Look at the figure below. Which of the following statements could be a step in a proof that vertical angles are congruent? \angle APC \cong \angle DPB because angles complementary to the same angle are congruent to each other. \angle APC \cong \angle DPB because angles adjacent to the same angle are congruent to each other. \angle APC \cong \angle DPB because angles formed by the same pair of intersecting lines are congruent to each other. \angle APC \cong \angle DPB because angles supplementary to the same angle are congruent to each other.
The Correct Answer and Explanation is:
The correct answer is:
∠APC ≅ ∠DPB because angles formed by the same pair of intersecting lines are congruent to each other.
Explanation
To understand why ∠APC ≅ ∠DPB, we need to recall the definition of vertical angles and a fundamental property of intersecting lines.
Vertical angles (also called opposite angles) are the angles that are across from each other when two lines intersect. When two lines cross, they form two pairs of vertical angles. These angles are not adjacent (they do not share a common side), but they share a common vertex, and the sides of the angles form two pairs of opposite rays.
For example, in the figure mentioned (though not shown here), if lines AC and BD intersect at point P, then ∠APC and ∠DPB are vertical angles. One angle is formed by rays PA and PC, and the other by rays PD and PB. These two angles are across from each other and share vertex P.
The key geometric rule is:
Vertical angles are always congruent.
This is because when two lines intersect, they form two pairs of adjacent angles that are supplementary (they add up to 180°). If each angle shares a supplementary relationship with its adjacent angle, and both adjacent pairs are the same (due to the same intersecting lines), then the opposite angles must be equal.
Thus, ∠APC and ∠DPB are congruent because they are formed by the same pair of intersecting lines.
Let’s look at the other options:
- Complementary angles add up to 90°, not relevant here.
- Adjacent angles can be supplementary or not, but not necessarily congruent.
- Angles supplementary to the same angle are not necessarily congruent.
Only the third option correctly identifies the reason for the congruence: Vertical angles formed by the same intersecting lines are always congruent.
