In the figure, two lines intersect at a point. Angle 1 and Angle 2 are vertical angles. The measure of...

1. TRANSLATE the problem information

• Given information:
  • Two lines intersect at a point
  • Angle 1 and angle 2 are vertical angles
  • Measure of angle 1 = \(72°\)

• What we need to find:
  • Measure of angle 2

2. INFER what property applies

• The problem tells us that angles 1 and 2 are vertical angles. This is the crucial piece of information.

• When two lines intersect, they form four angles at the intersection point. The angles directly across from each other (opposite sides of the intersection) are called vertical angles.

• Key property to apply: Vertical angles are always congruent, meaning they have equal measures.

3. Apply the vertical angles property

• Since angle 1 and angle 2 are vertical angles, they must be equal in measure.

• Therefore: measure of angle 2 = measure of angle 1 = \(72°\)

Answer: A. 72°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill - Confusing vertical angles with supplementary angles:

Students sometimes confuse the relationship between vertical angles with the relationship between supplementary angles. When two lines intersect, adjacent angles (angles that share a side) are supplementary, meaning they add up to 180°.

A student might mistakenly think: "Angle 1 and angle 2 must add to 180° because they're formed by intersecting lines." This leads to the calculation:

• Angle 2 = \(180° - 72°\)
• \(= 108°\)

This may lead them to select Choice B (108°).

Second Most Common Error:

Weak INFER skill - Adding angles instead of recognizing congruence:

Some students might think they need to perform an operation with both angles rather than recognizing they're equal. They might incorrectly add angle 1 to itself or add both angles together:

• \(72° + 72°\)
• \(= 144°\)

This may lead them to select Choice C (144°).

The Bottom Line:

The key to this problem is recognizing and correctly applying the vertical angles property. Students must distinguish between vertical angles (which are equal) and supplementary angles (which add to 180°). This is a fundamental geometry concept that appears frequently, so understanding the distinction is crucial.