Question

1. stair railing: a stair railing is designed as shown in the figure. use the angles identified in the figure to name two pairs of the indicated type of angle pair.

a) complementary angles ∠ & ∠ ∠ & ∠
b) supplementary angles ∠ & ∠ ∠ & ∠
c) vertical angles ∠ & ∠ ∠ & ∠
d) linear pair ∠ & ∠ ∠ & ∠
e) adjacent angles ∠ & ∠ ∠ & ∠

Explanation:

Step1: Recall angle - pair definitions

Complementary angles add up to 90°. Supplementary angles add up to 180°. Vertical angles are opposite each other when two lines intersect. A linear - pair is a pair of adjacent angles that form a straight line. Adjacent angles share a common side and a common vertex.

Step2: Identify complementary angles

In the figure, if we assume right - angled corners, $\angle1$ and $\angle2$ are complementary (since the corner is a right - angle, $\angle1+\angle2 = 90^{\circ}$), and also $\angle3$ and $\angle4$ are complementary.

Step3: Identify supplementary angles

$\angle1$ and $\angle5$ are supplementary (as they form a straight - line), and $\angle4$ and $\angle5$ are supplementary.

Step4: Identify vertical angles

$\angle5$ and $\angle7$ are vertical angles (opposite each other when the two diagonal lines intersect), and $\angle6$ and $\angle8$ are vertical angles.

Step5: Identify linear - pair

$\angle1$ and $\angle5$ form a linear - pair (adjacent and form a straight line), and $\angle4$ and $\angle5$ form a linear - pair.

Step6: Identify adjacent angles

$\angle2$ and $\angle3$ are adjacent angles (share a common side and a common vertex), and $\angle1$ and $\angle8$ are adjacent angles.

Answer:

a) $\angle1$ & $\angle2$, $\angle3$ & $\angle4$
b) $\angle1$ & $\angle5$, $\angle4$ & $\angle5$
c) $\angle5$ & $\angle7$, $\angle6$ & $\angle8$
d) $\angle1$ & $\angle5$, $\angle4$ & $\angle5$
e) $\angle2$ & $\angle3$, $\angle1$ & $\angle8$