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 16, \angle 17 \\)
d. \\( \angle 11, \angle 12 \\)

Explanation:

Step1: Use properties of parallel lines and transversals

• Vertical angles: Vertical angles are equal.
• Corresponding angles: When two parallel lines are cut by a transversal, corresponding angles are equal.
• Alternate interior angles: When two parallel lines are cut by a transversal, alternate interior angles are equal.
• Alternate exterior angles: When two parallel lines are cut by a transversal, alternate exterior angles are equal.

Since \(p\parallel q\) and \(r\parallel s\):

• \(\angle3=\angle7 = 65^{\circ}\) (corresponding angles as \(r\parallel s\) and transversal for \(\angle3\) and \(\angle7\))
• \(\angle3=\angle8 = 65^{\circ}\) (alternate exterior angles as \(p\parallel q\) and transversal)
• \(\angle3=\angle11 = 65^{\circ}\) (alternate interior angles as \(p\parallel q\) and transversal)
• \(\angle3=\angle12 = 65^{\circ}\) (vertical angles with \(\angle11\))
• \(\angle3=\angle16 = 65^{\circ}\) (corresponding angles to \(\angle8\) as \(r\parallel s\))
• \(\angle3=\angle17 = 65^{\circ}\) (vertical angles with \(\angle16\))

Step2: Check each option

• Option A: \(\angle5\) is \(90 - 65=25^{\circ}\) (complementary to \(\angle3\) if we consider right - angle formed in some cases, wrong)
• Option B: \(\angle15 = 180 - 65=115^{\circ}\) (supplementary to \(\angle8\) as \(p\parallel q\) and transversal, wrong)
• Option C: \(\angle7,\angle8,\angle11,\angle12,\angle16,\angle17\) all equal to \(65^{\circ}\) (correct)
• Option D: Only \(\angle11,\angle12\) is not sufficient (wrong)

Answer:

C. \(\angle7,\angle8,\angle11,\angle12,\angle16,\angle17\)