Question

In triangle, A, B, C, comma△ABC, start overline, B, C, end overline, \cong, start overline, A, B, end overline
BC
≅
AB
and m, angle, B, equals, 84, degrees, .m∠B=84
∘
. Find m, angle, A, .m∠A.

Answer 1

In triangle \( ABC \), we are given that \( \overline{BC} \cong \overline{AB} \) and \( m\angle B = 84^\circ \). Since two sides of the triangle are congruent, this is an isosceles triangle with \( BC = AB \).

For isosceles triangles, the angles opposite the equal sides are also equal. Therefore, angles \( A \) and \( C \) are equal:

\[
m\angle A = m\angle C
\]

Let \( m\angle A = m\angle C = x \). The sum of the angles in a triangle is \( 180^\circ \), so we have:

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

Substituting in the known values:

\[
x + 84^\circ + x = 180^\circ
\]

This simplifies to:

\[
2x + 84^\circ = 180^\circ
\]

Subtracting \( 84^\circ \) from both sides gives:

\[
2x = 180^\circ - 84^\circ
\]

\[
2x = 96^\circ
\]

Dividing both sides by 2 results in:

\[
x = 48^\circ
\]

Therefore, the measure of angle \( A \) is:

\[
m\angle A = 48^\circ
\]

In conclusion, the measure of angle \( A \) is \( \boxed{48^\circ} \).