Triangle A B C is shown with its exterior angles. Line C B extends through point D. Line B C extends to form exterior angle that is 135 degrees. Angle C A B is 75 degrees.

Question

What is m∠ABC?

m∠ABC = 15°
m∠ABC = 45°
m∠ABC = 60°
m∠ABC = 75°

Answer 1

To find \( m\angle ABC \), we can use the information provided about the exterior angle at point C.

The information given is as follows:

• The exterior angle at B (which is \( \angle DBC \)) measures \( 135^\circ \).
• The interior angle \( \angle CAB = 75^\circ \).

According to the exterior angle theorem, the measure of an exterior angle is equal to the sum of the measures of the two non-adjacent interior angles. Thus:

\[ m\angle DBC = m\angle CAB + m\angle ABC \]

Substituting the known values, we have:

\[ 135^\circ = 75^\circ + m\angle ABC \]

To isolate \( m\angle ABC \), we subtract \( 75^\circ \) from both sides:

\[ m\angle ABC = 135^\circ - 75^\circ = 60^\circ \]

Thus, the measure of angle \( ABC \) is:

\[ \boxed{60^\circ} \]