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The frame of a bridge is constructed of triangles, as shown below.

A large triangle is made up of 4 smaller triangles within the larger triangle. Vertical line T U cuts the large triangle in half. Lines U S and U V are diagonal from point U and go up to each side. Angle S U T is 45 degrees and angle T V U is 98 degrees. Line segment T V is 20 feet and line segment S U is 14 feet.
What additional information could you use to show that ΔSTU ≅ ΔVTU using SAS? Check all that apply.

UV = 14 ft and m∠TUV = 45°
TU = 26 ft
m∠STU = 37° and m∠VTU = 37°
ST = 20 ft, UV = 14 ft, and m∠UST = 98°
m∠UST = 98° and m ∠TUV = 45°

1 answer

To show that triangles \( \Delta STU \) and \( \Delta VTU \) are congruent using the Side-Angle-Side (SAS) postulate, we need to establish that:

  1. Two sides of one triangle are equal to two sides of the other triangle.
  2. The enclosed angle between those two sides is equal.

We can analyze the given options:

  1. UV = 14 ft and m∠TUV = 45°: This gives us information about one of the sides and an angle, but does not provide a sufficient angle between the sides for SAS.

  2. TU = 26 ft: This provides the length of one side \( TU \), which is common to both triangles, but it does not help in establishing the congruency since we need another side length or an angle.

  3. m∠STU = 37° and m∠VTU = 37°: This provides angle measures for the angles opposite the pairs of sides \( SU \) and \( TV \), and it suggests that this could help establish the congruency through SAS. However, it does not directly give us an angle between known sides, so it would be helpful but not sufficient alone.

  4. ST = 20 ft, UV = 14 ft, and m∠UST = 98°: This provides the length of side \( ST \) which is crucial, along with the length of \( UV \) and the measure of \( \angle UST \). This option gives us sufficient information to show SAS because we have:

    • One side \( ST = 20 \text{ ft} \)
    • Other side \( UV = 14 \text{ ft} \)
    • Angle \( \angle UST = 98° \) With these, we can show that the triangles share \( TU \) and \( ST \) and have \( \angle UST \), satisfying SAS.

  5. m∠UST = 98° and m∠TUV = 45°: While this gives us two angles, it doesn't provide the necessary information about the sides in a way that SAS could be confirmed.

Based on this assessment, the best answer that provides the additional information needed to prove \( \Delta STU \cong \Delta VTU \) using the SAS postulate is:

  • ST = 20 ft, UV = 14 ft, and m∠UST = 98°.

This is the only option that provides the necessary information to use SAS effectively.