Question

which angles are complementary to each other? select all that apply.
∠3 and ∠6 ∠3 and ∠2 ∠5 and ∠6 ∠4 and ∠3

Explanation:

Step1: Recall Complementary Angles

Complementary angles sum to \(90^\circ\) (a right angle). From the diagram, \(\angle 6\) is a right angle (marked with a right - angle symbol), so \(\angle 3+\angle 6 = 90^\circ\) (since \(\angle 3\) and \(\angle 6\) form the right angle), \(\angle 5+\angle 6=90^\circ\) (because \(\angle 5\) and \(\angle 6\) are adjacent and form the right angle at the intersection), and \(\angle 4+\angle 3 = 90^\circ\) (since \(\angle 4\) and \(\angle 3\) are adjacent and form part of the right - angle structure). For \(\angle 3\) and \(\angle 2\), there is no indication that their sum is \(90^\circ\).

Step2: Evaluate Each Option

• For \(\angle 3\) and \(\angle 6\): Since \(\angle 6\) is a right angle (\(90^\circ\)) and \(\angle 3\) and \(\angle 6\) are adjacent angles that form the right angle, \(\angle 3+\angle 6 = 90^\circ\), so they are complementary.
• For \(\angle 3\) and \(\angle 2\): There is no geometric relationship (like forming a right angle) that would make their sum equal to \(90^\circ\), so they are not complementary.
• For \(\angle 5\) and \(\angle 6\): Since \(\angle 6\) is a right angle (\(90^\circ\)) and \(\angle 5\) and \(\angle 6\) are adjacent angles that form the right angle, \(\angle 5+\angle 6=90^\circ\), so they are complementary.
• For \(\angle 4\) and \(\angle 3\): \(\angle 4\) and \(\angle 3\) are adjacent angles that form part of the right - angle structure, so \(\angle 4+\angle 3 = 90^\circ\), so they are complementary.

Answer:

\(\angle 3\) and \(\angle 6\), \(\angle 5\) and \(\angle 6\), \(\angle 4\) and \(\angle 3\) (i.e., the options: \(\boldsymbol{\angle 3}\) and \(\boldsymbol{\angle 6}\), \(\boldsymbol{\angle 5}\) and \(\boldsymbol{\angle 6}\), \(\boldsymbol{\angle 4}\) and \(\boldsymbol{\angle 3}\))