Question
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.
What is m∠ABC?
m∠ABC = 15°
m∠ABC = 45°
m∠ABC = 60°
m∠ABC = 75°
What is m∠ABC?
m∠ABC = 15°
m∠ABC = 45°
m∠ABC = 60°
m∠ABC = 75°
Answers
GPT-4o mini
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:
\[
\boxed{60^\circ}
\]
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:
\[
\boxed{60^\circ}
\]