no, vertical angles are formed when two lines intersect. So the transversal and the bottom line intersect, forming $\angle 2$ and $\angle 3$ as vertical angles? Wait, no, $\angle 2$ and $\angle 3$: if the transversal crosses the bottom line, then $\angle 2$ and $\angle 3$ are adjacent and form a linear pair? No, that can't be. Wait, maybe $\angle 2$ and $\angle 3$ are equal because they are alternate interior angles? Wait, the top and bottom lines are parallel, so $\angle 1$ and $\angle 3$ are corresponding angles, so $\angle 1\cong\angle 3$. But that's not helpful. Wait, let's start over.

Given: $\angle 1$ and $\angle 4$ form a linear pair (so they are supplementary: $\angle 1 + \angle 4 = 180^\circ$). $\angle 1$ and $\angle 2$ are supplementary ($\angle 1 + \angle 2 = 180^\circ$). Prove $\angle 3\cong\angle 4$.

So:

1. $\angle 1$ and $\angle 4$ form a linear pair - Given
2. $\angle 1 + \angle 4 = 180^\circ$ - Linear Pair Postulate (supplementary)
3. $\angle 1 + \angle 2 = 180^\circ$ - Given (supplementary)
4. $\angle 2 = \angle 4$ - Substitution (from 2 and 3: $\angle 1 + \angle 4 = \angle 1 + \angle 2$ ⇒ $\angle 4 = \angle 2$)
5. $\angle 2 = \angle 3$ - Vertical Angles Theorem (since $\angle 2$ and $\angle 3$ are vertical angles)
6. $\angle 3 = \angle 4$ - Substitution (from 4 and 5: $\angle 3 = \angle 2 = \angle 4$)

Ah, that makes sense. So step 4: $\angle 2 = \angle 4$ (or $\angle 2\cong\angle 4$), step 5: $\angle 2\cong\angle 3$ (Vertical Angles Theorem), step 6: $\angle 3\cong\angle 4$ (Transitive Property).

So filling the table:

1. Reason: Given
2. Statement: $\angle 1 + \angle 4 = 180^\circ$ (or $\angle 1$ and $\angle 4$ are supplementary); Reason: Linear Pair Postulate
3. Statement: $\angle 1 + \angle 2 = 180^\circ$ (or $\angle 1$ and $\angle 2$ are supplementary); Reason: Given
4. Statement: $\angle 2\cong\angle 4$; Reason: Congruent Supplements Theorem (since both are supplementary to $\angle 1$)
5. Reason: Vertical Angles Theorem
6. Statement: $\angle 3\cong\angle 4$; Reason: Transitive Property (if $\angle 2\cong\angle 4$ and $\angle 2\cong\angle 3$, then $\angle 3\cong\angle 4$)

So the correct statements and reasons are:

1. $\angle 1$ and $\angle 4$ form a linear pair - Given
2. $\angle 1$ and $\angle 4$ are supplementary - Linear Pair Postulate (Supplement Theorem)
3. $\angle 1$ and $\angle 2$ are supplementary - Given
4. $\angle 2\cong\angle 4$ - Congruent Supplements Theorem
5. $\angle 2\cong\angle 3$ - Vertical Angles Theorem
6. $\angle 3\cong\angle 4$ - Transitive Property

Now, let's answer the specific parts (assuming the question is to fill the table):

1. Reason: Given
2. Statement: $\angle 1$ and $\angle 4$ are supplementary
3. Statement: $\angle 1$ and $\angle 2$ are supplementary
4. Statement: $\angle 2\cong\angle 4$
5. Reason: Vertical Angles Theorem
6. Statement: $\angle 3\cong\angle 4$

But the original table has step 5 as $\angle 2\cong\angle 3$, so reason for step 5 is Vertical Angles Theorem. Then step 6: using Transitive Property, since $\angle 2\cong\angle 3$ (step 5) and $\angle 2\cong\angle 4$ (step 4), then $\angle 3\cong\angle 4$.

Yes, that makes sense. So the key steps are:

- Linear pair implies supplementary (step 2)
- Given supplementary (step 3)
- Congruent supplements (step 4: $\angle 2\cong\angle 4$)
- Vertical angles (step 5: $\angle 2\cong\angle 3$)
- Transitive property (step 6: $\angle 3\cong\angle 4$)

Answer:

1. Reason: Given
2. Statement: $\boldsymbol{\angle 1}$ and $\boldsymbol{\angle 4}$ are supplementary
3. Statement: $\boldsymbol{\angle 1}$ and $\boldsymbol{\angle 2}$ are supplementary
4. Statement: $\boldsymbol{\angle 2 \cong \angle 4}$
5. Reason: Vertical Angles Theorem
6. Statement: $\boldsymbol{\angle 3 \cong \angle 4}$