Question

given: x ∥ y and w is a transversal
prove: ∠3 ≅ ∠6
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

Explanation:

Step1: Recall Properties

We know from steps 2 and 3 that \( \angle 2 \cong \angle 3 \) and \( \angle 2 \cong \angle 6 \). The transitive property of congruence states that if \( a \cong b \) and \( b \cong c \), then \( a \cong c \). Here, let \( a=\angle 3 \), \( b = \angle 2 \), \( c=\angle 6 \). So since \( \angle 3\cong\angle 2 \) and \( \angle 2\cong\angle 6 \), by transitive property, \( \angle 3\cong\angle 6 \).

Step2: Eliminate Other Options

• Symmetric property: It states if \( a\cong b \), then \( b\cong a \), which is not relevant here.
• Vertical angles are congruent: This was used in step 2, not step 4.
• Definition of supplementary angles: Supplementary angles sum to \( 180^\circ \), not related to this congruence chain.

Answer:

transitive property