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° .(1 point)
Responses
15°
15 degrees
105°
105 degrees
180°
180 degrees
75°
The sum of the measures of the four angles around a point is always 360 degrees. Since angle C is 75 degrees, the sum of angles C, A, B, and D is 360 degrees.
So, 75 + A + B + D = 360.
Since angle B is adjacent to angles A and C, it must be supplementary to both of them. Therefore, angle B is also 75 degrees.
Substituting this value into the equation, we have 75 + A + 75 + D = 360.
Combining like terms, this simplifies to A + D = 210.
Since angles A and D are opposite each other, they must be congruent. Therefore, A = D.
Substituting this into the equation, we have A + A = 210.
Simplifying further, we have 2A = 210.
Dividing both sides by 2, we find that A = 105 degrees.
Therefore, m∠A = 105 degrees.