Question

the frame of a bridge is constructed of triangles, as shown below. what additional information could you use to show that \\( \triangle stu \cong \triangle vtu \\) using sas? check all that apply. \\( \square uv = 14 \mathrm{ft} \\) and \\( m \angle tuv = 45 ^ { \circ } \\) \\( \square tu = 26 \mathrm{ft} \\) \\( \square m \angle stu = 37 ^ { \circ } \\) and \\( m \angle vtu = 37 ^ { \circ } \\) \\( \square st = 20 \mathrm{ft}, uv = 14 \mathrm{ft} \\), and \\( m \angle ust = 98 ^ { \circ } \\) \\( \square m \angle ust = 98 ^ { \circ } \\) and \\( m \angle tuv = 45 ^ { \circ } \\)

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

For two triangles \(\triangle STU\) and \(\triangle VTU\), the SAS criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent.

Step2: Analyze each option

• Option 1: \(UV = 14\) ft and \(m\angle TUV=45^{\circ}\)

In \(\triangle STU\) and \(\triangle VTU\), we know \(SU = 14\) ft (given). If \(UV = 14\) ft (so \(SU=UV\)) and \(m\angle TUS=m\angle TUV = 45^{\circ}\) (common side \(TU\)), by SAS \(\triangle STU\cong\triangle VTU\).

• Option 2: \(TU = 26\) ft

Just knowing the length of \(TU\) (a common side) does not give us information about the other sides and the included angle relationship required for SAS.

• Option 3: \(m\angle STU=37^{\circ}\) and \(m\angle VTU = 37^{\circ}\)

If \(m\angle STU=m\angle VTU\) (the included angle), and \(SU = 14\) ft (assume we can relate sides, for example, if we consider \(SU\) and \(UV\) (if \(UV = 14\) ft from other information in the problem - like in option 1 context) and common side \(TU\)), by SAS \(\triangle STU\cong\triangle VTU\).

• Option 4: \(ST = 20\) ft, \(UV = 14\) ft, and \(m\angle UST=98^{\circ}\)

This does not give the relationship of two sides and the included angle between the two triangles \(\triangle STU\) and \(\triangle VTU\). The angle \(m\angle UST\) is not the included angle for the sides relevant to the two triangles \(\triangle STU\) and \(\triangle VTU\).

• Option 5: \(m\angle UST = 98^{\circ}\) and \(m\angle TUV=45^{\circ}\)

This does not give the two - side and included - angle relationship. The angles \(m\angle UST\) and \(m\angle TUV\) are not the included angles for the sides of \(\triangle STU\) and \(\triangle VTU\) that we need for SAS.

Answer:

UV = 14 ft and \(m\angle TUV = 45^{\circ}\), \(m\angle STU=37^{\circ}\) and \(m\angle VTU = 37^{\circ}\)