Question

One angle of a triangle measures 24° more than the smallest angle in the triangle. The third angle measures twice the measure of the smallest angle.



What is the measure of the smallest angle?

Answer 1

Let's denote the smallest angle as \( x \).

According to the problem:
- The second angle measures \( x + 24^\circ \).
- The third angle measures \( 2x \).

Since the sum of the angles in a triangle is \( 180^\circ \), we can set up the following equation:

\[
x + (x + 24) + 2x = 180
\]

Combining like terms, we have:

\[
x + x + 24 + 2x = 180
\]
\[
4x + 24 = 180
\]

Now, subtract 24 from both sides:

\[
4x = 180 - 24
\]
\[
4x = 156
\]

Next, divide both sides by 4:

\[
x = \frac{156}{4}
\]
\[
x = 39
\]

Thus, the measure of the smallest angle is \( \boxed{39^\circ} \).

To verify, we can calculate the measures of the other angles:
- The second angle is \( 39 + 24 = 63^\circ \),
- The third angle is \( 2 \times 39 = 78^\circ \).

Now, checking the sum:

\[
39 + 63 + 78 = 180
\]

Since this holds true, our solution is confirmed. The measure of the smallest angle is indeed \( \boxed{39^\circ} \).