Question

LN and OQ are parallel lines.

K
M
P
ZNMP and ZOPM
R
Which angles are supplementary angles?
ZNMP and 2LMP
N
ZNMP and LMK
ZNMP and QPR

Answer 1

∠NMP and ∠LMP

Explanation:

The given diagram shows two parallel lines, LN and OQ. A transversal line, KR, intersects these parallel lines at two distinct points: point M lies on line LN, while point P is situated on line OQ.

Supplementary angles are a pair of angles whose measures add up to 180°.

To determine which pair of angles are supplementary, we can analyse each pair.

`\boxed{\angle NMP\;\textsf{and}\;\angle OPM}`

Alternate interior angles are pairs of angles that lie on opposite sides of a transversal line and inside a pair of parallel lines. These angles are always congruent. Since ∠NMP and ∠OPM are alternate interior angles, they are congruent.

`\boxed{\angle NMP\;\textsf{and}\;\angle LMK}`

Vertical angles are a pair of non-adjacent (opposite) angles formed by the intersection of two lines, and they are always congruent. Since ∠NMP and ∠LMK are vertical angles, they are congruent.

`\boxed{\angle NMP\;\textsf{and}\;\angle LMP}`

A linear pair consists of two adjacent angles formed by two intersecting lines, whose measures sum to 180°. Since ∠NMP and ∠LMP are a linear pair, their measures add up to 180°.

`\boxed{\angle NMP\;\textsf{and}\;\angle QPR}`

When a transversal intersects two parallel lines, the pairs of angles formed that are in the same relative position are called corresponding angles. These angles are always congruent. Since ∠NMP and ∠QPR are corresponding angles, they are congruent.

Conclusion

The supplementary angles are ∠NMP and ∠LMP, as they form a linear pair, and so their measures add up to 180°.

Answer 2

∠NMP and ∠LMP

Explanation:

"Supplementary angles are two angles that add up to 180 degrees. When two supplementary angles are placed next to each other, they form a straight line."

Let's see the options:

∠NMP and ∠OPM

Since ∠NMP and ∠OPM are Alternate angles, they are equal to each other.

So, ∠NMP and ∠OPM are not supplementary.

∠NMP and ∠LMK

Since ∠NMP and ∠LMK are vertically opposite angles, they are equal to each other.

So, ∠NMP and ∠LMK are not supplementary.

∠NMP and ∠LMP

Since ∠NMP and ∠LMK are pair angles of linear pair.

So, ∠NMP and ∠LMP are supplementary.

∠NMP and ∠QPR

Since ∠NMP and ∠LMK are corresponding angles, they are equal to each other.

So, ∠NMP and ∠LMK are not supplementary.

Therefore, supplementary angles are:

∠NMP and ∠LMP