Question

In triangle, U, V, W, comma△UVW, start overline, U, V, end overline, \cong, start overline, W, U, end overline
UV
≅
WU
and m, angle, W, equals, 38, degrees, .m∠W=38
∘
. Find m, angle, U, .m∠U.
Answer
Attempt 1 out of 2

Answer 1

In triangle \( \triangle UVW \), you are given that \( \overline{UV} \cong \overline{WU} \) and \( m \angle W = 38^\circ \). Since the sides \( UV \) and \( WU \) are congruent, triangle \( UVW \) is an isosceles triangle with sides \( UV \) and \( WU \) being equal.

In an isosceles triangle, the angles opposite the equal sides are also equal. Therefore:

\[
m \angle U = m \angle V
\]

We can use the fact that the sum of the angles in any triangle adds up to \( 180^\circ \):

\[
m \angle U + m \angle V + m \angle W = 180^\circ
\]

Let \( m \angle U = m \angle V = x \). Thus, we can write the equation as:

\[
x + x + 38^\circ = 180^\circ
\]

This simplifies to:

\[
2x + 38^\circ = 180^\circ
\]

Now, subtract \( 38^\circ \) from both sides:

\[
2x = 180^\circ - 38^\circ
\]

\[
2x = 142^\circ
\]

Now, divide by 2:

\[
x = 71^\circ
\]

So,

\[
m \angle U = 71^\circ
\]

Thus, the measure of angle \( U \) is

\[
\boxed{71^\circ}
\]