Question

Decide whether the triangles are similar. If so, determine the appropriate expression to solve for x.

Triangles ABC and EDF, where triangle ABC has angle A measuring 47 degrees, angle C measuring 62 degrees, side AC labeled as y, side AB labeled as w, and side BC labeled as x and triangle EDF has angle D measuring 71 degrees, angle E measuring 47 degrees, side DE labeled z, side EF labeled u, and side DF labeled r

The triangles are not similar; no expression for x can be found.
ΔABC ~ ΔDEF; x equals r times w over u
ΔABC ~ ΔEDF; x equals r times w over u
ΔABC ~ ΔEDF; x equals r times w over z

Answer 1

Δ ABC is similar to Δ EDF and
`x = r(w)/(z)`

Explanation:

In Δ ABC, ∠ A = 47° and ∠ C = 62°

So, ∠ B = 180° - 62° - 47° = 71°

Again, in Δ EDF, ∠ D = 71°, and ∠ E = 47°

So, ∠ F = 180° - 71° - 47° = 62°

Hence, Δ ABC is similar to Δ EDF

Now, for two similar triangles the ratio of corresponding sides will be in the same ratio.

So,
`(BC)/(DF) = (AB)/(ED)`

⇒
`(x)/(r) = (w)/(z)`

⇒
`x = r(w)/(z)`.

Therefore, the correct option is 3. (Answer)

Answer 2

`\triangle ABC \sim \triangle ED\ F\\`

`x=r * (w)/(z)`

Explanation:

The given triangles are similar by Angle-Angle postulate, because all three pairs of corresponding angles are congruent.

Triangle ABC

`\angle A + \angle B + \angle C = 180\°\\47\° + \angle B + 62\° = 180\°\\\angle B = 180\° - 62\° - 47\°= 71\°`

Triangle DEF

`\angle D + \angle E + \angle F = 180\°\\71\° + 47\° + \angle F = 180\°\\\angle F = 180\° - 71\° - 47\°=62\°`

As you can see,

`\angle A = \angle E\\\angle B = \angle D\\\angle C = \angle F`

Therefore, triangles are similar, that is

`\triangle ABC \sim \triangle ED\ F`

From the similarity, we deduct the following proportions

`(AB)/(ED)=(BC)/(DF)=(AC)/(EF)`

According to the problem,
`x = BC; y= AC;w=AB\\r = DF; u=EF; z=De`

From proportions, we have

`(BC)/(DF)=(AB)/(ED)\\(x)/(r)=(w)/(z)\\ x=r * (w)/(z)`

Therefore, the right answer is the last choice.