Question

which angles are complementary to each other?
∠5 and ∠1
∠6 and ∠4
∠3 and ∠4
∠4 and ∠1

Explanation:

Step1: Recall the definition of complementary angles

Complementary angles are two angles whose sum is \(90^{\circ}\).

Step2: Analyze \(\angle3\) and \(\angle4\)

From the figure, \(\angle3 = 90^{\circ}\) (right - angle). But \(\angle3+\angle4
eq90^{\circ}\).

Step3: Analyze \(\angle4\) and \(\angle1\)

They are vertical angles (opposite angles formed by the intersection of two lines), \(\angle4=\angle1\), sum is not \(90^{\circ}\).

Step4: Analyze \(\angle6\) and \(\angle4\)

They are not adjacent in a way to form \(90^{\circ}\) sum.

Step5: Analyze \(\angle5\) and \(\angle1\)

We know that \(\angle3 = 90^{\circ}\), and \(\angle1+\angle2+\angle3+\angle4 = 180^{\circ}\) (straight - line angle). Also, \(\angle2=\angle4\) (vertical angles). Since \(\angle3 = 90^{\circ}\), and \(\angle1+\angle2= 90^{\circ}\) (because \(\angle1+\angle2+\angle3+\angle4=( \angle1+\angle2)+( \angle3+\angle4)=180^{\circ}\) and \(\angle3 = 90^{\circ},\angle4=\angle2\)), \(\angle1+\angle5 = 90^{\circ}\) (because \(\angle2=\angle5\) (vertical angles))

Answer:

\(\angle5\) and \(\angle1\)