Question

identifying additional information needed
the frame of a bridge is constructed of triangles, as shown
what additional information could you use to show that \\( \triangle stu \cong \triangle vtu \\) using sas? choose two correct
answers.
\\( uv = 14 \mathrm { ft } \\) and
\\( m \angle tuv = 45 ^ { \circ } \\)
\\( tu = 26 \mathrm { ft } \\)
\\( st = 20 \mathrm { ft }, uv = 14 \mathrm { ft } \\),
and \\( m \angle ust = 98 ^ { \circ } \\)
\\( m \angle stu = 37 ^ { \circ } \\) and
\\( m \angle vtu = 37 ^ { \circ } \\)
\\( 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 \(\triangle VTU\), we need two sides and the included angle to be equal.
We know that \(SU = 14\) ft (given). If \(UV=14\) ft (so \(SU = UV\)), and if the included angles \(\angle TUS\) and \(\angle TUV\) are equal. Also, \(TU\) is a common side.
Another way: If \(\angle STU=\angle VTU\) (the included angles) and we know \(SU = UV\) (sides) and \(TU\) is common.

Step2: Analyze each option

• Option 1: \(UV = 14\) ft and \(m\angle TUV=45^{\circ}\). Since \(SU = 14\) ft (given in the figure for \(SU\)), \(SU = UV\), \(TU\) is common, and if \(\angle TUS=\angle TUV = 45^{\circ}\) (included angles), by SAS \(\triangle STU\cong\triangle VTU\).
• Option 2: \(TU = 26\) ft. Just knowing the length of \(TU\) (a common side) doesn't give us the required two - side - and - included - angle information for SAS.
• Option 3: \(ST = 20\) ft, \(UV = 14\) ft, and \(m\angle UST=98^{\circ}\). This does not give us the two sides and included angle for \(\triangle STU\) and \(\triangle VTU\) (the angles and sides are not in the right proportion for SAS).
• Option 4: \(m\angle STU = 37^{\circ}\) and \(m\angle VTU=37^{\circ}\). Since \(TU\) is common, \(SU = UV\) (if we consider the side - angle - side, \(SU\) and \(UV\) are sides, \(\angle STU\) and \(\angle VTU\) are included angles).
• Option 5: \(m\angle UST = 98^{\circ}\) and \(m\angle TUV=45^{\circ}\). This does not give us the two sides and included angle for SAS.

Answer:

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