Question

Two intersecting lines create four angles with the same vertex, labeled A, B,C, and D. angle B is adjacent to <A and <C. Find m<A if m<C = 75 .
A. 15
B. 180
C. 105
D. 75

Answer 1

When two lines intersect, they create pairs of vertical angles, which are equal, and adjacent angles that are supplementary (add up to 180 degrees).

Given that \( m\angle C = 75^\circ \), we know that the angle adjacent to angle C, which is angle B, must be:

\[
m\angle B + m\angle C = 180^\circ
\]

Substituting \( m\angle C \):

\[
m\angle B + 75^\circ = 180^\circ
\]

Solving for \( m\angle B \):

\[
m\angle B = 180^\circ - 75^\circ = 105^\circ
\]

Now, angle A is adjacent to angle B, so we apply the same supplementary relationship:

\[
m\angle A + m\angle B = 180^\circ
\]

Substituting \( m\angle B \):

\[
m\angle A + 105^\circ = 180^\circ
\]

Solving for \( m\angle A \):

\[
m\angle A = 180^\circ - 105^\circ = 75^\circ
\]

Thus, the measure of angle A, \( m\angle A \), is:

\[
\boxed{75}
\]