Question

determine whether each pair of angles is complementary or supplementary.
∠2 and ∠3
∠1 and ∠2
∠3 and ∠4
(there is a diagram with point k, lines pc, ps, th intersecting at k, angles labeled 1,2,3,4,5 as shown.)

Explanation:

Step1: Recall angle definitions

Complementary angles sum to \(90^\circ\), supplementary to \(180^\circ\).

Step2: Analyze \(\angle 2\) and \(\angle 3\)

\(\angle 2\) and \(\angle 3\) form a linear pair (adjacent, straight line), so \(\angle 2 + \angle 3 = 180^\circ\) (supplementary).

Step3: Analyze \(\angle 1\) and \(\angle 2\)

\(\angle 1\) and \(\angle 2\) are adjacent, with \(\angle 5 = 90^\circ\) (right angle), so \(\angle 1 + \angle 2 = 90^\circ\) (complementary).

Step4: Analyze \(\angle 3\) and \(\angle 4\)

\(\angle 3\) and \(\angle 4\) are vertical angles? No, linear pair? Wait, \(\angle 3\) and \(\angle 4\): \(\angle 3 + \angle 4\)? Wait, \(\angle 3\) and \(\angle 2\) are supplementary, \(\angle 4\) and \(\angle 2\) are vertical? Wait, no, \(\angle 3\) and \(\angle 4\): actually, \(\angle 3\) and \(\angle 4\) form a linear pair? Wait, no, \(\angle 3\) and \(\angle 2\) are adjacent on a straight line (\(PS\)), so \(\angle 2 + \angle 3 = 180^\circ\). \(\angle 4\) and \(\angle 2\) are vertical angles (equal), so \(\angle 3 + \angle 4 = \angle 3 + \angle 2 = 180^\circ\) (supplementary). Wait, no: \(\angle 3\) and \(\angle 4\): let's see, \(PH\) and \(TS\) intersect at \(K\), so \(\angle 4\) and \(\angle 2\) are vertical (equal), \(\angle 3\) and \(\angle 2\) are supplementary, so \(\angle 3\) and \(\angle 4\) are supplementary.

Wait, the table:

• \(\angle 2\) and \(\angle 3\): supplementary (circle supplementary)
• \(\angle 1\) and \(\angle 2\): complementary (circle complementary)
• \(\angle 3\) and \(\angle 4\): supplementary (circle supplementary)

Answer:

For \(\angle 2\) and \(\angle 3\): Supplementary
For \(\angle 1\) and \(\angle 2\): Complementary
For \(\angle 3\) and \(\angle 4\): Supplementary

(Assuming the task is to mark each pair: complementary or supplementary. So for each row, select the correct column.)