In the triangle $\triangle K L M$ we have $K L=K M, K T=K S$ and $L \widehat{K} S=30^{\circ}$. What is the measure of the angle $x=T \widehat{S} M$?
(a) $10^{\circ}$
(b) $15^{\circ}$
(c) $20^{\circ}$
(d) $25^{\circ}$
(e) $30^{\circ}$

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-015.jpg?height=277&width=600&top_left_y=1689&top_left_x=1279)
The correct option is (b).

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-057.jpg?height=343&width=751&top_left_y=545&top_left_x=687)

Let \( T \widehat{S} M = x \), \( S \widehat{K} T = y \), \( K \widehat{L} S = \alpha \), and \( K \widehat{T} S = \beta \). The triangle \( \triangle K L M \) is isosceles because it has two equal sides; consequently, its base angles are equal, i.e., \( K \widehat{M} S = K \widehat{L} S = \alpha \). Similarly, the triangle \( \triangle K S T \) is also isosceles and therefore \( K \widehat{S} T = K \widehat{T} S = \beta \). We will now use the fact that the sum of the interior angles of a triangle is \( 180^{\circ} \). Follow the figure:

- In the triangle \( \triangle S M T \), we have \( x + \alpha + 180^{\circ} - \beta = 180^{\circ} \), therefore, \( x = \beta - \alpha \).
- In the triangle \( \triangle K L M \), we have \( \alpha + \alpha + 30^{\circ} + y = 180^{\circ} \), therefore, \( y = 150^{\circ} - 2 \alpha \).
- In the triangle \( \triangle K S T \), we have \( \beta + \beta + 150^{\circ} - 2 \alpha = 180^{\circ} \), therefore, \( \beta - \alpha = 15^{\circ} \).

Thus, \( x = 15^{\circ} \).