Question

what are the values of x and y?
(there is a triangle diagram with vertices u, v, w. angle at u is 30°, sides uw and uv have markings indicating they are equal? wait, no, the sides with red marks: one on wv and one on uv? wait, the diagram: triangle uwv? wait, vertices u, w, v. side uw and side uv? wait, the red marks: one on wv (the side from w to v) and one on uv (the side from u to v). so triangle uwv, with uw and uv? wait, no, the angle at u is 30°, side wv has a red mark, side uv has a red mark? wait, maybe its an isosceles triangle. then x is the angle at w, y is the angle at v. then we need to find x and y. the ocr text also has x = box°, y = box°)

Explanation:

Step1: Identify the triangle type

The triangle \( \triangle UWV \) has two sides marked as equal (the tick marks), so it is an isosceles triangle. In an isosceles triangle, the base angles are equal. The angle at \( U \) is \( 30^\circ \), and the two equal sides are \( UW \) and \( UV \)? Wait, no, looking at the diagram, the equal sides are \( WV \) and \( UV \)? Wait, the tick marks: one on \( WV \) and one on \( UV \)? Wait, no, the diagram shows tick marks on \( WV \) (the side from \( W \) to \( V \)) and on \( UV \) (the side from \( U \) to \( V \))? Wait, no, let's re-examine. The triangle has vertices \( U \), \( V \), \( W \). The side \( WV \) has a tick mark, and the side \( UV \) has a tick mark? Wait, no, maybe the two equal sides are \( UW \) and \( UV \)? Wait, no, the angle at \( U \) is \( 30^\circ \), and the two equal sides are \( WV \) and \( UV \)? Wait, no, in an isosceles triangle, the angles opposite the equal sides are equal. So if sides \( UW \) and \( UV \) are equal? Wait, no, the tick marks: one on \( WV \) (from \( W \) to \( V \)) and one on \( UV \) (from \( U \) to \( V \))? Wait, maybe the two equal sides are \( WV \) and \( UV \), so the angles opposite them are \( \angle U \) and \( \angle W \). Wait, no, angle at \( U \) is \( 30^\circ \), so angle at \( W \) (which is \( x \)) should be equal to angle at \( U \) if the sides opposite are equal? Wait, no, let's clarify.

Wait, the triangle has sides: \( UW \), \( UV \), and \( WV \). The tick marks are on \( WV \) and \( UV \)? Wait, no, the diagram: \( W \) at the top, \( V \) at the bottom right, \( U \) at the bottom left. The side from \( W \) to \( V \) (vertical) has a tick mark, and the side from \( U \) to \( V \) (horizontal? No, slanting) has a tick mark. So \( WV = UV \). Therefore, the angles opposite these sides are \( \angle U \) (opposite \( WV \)) and \( \angle W \) (opposite \( UV \)). Therefore, \( \angle U = \angle W \). Wait, angle at \( U \) is \( 30^\circ \), so \( x = 30^\circ \).

Step2: Calculate \( y \)

The sum of angles in a triangle is \( 180^\circ \). So \( \angle U + \angle W + \angle V = 180^\circ \). We know \( \angle U = 30^\circ \), \( \angle W = x = 30^\circ \), so \( 30^\circ + 30^\circ + y = 180^\circ \). Wait, no, wait: \( \angle V \) is \( y \). So \( 30 + x + y = 180 \). But since it's isosceles, \( x = 30^\circ \) (because the two equal sides are \( UV \) and \( WV \), so angles opposite are \( \angle W \) (opposite \( UV \)) and \( \angle U \) (opposite \( WV \))? Wait, no, angle opposite \( UV \) is \( \angle W \), and angle opposite \( WV \) is \( \angle U \). So if \( UV = WV \), then \( \angle W = \angle U = 30^\circ \), so \( x = 30^\circ \). Then, sum of angles: \( 30 + 30 + y = 180 \). So \( 60 + y = 180 \), so \( y = 180 - 60 = 120^\circ \).

Wait, that makes sense. So:

Step1: Determine \( x \)

Since the triangle is isosceles with \( UV = WV \), the angles opposite these sides are equal. The angle opposite \( UV \) is \( \angle W \) (which is \( x \)), and the angle opposite \( WV \) is \( \angle U \) (which is \( 30^\circ \)). Therefore, \( x = 30^\circ \).

Step2: Calculate \( y \)

The sum of the interior angles of a triangle is \( 180^\circ \). So:
\[
\angle U + \angle W + \angle V = 180^\circ
\]
Substituting \( \angle U = 30^\circ \), \( \angle W = x = 30^\circ \), and \( \angle V = y \):
\[
30^\circ + 30^\circ + y = 180^\circ
\]
\[
60^\circ + y = 180^\circ
\]
\[
y = 180^\circ - 60^\circ = 120^\circ
\]

Answer:

\( x = 30^\circ \), \( y = 120^\circ \)

So, \( x = \boxed{30} \), \( y = \boxed{120} \)