Question

consider the two triangles.
how can the triangles be proven similar by the sss similarity theorem?
show that the ratios \\(\frac{uv}{xy}\\) and \\(\frac{wv}{zy}\\) are equivalent, and \\(\angle v \cong \angle y\\).
show that the ratios \\(\frac{uv}{zy}\\) and \\(\frac{wu}{zx}\\) are equivalent, and \\(\angle u \cong \angle z\\).
show that the ratios \\(\frac{uv}{xy}\\), \\(\frac{wu}{zx}\\), and \\(\frac{wv}{zy}\\) are equivalent.
show that the ratios \\(\frac{uv}{zy}\\), \\(\frac{wu}{zx}\\), and \\(\frac{wv}{xy}\\) are equivalent.

Explanation:

Step1: Recall SSS Similarity Theorem

The SSS (Side - Side - Side) similarity theorem states that if the corresponding sides of two triangles are in proportion (i.e., the ratios of the lengths of corresponding sides are equal), then the two triangles are similar. So we need to find the ratios of the corresponding sides of the two triangles and check which option shows that all three pairs of corresponding sides have equal ratios.

Step2: Identify Corresponding Sides

First, let's label the sides of the two triangles. In triangle \(UVW\), the sides are \(UV = 50\), \(WU=40\), \(WV = 60\). In triangle \(XYZ\), the sides are \(ZY = 48\), \(ZX = 32\), \(XY=40\).

Now, let's find the ratios of corresponding sides:

- For \(UV\) and \(ZY\): \(\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}\) (Wait, no, maybe I mixed up the correspondence. Wait, let's check the angles. The marked angles: \(\angle U\) in triangle \(UVW\) and \(\angle X\) in triangle \(XYZ\) are right angles? Wait, no, the marked arcs: \(\angle V\) has two arcs, \(\angle Y\) has two arcs; \(\angle W\) has one arc, \(\angle X\) has one arc? Wait, maybe the correspondence is \(U - Z\), \(V - Y\), \(W - X\)? Wait, no, let's calculate the ratios properly.

Wait, let's list the sides:

Triangle \(UVW\): \(WU = 40\), \(UV=50\), \(WV = 60\)

Triangle \(XYZ\): \(ZX = 32\), \(ZY = 48\), \(XY = 40\)

Now, let's find the ratios:

\(\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}\)

\(\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}\)? No, that can't be. Wait, maybe I got the correspondence wrong. Wait, \(WU = 40\), \(XY = 40\); \(WV=60\), \(ZY = 48\); \(UV = 50\), \(ZX=32\)? No, that's not. Wait, let's check the option D: \(\frac{UV}{ZY}\), \(\frac{WU}{ZX}\), \(\frac{WV}{XY}\)

Wait, \(\frac{UV}{ZY}=\frac{50}{48}=\frac{25}{24}\)? No, wait \(ZY = 48\), \(UV = 50\); \(WU = 40\), \(ZX = 32\), \(\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}\); \(WV = 60\), \(XY = 40\), \(\frac{WV}{XY}=\frac{60}{40}=\frac{3}{2}\). No, that's not. Wait, maybe the correct correspondence is:

Wait, the fourth option is \(\frac{UV}{ZY}\), \(\frac{WU}{ZX}\), \(\frac{WV}{XY}\)? No, wait the fourth option is "Show that the ratios \(\frac{UV}{ZY}\), \(\frac{WU}{ZX}\), and \(\frac{WV}{XY}\) are equivalent." Wait, let's recalculate:

Wait, \(WU = 40\), \(ZX = 32\), so \(\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}\)

\(UV = 50\), \(ZY = 48\)? No, \(ZY = 48\), \(UV = 50\), \(\frac{50}{48}=\frac{25}{24}\). No, that's not. Wait, maybe I made a mistake. Wait, let's check the third option: "Show that the ratios \(\frac{UV}{XY}\), \(\frac{WU}{ZX}\), and \(\frac{WV}{ZY}\) are equivalent."

\(\frac{UV}{XY}=\frac{50}{40}=\frac{5}{4}\)

\(\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}\)

\(\frac{WV}{ZY}=\frac{60}{48}=\frac{5}{4}\)

Ah! There we go. So \(UV\) corresponds to \(XY\), \(WU\) corresponds to \(ZX\), \(WV\) corresponds to \(ZY\). So the ratios \(\frac{UV}{XY}=\frac{50}{40}=\frac{5}{4}\), \(\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}\), \(\frac{WV}{ZY}=\frac{60}{48}=\frac{5}{4}\). So all three ratios are equal (\(\frac{5}{4}\)). So the option that says "Show that the ratios \(\frac{UV}{XY}\), \(\frac{WU}{ZX}\), and \(\frac{WV}{ZY}\) are equivalent" is the correct one, which is the third option (the one with three ratios: \(\frac{UV}{XY}\), \(\frac{WU}{ZX}\), \(\frac{WV}{ZY}\)). Wait, looking at the options:

Option 3: "Show that the ratios \(\frac{UV}{XY}\), \(\frac{WU}{ZX}\), and \(\frac{WV}{ZY}\) are equivalent."

Yes, because when we calculate \(\frac{UV}{XY}=\frac{50}{40}=\frac{5}{4}\), \(\frac{WU}{ZX}=\frac{40}{32}=\frac{5}{4}\), \(\frac{WV}{ZY}=\frac{60}{48}=\frac{5}{4}\), all ratios are equal, so by SSS similarity, the triangles are similar.

The first option uses an angle (SAS), the second option uses an angle (SAS), the third option uses three side ratios (SSS), the fourth option has the wrong ratios. Wait, the fourth option is \(\frac{UV}{ZY}\), \(\frac{WU}{ZX}\), \(\frac{WV}{XY}\). Let's check: \(\frac{UV}{ZY}=\frac{50}{48}\approx1.04\), \(\frac{WU}{ZX}=\frac{40}{32}=1.25\), \(\frac{WV}{XY}=\frac{60}{40}=1.5\), not equal. The third option: \(\frac{UV}{XY}=\frac{50}{40}=1.25\), \(\frac{WU}{ZX}=\frac{40}{32}=1.25\), \(\frac{WV}{ZY}=\frac{60}{48}=1.25\), equal. So the correct option is the third one: "Show that the ratios \(\frac{UV}{XY}\), \(\frac{WU}{ZX}\), and \(\frac{WV}{ZY}\) are equivalent."

Answer:

The option: "Show that the ratios \(\frac{UV}{XY}\), \(\frac{WU}{ZX}\), and \(\frac{WV}{ZY}\) are equivalent." (the third option in the list of options)