Question

determine if triangle uvw and triangle xyz are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

answer attempt 1 out of 2
the triangles similar.

Explanation:

Step1: Identify corresponding sides and angle

First, we check the sides around the given angle (49° in triangle UVW). The sides adjacent to the 49° angle in triangle UVW are \( UV = 28 \) and \( VW = 16 \)? Wait, no, wait. Wait, in triangle UVW, the angle at V is 49°, with sides \( UV = 28 \), \( VW = 16 \)? Wait, no, looking at the diagram: triangle UVW has sides: UV = 28, VW = 16, and UW = 22? Wait, no, maybe I misread. Wait, triangle UVW: vertices U, V, W. So UV is 28, VW is 16, and UW is 22? Wait, no, the other triangle XYZ: sides YZ = 80, XY = 140, XZ = 110? Wait, no, let's list the sides properly.

Wait, triangle UVW: sides: UV = 28, VW = 16, and UW = 22? Wait, no, maybe the sides are UV = 28, VW = 16, and the angle at V is 49°. Then triangle XYZ: let's see the sides. Let's check the ratios of the sides. Let's pair the sides. Let's see:

In triangle UVW, the sides around the 49° angle (angle at V) are UV = 28 and VW = 16? Wait, no, maybe UV is 28, VW is 16, and the other side? Wait, no, maybe the sides are UV = 28, UW = 22, VW = 16? Wait, no, let's check the other triangle. Triangle XYZ: sides YZ = 80, XY = 140, XZ = 110? Wait, no, maybe YZ = 80, XZ = 110, XY = 140. Wait, let's check the ratios.

Wait, let's list the sides of triangle UVW: UV = 28, VW = 16, UW = 22? Wait, no, maybe I got the sides wrong. Wait, the first triangle: U to V is 28, V to W is 16, W to U is 22. The second triangle: Z to X is 110, X to Y is 140, Y to Z is 80. Wait, let's check the ratios of the sides. Let's see:

If we consider the sides adjacent to the angle (maybe the included angle). Wait, maybe the angle at V (49°) in triangle UVW, and in triangle XYZ, is there a corresponding angle? Wait, maybe we need to check the ratios of the sides. Let's compute the ratios of the sides.

Wait, let's list the sides of triangle UVW: UV = 28, VW = 16, UW = 22.

Triangle XYZ: let's see the sides. Let's assume the sides are YZ = 80, XY = 140, XZ = 110. Wait, no, maybe YZ = 80, XZ = 110, XY = 140. Wait, let's check the ratios:

First, check the ratio of UV to YZ: 28 / 80 = 7 / 20 ≈ 0.35

VW to XY: 16 / 140 = 4 / 35 ≈ 0.114, which is not equal. Wait, that can't be. Wait, maybe I paired the wrong sides.

Wait, maybe the sides are UW = 22, UV = 28, and the other triangle: XZ = 110, XY = 140, YZ = 80. Wait, 22 / 110 = 1/5 = 0.2; 28 / 140 = 1/5 = 0.2; 16 / 80 = 1/5 = 0.2. Oh! Wait, that's the key. Let's check:

UW = 22, XZ = 110: 22 / 110 = 1/5

UV = 28, XY = 140: 28 / 140 = 1/5

VW = 16, YZ = 80: 16 / 80 = 1/5

So all three sides are in the ratio 1/5. Wait, but also, is there an included angle? Wait, the angle at V is 49°, but in triangle XYZ, is there a corresponding angle? Wait, no, maybe the triangles are similar by SSS (Side-Side-Side) similarity, because all three sides are in proportion. Wait, but wait, the angle at V is 49°, but in the other triangle, is there a corresponding angle? Wait, maybe I made a mistake. Wait, let's re-express:

Triangle UVW: sides: UW = 22, UV = 28, VW = 16.

Triangle XYZ: sides: XZ = 110, XY = 140, YZ = 80.

Now, check the ratios:

UW / XZ = 22 / 110 = 1/5

UV / XY = 28 / 140 = 1/5

VW / YZ = 16 / 80 = 1/5

So all three sides are proportional (ratio 1/5). Therefore, by the SSS (Side-Side-Side) similarity criterion, the triangles are similar. Wait, but also, is there an included angle? Wait, no, SSS similarity says that if all three sides are in proportion, the triangles are similar. Alternatively, maybe SAS? Wait, but we have a 49° angle in triangle UVW. Wait, maybe the angle is included between the sides. Wait, in triangle UVW, the angle at V is 49°, between sides UV (28) and VW (16). In triangle XYZ, is there an angle between sides XY (140) and YZ (80)? Wait, if the ratio of UV to XY is 28/140 = 1/5, and VW to YZ is 16/80 = 1/5, and the included angle (angle at V and angle at Y, maybe) is equal? Wait, but we don't know the angle at Y. Wait, but if all three sides are proportional, then by SSS, they are similar. Alternatively, maybe the angle is included. Wait, let's check the ratios again.

Wait, 28/140 = 1/5, 16/80 = 1/5, and the included angle (angle at V and angle at Y) – but we don't know angle at Y. Wait, but maybe the angle at V (49°) and the angle at Y are equal? Wait, no, maybe I messed up the side pairing. Wait, maybe the sides are UV = 28, UW = 22, and in triangle XYZ, XZ = 110, XY = 140. Wait, 28/140 = 1/5, 22/110 = 1/5, and the third side VW = 16, YZ = 80: 16/80 = 1/5. So all three sides are in ratio 1/5, so by SSS similarity, the triangles are similar. Alternatively, maybe the included angle. Wait, the angle at V is 49°, between UV (28) and VW (16). In triangle XYZ, if we take the angle between XY (140) and YZ (80), and if the ratio of UV/XY = 28/140 = 1/5, VW/YZ = 16/80 = 1/5, and the included angle is equal, then by SAS similarity. But we know the angle at V is 49°, but do we know the angle at Y? Wait, maybe the problem is that the angle is included, and the sides around it are in proportion. Wait, let's check the ratios:

UV = 28, XY = 140: 28/140 = 1/5

VW = 16, YZ = 80: 16/80 = 1/5

So the sides around the angle (if angle at V and angle at Y are included) – but we don't know angle at Y. Wait, but maybe the angle is equal? Wait, no, maybe the triangles are similar by SSS. Wait, let's confirm the side lengths:

Triangle UVW: sides 16, 28, 22

Triangle XYZ: sides 80, 140, 110

Check ratios:

16/80 = 1/5

28/140 = 1/5

22/110 = 1/5

Yes! All three sides are in the ratio 1/5. Therefore, by the SSS (Side-Side-Side) similarity criterion, the triangles are similar. Alternatively, if we consider the included angle: the angle at V is 49°, and if the sides around it (UV = 28, VW = 16) and the sides around the corresponding angle in XYZ (say, angle at Y, with sides XY = 140, YZ = 80) have ratios 28/140 = 1/5 and 16/80 = 1/5, so the included angles would be equal (since the sides are proportional), so by SAS similarity. But since all three sides are proportional, SSS also applies.

Step2: Conclude similarity

Since the ratios of all corresponding sides are equal (1/5), the triangles are similar by the SSS (or SAS, if we consider the included angle) similarity criterion.

Answer:

The triangles are similar. We know this because the ratios of all corresponding sides are equal ( \( \frac{28}{140} = \frac{16}{80} = \frac{22}{110} = \frac{1}{5} \) ), so by the Side - Side - Side (SSS) similarity criterion (or Side - Angle - Side (SAS) if considering the included angle with the given 49° angle and proportional sides around it), the triangles are similar.