Question

are these scalene triangles similar?
u
32°
w v
t
103°
32°
s
yes
no

Explanation:

Step1: Find angle at U in triangle UVW

In triangle \( TSW \), angles are \( 103^\circ \), \( 32^\circ \), so the third angle \( \angle TWS = 180^\circ - 103^\circ - 32^\circ = 45^\circ \)? Wait, no, wait. Wait, vertical angles: \( \angle UWV \) and \( \angle TWS \) are vertical angles, so they are equal. Wait, first, let's find the third angle in triangle \( TSW \). Sum of angles in a triangle is \( 180^\circ \). So \( \angle TSW = 32^\circ \), \( \angle T = 103^\circ \), so \( \angle TWS = 180 - 103 - 32 = 45^\circ \). Now, in triangle \( UVW \), angle at \( U \) is \( 32^\circ \), angle at \( W \) is equal to \( \angle TWS \) (vertical angles) which is \( 45^\circ \)? Wait, no, wait, maybe I made a mistake. Wait, let's check triangle \( UVW \): angle at \( U \) is \( 32^\circ \), angle at \( W \) is vertical to \( \angle TWS \), but let's find angle at \( V \). Wait, no, maybe another approach. Wait, in triangle \( TSW \), angles are \( 103^\circ \) (at \( T \)), \( 32^\circ \) (at \( S \)), so angle at \( W \) is \( 180 - 103 - 32 = 45^\circ \). Now, in triangle \( UVW \), angle at \( U \) is \( 32^\circ \), angle at \( W \) is \( 45^\circ \) (vertical angles), so angle at \( V \) is \( 180 - 32 - 45 = 103^\circ \). Wait, so triangle \( UVW \) has angles \( 32^\circ \), \( 45^\circ \), \( 103^\circ \)? No, wait, no, maybe I messed up the triangles. Wait, the two triangles are \( TSW \) and \( UVW \), with \( W \) being the intersection point. So angle at \( S \) is \( 32^\circ \), angle at \( U \) is \( 32^\circ \) (so that's a pair of equal angles). Angle at \( T \) is \( 103^\circ \), let's find angle at \( V \). Wait, in triangle \( TSW \), angles: \( \angle T = 103^\circ \), \( \angle S = 32^\circ \), so \( \angle W = 45^\circ \). In triangle \( UVW \), angle at \( U = 32^\circ \), angle at \( W \) is equal to \( \angle TWS \) (vertical angles) which is \( 45^\circ \), so angle at \( V = 180 - 32 - 45 = 103^\circ \). So now, triangle \( TSW \) has angles \( 103^\circ \), \( 32^\circ \), \( 45^\circ \). Triangle \( UVW \) has angles \( 32^\circ \), \( 45^\circ \), \( 103^\circ \). Wait, but \( 103^\circ \) is present in both? Wait, no, triangle \( TSW \) has \( 103^\circ \) at \( T \), and triangle \( UVW \) has \( 103^\circ \) at \( V \). And both have \( 32^\circ \) at \( S \) and \( U \). So that's two pairs of equal angles: \( \angle S = \angle U = 32^\circ \), and \( \angle T = \angle V = 103^\circ \). Wait, so by AA (Angle-Angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. So \( \angle S = \angle U = 32^\circ \), \( \angle T = \angle V = 103^\circ \), so AA similarity applies. Therefore, the triangles are similar. Wait, but let's confirm. Let's list the angles:

For triangle \( TSW \):
- \( \angle T = 103^\circ \)
- \( \angle S = 32^\circ \)
- \( \angle W = 180 - 103 - 32 = 45^\circ \)

For triangle \( UVW \):
- \( \angle U = 32^\circ \) (equal to \( \angle S \))
- \( \angle V = 103^\circ \) (equal to \( \angle T \))
- \( \angle W = 45^\circ \) (equal to \( \angle W \) in \( TSW \), but actually, the vertical angles: \( \angle TWS = \angle UVW \)? No, wait, \( \angle TWS \) and \( \angle UVW \)'s angle at \( W \) are vertical angles, so they are equal. Wait, maybe I confused the angle labels. But the key is: \( \angle S = \angle U = 32^\circ \), and \( \angle T = \angle V = 103^\circ \), so two angles are equal, so by AA similarity, the triangles are similar. Therefore, the answer is yes.

Wait, maybe a simpler way: in triangle \( TSW \), angles are \( 103^\circ \), \( 32^\circ \), and \( 45^\circ \). In triangle \( UVW \), angle at \( U \) is \( 32^\circ \), angle at \( V \) would be \( 103^\circ \) (since \( 180 - 32 - 45 = 103 \)), so that's equal to angle at \( T \). So two angles are equal (32° and 103°), so AA similarity holds. Therefore, the triangles are similar.

Step1: Identify equal angles

- In triangle \( TSW \): \( \angle S = 32^\circ \), \( \angle T = 103^\circ \).
- In triangle \( UVW \): \( \angle U = 32^\circ \) (equal to \( \angle S \)).
- Calculate \( \angle V \) in \( UVW \): Sum of angles in a triangle is \( 180^\circ \). Let \( \angle W \) (vertical angle to \( \angle TWS \)) be \( x \). In \( TSW \), \( x = 180 - 103 - 32 = 45^\circ \). Then \( \angle V = 180 - 32 - 45 = 103^\circ \), which equals \( \angle T \).

Step2: Apply AA Similarity Criterion

- AA (Angle-Angle) similarity: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
- Here, \( \angle S = \angle U = 32^\circ \) and \( \angle T = \angle V = 103^\circ \), so two angles are equal. Thus, the triangles are similar.

Answer:

yes