Question

based on the diagram, \\( \angle 3 \\) measures \\( 65 ^ { \circ } \\) by the vertical angles theorem. given that \\( p \parallel q \\) and \\( r \parallel s \\), identify all the other angles that also measure \\( 65 ^ { \circ } \\).
a. \\( \angle 5, \angle 7, \angle 8, \angle 11, \angle 12, \angle 14, \angle 16, \angle 17 \\)
b. \\( \angle 7, \angle 8, \angle 11, \angle 12, \angle 15, \angle 16, \angle 17, \angle 18 \\)
c. \\( \angle 7, \angle 8, \angle 11, \angle 12, \angle 15, \angle 16, \angle 17, \angle 18 \\)
d. \\( \angle 11, \angle 12 \\)

Explanation:

Step1: Identify angles equal to ∠3 due to parallel lines and transversals

• Since \(p\parallel q\) and using the properties of parallel lines (corresponding angles, alternate interior angles), \(\angle7 = \angle3\) (corresponding angles), \(\angle8=\angle3\) (alternate exterior angles).
• Since \(r\parallel s\) and using the properties of parallel lines, \(\angle11=\angle3\) (alternate exterior angles), \(\angle12 = \angle3\) (corresponding angles).
• Also, \(\angle16=\angle11\) (vertical angles) and \(\angle17=\angle12\) (vertical angles).

Answer:

D. \(\angle11,\angle12,\angle16,\angle17\)