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: \(\angle 2 \cong \angle 3\) (definition of vertical angles)
3. Statement 3: \(\angle 2 \cong \angle 6\) (definition of corresponding angles)
4. Statement 4: \(\angle 3 \cong \angle 6\) (missing reason)

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

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. \(\angle 2 \cong \angle 3\)
3. \(\angle 2 \cong \angle 6\)
4. \(\angle 3 \cong \angle 6\)

Reason

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