Question

Given
\[\qquad m \angle ABD\] is a straight angle.
\[\qquad m \angle CBD = 8x + 16^\circ\]
\[\qquad m \angle ABC = 6x - 60^\circ\]
Find
\[m\angle ABC\]:
Line AD with ray BC. Point B lies on line AD. The sum of the acute angle ABC and the obtuse angle CBD equals 180 degrees.
\[B\]
\[A\]
\[D\]
\[C\]
\[{}^{\circ}\]

Answer 1

Given that \( m \angle ABD \) is a straight angle, we know that the sum of angles \( m \angle ABC \) and \( m \angle CBD \) must equal \( 180^\circ \):

\[
m \angle ABC + m \angle CBD = 180^\circ
\]

We are given:

- \( m \angle CBD = 8x + 16^\circ \)
- \( m \angle ABC = 6x - 60^\circ \)

Substituting these expressions into the equation gives us:

\[
(6x - 60^\circ) + (8x + 16^\circ) = 180^\circ
\]

Now, we combine like terms:

\[
6x + 8x - 60^\circ + 16^\circ = 180^\circ
\]

This simplifies to:

\[
14x - 44^\circ = 180^\circ
\]

Next, we solve for \( x \):

\[
14x = 180^\circ + 44^\circ
\]
\[
14x = 224^\circ
\]
\[
x = \frac{224^\circ}{14} = 16^\circ
\]

Now that we have \( x \), we can find \( m \angle ABC \):

\[
m \angle ABC = 6x - 60^\circ
\]

Substituting \( x = 16^\circ \):

\[
m \angle ABC = 6(16^\circ) - 60^\circ
\]

Calculating \( 6(16^\circ) \):

\[
6(16^\circ) = 96^\circ
\]

Now substituting back:

\[
m \angle ABC = 96^\circ - 60^\circ = 36^\circ
\]

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

\[
\boxed{36^\circ}
\]