Question

work by pair
fill in the missing reasons in the two - column proof below. choose from the reason bank provided.

statement	reasons
2. $\angle 1$ and $\angle 2$ are formed
3. $\angle 1$ and $\angle 2$ are opposite angles
4. $\angle 1$ and $\angle 2$ share a common vertex (e)
5. the sides of $\angle 1$ and $\angle 2$ form two pairs of opposite rays
6. $\angle 1$ and $\angle 2$ are vertical angles
7. vertical angles are congruent
8. $\angle 1 \cong \angle 2$

reason bank (use each reason only once):
definition of vertical angles
given
definition of opposite rays
definition of intersecting lines
vertical angles are congruent
definition of congruent angles
angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles
when two lines intersect, they form two pairs of vertical angles

Answer:

Step 1: Analyze Statement 1

• Statement: Line AB intersects line CD at point E
• Reason: This is given information, so the reason is "Given".

Step 2: Analyze Statement 2

• Statement: ∠1 and ∠2 are formed
• Reason: When two lines intersect, they form angles. The reason from the bank is "When two lines intersect, they form two pairs of vertical angles" (this explains angle formation on intersection).

Step 3: Analyze Statement 3

• Statement: ∠1 and ∠2 are opposite angles
• Reason: From the intersection of lines, opposite angles are formed. The reason related to intersecting lines and angle positions: "Definition of intersecting lines" (intersecting lines create opposite angles as part of their intersection). Wait, no—better: "When two lines intersect, they form two pairs of vertical angles" is for formation, but for "opposite angles"—actually, the reason for opposite angles when lines intersect: Let's re - check. The correct reason for ∠1 and ∠2 being opposite angles (formed by intersecting lines) is related to the intersection. Wait, the Reason Bank has "When two lines intersect, they form two pairs of vertical angles" which also implies angle formation and their positions. Alternatively, "Definition of intersecting lines" says that intersecting lines meet at a point and form angles, including opposite ones. But more accurately, the reason for ∠1 and ∠2 being opposite angles (after lines intersect) is "When two lines intersect, they form two pairs of vertical angles" (since vertical angles are opposite angles formed by intersecting lines). Wait, maybe I made a mistake. Let's go step by step.

1. Statement 1: Line AB intersects line CD at point E

• Reason: Given (from Reason Bank: "Given")

1. Statement 2: ∠1 and ∠2 are formed

• Reason: When two lines intersect, they form two pairs of vertical angles (from Reason Bank: "When two lines intersect, they form two pairs of vertical angles")

1. Statement 3: ∠1 and ∠2 are opposite angles

• Reason: Definition of intersecting lines (intersecting lines create opposite angles at the intersection point) or more precisely, from the intersection, the angles opposite each other are formed. Wait, the Reason Bank has "Definition of intersecting lines" which states that two lines intersect if they meet at a point, and when they do, they form angles, including opposite angles. But maybe a better fit: "When two lines intersect, they form two pairs of vertical angles" - but we used that for statement 2. Wait, no—statement 2 is about the formation of ∠1 and ∠2, so "When two lines intersect, they form two pairs of vertical angles" explains that angles are formed. Then statement 3: ∠1 and ∠2 are opposite angles. The reason could be related to the intersection—"Definition of intersecting lines" (since intersecting lines create opposite angles). Alternatively, maybe "Definition of vertical angles" but no, vertical angles are a type of opposite angles. Wait, let's check the Reason Bank again.

Reason Bank:

• Definition of vertical angles
• Given
• Definition of opposite rays
• Definition of intersecting lines
• Vertical angles are congruent
• Definition of congruent angles
• Angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles
• When two lines intersect, they form two pairs of vertical angles

Let's re - assign:

1. Statement 1: Line AB intersects line CD at point E

• Reason: Given (this is the starting information)

1. Statement 2: ∠1 and ∠2 are formed

• Reason: When two lines intersect, they form two pairs of vertical angles (this explains that intersection of lines leads to angle formation)

1. Statement 3: ∠1 and ∠2 are opposite angles

• Reason: Definition of intersecting lines (intersecting lines create opposite angles at the point of intersection)

1. Statement 4: ∠1 and ∠2 share a common vertex (E)

• Reason: (From the intersection of lines, the angles formed share the intersection point as vertex) - but from the Reason Bank, maybe no direct reason, but wait, the angles formed by intersecting lines share the vertex (the intersection point). But looking at the Reason Bank, maybe "Angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles" - no, that's for statement 6. Wait, maybe I messed up. Let's try a different approach.

Let's list the statements and match with reasons:

• Statement 1: Line AB intersects line CD at point E
• Reason: Given (because it's the initial information)

• Statement 2: ∠1 and ∠2 are formed
• Reason: When two lines intersect, they form two pairs of vertical angles (this is the reason for angle formation on intersection)

• Statement 3: ∠1 and ∠2 are opposite angles
• Reason: (From the intersection, the angles are opposite) - the Reason "Definition of intersecting lines" says that intersecting lines meet at a point and form angles, including opposite ones. So "Definition of intersecting lines"

• Statement 4: ∠1 and ∠2 share a common vertex (E)
• Reason: (Since they are formed by intersecting lines at E, they share E as vertex) - but from the Reason Bank, maybe no direct reason, but wait, the next statement is about opposite rays. Let's move to statement 5.

• Statement 5: The sides of ∠1 and ∠2 form two pairs of opposite rays
• Reason: Definition of opposite rays (this defines what opposite rays are, and here the sides of the angles are opposite rays)

• Statement 6: ∠1 and ∠2 are vertical angles
• Reason: Angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles (this is the definition of vertical angles, matching the statement)

• Statement 7: Vertical angles are congruent
• Reason: Vertical angles are congruent (from Reason Bank)

• Statement 8: ∠1 ≅ ∠2
• Reason: Vertical angles are congruent (using the theorem from statement 7, so by substitution or direct application)

Now, let's formalize the two - column proof with reasons:

Statement	Reason
2. ∠1 and ∠2 are formed	When two lines intersect, they form two pairs of vertical angles
3. ∠1 and ∠2 are opposite angles	Definition of intersecting lines

| 4. ∠1 and ∠2 share a common vertex (E) | (Implicit from intersection, but since we need to use the bank, maybe no - wait, maybe I made a mistake. Let's re - check. The Reason "Angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles" - no, that's for statement 6. Wait, maybe statement 4's reason is just part of the intersection, but since the Reason Bank has to be used, maybe we can adjust. Alternatively, statement 4: ∠1 and ∠2 share a common vertex (E) - Reason: (From the intersection, the vertex is E) but since the Reason Bank has "Angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles" - no. Maybe I was wrong in statement 3. Let's try again.

Correct matching:

1. Statement 1: Line AB intersects line CD at point E

• Reason: Given

1. Statement 2: ∠1 and ∠2 are formed

• Reason: When two lines intersect, they form two pairs of vertical angles

1. Statement 3: ∠1 and ∠2 are opposite angles

• Reason: (From the intersection, the angles are opposite) - the Reason "Definition of vertical angles" is not right. Wait, the Reason "When two lines intersect, they form two pairs of vertical angles" also implies that the angles formed are vertical (opposite) angles. Maybe statement 3's reason is "When two lines intersect, they form two pairs of vertical angles" but we used that for statement 2. No, statement 2 is about formation, statement 3 is about being opposite.

Alternative correct matching (more accurate):

• Statement 1: Line AB intersects line CD at point E
• Reason: Given

• Statement 2: ∠1 and ∠2 are formed
• Reason: When two lines intersect, they form two pairs of vertical angles

• Statement 3: ∠1 and ∠2 are opposite angles
• Reason: Definition of vertical angles (no, vertical angles are opposite angles formed by intersecting lines. So "Definition of vertical angles" says that vertical angles are opposite angles formed by intersecting lines. So statement 3: ∠1 and ∠2 are opposite angles - Reason: Definition of vertical angles

• Statement 4: ∠1 and ∠2 share a common vertex (E)
• Reason: (From the intersection, the vertex is E) - but from the Reason Bank, maybe "Angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles" - no, that's for statement 6. Wait, statement 4: ∠1 and ∠2 share a common vertex (E) - Reason: (Implicit, but since we need to use the bank, maybe we can say that from the intersection of lines, the angles share the vertex. But the Reason Bank has "Definition of intersecting lines" which says that intersecting lines meet at a point, so the angles formed share that point as vertex. So "Definition of intersecting lines"

• Statement 5: The sides of ∠1 and ∠2 form two pairs of opposite rays
• Reason: Definition of opposite rays

• Statement 6: ∠1 and ∠2 are vertical angles
• Reason: Angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles

• Statement 7: Vertical angles are congruent
• Reason: Vertical angles are congruent

• Statement 8: ∠1 ≅ ∠2
• Reason: Vertical angles are congruent (by applying the theorem from statement 7)

Now, let's present the final two - column proof with correct reasons:

Statement	Reason
2. ∠1 and ∠2 are formed	When two lines intersect, they form two pairs of vertical angles
3. ∠1 and ∠2 are opposite angles	Definition of vertical angles
4. ∠1 and ∠2 share a common vertex (E)	Definition of intersecting lines
5. The sides of ∠1 and ∠2 form two pairs of opposite rays	Definition of opposite rays
6. ∠1 and ∠2 are vertical angles	Angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles
7. Vertical angles are congruent	Vertical angles are congruent
8. ∠1 ≅ ∠2	Vertical angles are congruent

Final Answer (Filled Two - Column Proof)

Statement	Reason
2. ∠1 and ∠2 are formed	When two lines intersect, they form two pairs of vertical angles
3. ∠1 and ∠2 are opposite angles	Definition of vertical angles
4. ∠1 and ∠2 share a common vertex (E)	Definition of intersecting lines
5. The sides of ∠1 and ∠2 form two pairs of opposite rays	Definition of opposite rays
6. ∠1 and ∠2 are vertical angles	Angles that share a vertex and whose sides form two pairs of opposite rays are vertical angles
7. Vertical angles are congruent	Vertical angles are congruent
8. ∠1 ≅ ∠2	Vertical angles are congruent