In triangle MNO, MN is extended to point P. The measure of angleMON is 119°, and the measure of angleOMP...

1. TRANSLATE the problem information

Given:

• Triangle MNO
• MN is extended to point P (so P-M-N are collinear with M between P and N)
• \(\angle\mathrm{MON} = 119°\) (interior angle at vertex O)
• \(\angle\mathrm{OMP} = 156°\) (angle at M on the exterior side)
• Need to find: \(\angle\mathrm{OMN}\) (interior angle at vertex M)

2. INFER the geometric relationship

Key insight: Look at point M carefully. The line segment MN is extended backward to point P. This means:

• Points P, M, and N all lie on the same straight line
• Angle \(\angle\mathrm{OMN}\) is inside the triangle (interior angle)
• Angle \(\angle\mathrm{OMP}\) is outside the triangle (exterior angle)
• These two angles are adjacent and their outer sides (MP and MN) form a straight line

Therefore: \(\angle\mathrm{OMN}\) and \(\angle\mathrm{OMP}\) form a linear pair, which means they're supplementary angles that sum to 180°.

3. SIMPLIFY to find the answer

Set up the equation:

\(\angle\mathrm{OMN} + \angle\mathrm{OMP} = 180°\)

Substitute the known value:

\(\angle\mathrm{OMN} + 156° = 180°\)

Solve:

\(\angle\mathrm{OMN} = 180° - 156° = 24°\)

Answer: 24° (Choice B)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the simple linear pair relationship and instead attempting to use the triangle angle sum theorem (all angles in a triangle sum to 180°).

A student might think: "I know \(\angle\mathrm{MON} = 119°\). If I can find \(\angle\mathrm{MNO}\), then I can use the triangle angle sum to find \(\angle\mathrm{OMN}\)." They might incorrectly reason that \(\angle\mathrm{MNO}\) and \(\angle\mathrm{OMP}\) have some direct relationship, or make calculation errors trying to work through multiple steps. Through faulty reasoning about the angle relationships, they might calculate: \(\angle\mathrm{OMN} = 180° - 119° - 24° = 37°\).

This may lead them to select Choice D (37°).

Second Most Common Error:

Poor TRANSLATE reasoning: Confusing which angle is which, particularly mixing up \(\angle\mathrm{OMN}\) (interior) with \(\angle\mathrm{OMP}\) (exterior), or not understanding what "MN is extended to point P" means geometrically.

If a student doesn't correctly identify the angles from the diagram and description, they might subtract the two given angles: \(156° - 119° = 37°\), thinking this represents some meaningful relationship.

This may also lead them to select Choice D (37°).

The Bottom Line:

This problem tests whether you can identify the most direct path to the solution. The key is recognizing that when a line segment is extended, the interior and exterior angles at that vertex form a linear pair - a much simpler relationship than trying to use triangle angle sum. Once you see this geometric relationship, the problem becomes a simple one-step subtraction from 180°.