Question

Consider the two triangles.

Triangles W U V and X Z Y are shown. Angles V U W and Y X Z are congruent. Angles U W V and X Z Y are congruent. Angles U V W and Z Y X are congruent. The length of side V W is 60 and the length of side Z Y is 48. The length of side Y X is 40 and the length of V U is 50. The length of side U W is 40 and the length of X Z is 32.

How can the triangles be proven similar by the SAS similarity theorem?

Show that the ratios StartFraction X Y Over V U EndFraction and StartFraction Y Z Over V W EndFraction are equivalent, and ∠U ≅ ∠X.
Show that the ratios StartFraction U V Over X Y EndFraction and StartFraction W V Over Z Y EndFraction are equivalent, and ∠V ≅ ∠Y.
Show that the ratios StartFraction U W Over Z X EndFraction and StartFraction X Y Over W V EndFraction are equivalent, and ∠W ≅ ∠X.
Show that the ratios StartFraction X Z Over W U EndFraction and StartFraction Z Y Over W V EndFraction are equivalent, and ∠U ≅ ∠Z.Consider the two triangles.

Triangles W U V and X Z Y are shown. Angles V U W and Y X Z are congruent. Angles U W V and X Z Y are congruent. Angles U V W and Z Y X are congruent. The length of side V W is 60 and the length of side Z Y is 48. The length of side Y X is 40 and the length of V U is 50. The length of side U W is 40 and the length of X Z is 32.

How can the triangles be proven similar by the SAS similarity theorem?

Show that the ratios StartFraction X Y Over V U EndFraction and StartFraction Y Z Over V W EndFraction are equivalent, and ∠U ≅ ∠X.
Show that the ratios StartFraction U V Over X Y EndFraction and StartFraction W V Over Z Y EndFraction are equivalent, and ∠V ≅ ∠Y.
Show that the ratios StartFraction U W Over Z X EndFraction and StartFraction X Y Over W V EndFraction are equivalent, and ∠W ≅ ∠X.
Show that the ratios StartFraction X Z Over W U EndFraction and StartFraction Z Y Over W V EndFraction are equivalent, and ∠U ≅ ∠Z.

Answer 1

After checking the side ratios and using the SAS similarity theorem with congruent included angles, we can confirm that triangles WUV and XZY are similar.

Step-by-step explanation:

To prove the triangles similar by the SAS similarity theorem, we need to confirm two ratios are equal and that the included angles are congruent.

Given that angles ∠U ≅ ∠X, ∠V ≅ ∠Y, and ∠W ≅ ∠Z, we examine the triangle side ratios provided:

• ∠U ≅ ∠X: Corresponding sides are UV = 50 and XY = 40, UW = 40 and XZ = 32. Check the ratios:

UV / XY = 50 / 40

= 5 / 4

UW / XZ = 40 / 32

= 5 / 4

Since the ratios are equal and the included angles are congruent, by SAS theorem, triangles WUV and XZY are similar triangles.