Given: x ∥ y and w is a transversal

Question

Prove: ∠3 ≅ ∠6

Parallel lines x and y are cut by transversal w. On line x where it intersects with line w, 4 angles are created. Labeled clockwise, from uppercase left, the angles are: 2, 4, 3, 1. On line y where it intersects with line w, 4 angles are created. Labeled clockwise, from uppercase left, the angles are: 6, 8, 7, 5.

What is the missing reason in the proof?


Statement

Reason
1. x ∥ y
w is a transversal 1. given
2. ∠2 ≅ ∠3 2. def. of vert. ∠s
3. ∠2 ≅ ∠6 3. def. of corr. ∠s
4. ∠3 ≅ ∠6 4.
transitive property
symmetric property
vertical angles are congruent
definition of supplementary angles

Answer 1

To complete the proof, let's analyze the statements provided and determine what the missing reason should be:

1. Statement 1: $x\parallel y$ (given)
2. Statement 2: $\mathrm{\angle }2\cong \mathrm{\angle }3$ (definition of vertical angles)
3. Statement 3: $\mathrm{\angle }2\cong \mathrm{\angle }6$ (definition of corresponding angles)
4. Statement 4: $\mathrm{\angle }3\cong \mathrm{\angle }6$ (missing reason)

In order to correctly conclude that $\mathrm{\angle }3\cong \mathrm{\angle }6$, we can see that this follows from the Transitive Property of Congruence. According to the Transitive Property, if $\mathrm{\angle }2\cong \mathrm{\angle }3$ and $\mathrm{\angle }2\cong \mathrm{\angle }6$, then it can be concluded that $\mathrm{\angle }3\cong \mathrm{\angle }6$.

So, the appropriate missing reason for Statement 4 is:

Reason: Transitive Property of Congruence.

Thus, the completed proof will look like this:

Statement

1. $x\parallel y$
2. $\mathrm{\angle }2\cong \mathrm{\angle }3$
3. $\mathrm{\angle }2\cong \mathrm{\angle }6$
4. $\mathrm{\angle }3\cong \mathrm{\angle }6$

Reason

1. Given
2. Definition of vertical angles
3. Definition of corresponding angles
4. Transitive Property of Congruence.