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

Question

Accepted Answer

To solve this, we recall the SAS (Side - Angle - Side) congruence criterion: For two triangles to be congruent by SAS, two sides and the included angle of one triangle must be equal to the corresponding two sides and the included angle of the other triangle.

### Analyzing each option:
- **Option 1: $UV = 14$ ft and $m\angle TUV=45^{\circ}$**
In $\triangle STU$ and $\triangle VTU$, we know that $SU = 14$ ft (from the diagram), $m\angle STU = 45^{\circ}$ (from the diagram). If $UV = 14$ ft and $m\angle TUV = 45^{\circ}$, then $SU=UV$, $\angle STU=\angle VTU$ and $TU$ is common. So, we have two sides ($SU = UV$, $TU = TU$) and the included angle ($\angle STU=\angle VTU$) equal. This satisfies SAS.
- **Option 2: $TU = 26$ ft**
Just knowing the length of $TU$ does not give us two sides and the included angle for the two triangles. We need information about the other sides and the included angle. So, this does not help in proving congruence by SAS.
- **Option 3: $m\angle STU = 37^{\circ}$ and $m\angle VTU = 37^{\circ}$**
This gives us information about the angles, but we do not have information about the sides. To apply SAS, we need two sides and the included angle. So, this is not sufficient.
- **Option 4: $ST = 20$ ft, $UV = 14$ ft, and $m\angle UST = 98^{\circ}$**
We know that $TV = 20$ ft (from the diagram). If $ST = 20$ ft, then $ST=TV$, $SU = UV = 14$ ft (from the diagram and given $UV = 14$ ft) and we can find the included angle. Also, we know that the sum of angles in a triangle is $180^{\circ}$. If $m\angle UST=98^{\circ}$ and $m\angle STU = 45^{\circ}$, then $m\angle SUT=180-(98 + 45)=37^{\circ}$. Similarly, we can find the angles in $\triangle VTU$. But more directly, we have $ST = TV = 20$ ft, $SU=UV = 14$ ft and we can show that the included angles are equal. However, a more straightforward way is that with $ST = 20$ ft, $UV = 14$ ft and the given angle - related information, we can use SAS. Wait, actually, if $ST = 20$ (so $ST=TV$ since $TV = 20$), $SU = 14$ (so $SU = UV$ if $UV = 14$) and we can check the included angle. But let's re - evaluate. The SAS requires two sides and the included angle. If $ST = 20$, $SU = 14$ and $m\angle UST = 98^{\circ}$, and $TV = 20$, $UV = 14$ and we can find the included angle between $TV$ and $UV$ to be equal to $\angle UST$ (by angle - sum property). So, this option can be used.
- **Option 5: $m\angle UST = 98^{\circ}$ and $m\angle TUV = 45^{\circ}$**
This gives us information about two non - included angles. We do not have information about the sides required for SAS. So, this is not sufficient.

Wait, let's re - check the first option. In $\triangle STU$ and $\triangle VTU$:
- $SU = 14$ (given in the diagram), $UV = 14$ (from option 1)
- $\angle STU=45^{\circ}$ (diagram), $\angle VTU = 45^{\circ}$ (from $m\angle TUV = 45^{\circ}$ and triangle angle - sum, or by the fact that if $UV = 14$ and $SU = 14$, and $TU$ is common, the included angle is equal)
- $TU$ is common. So, $SU = UV$, $\angle STU=\angle VTU$, $TU = TU$ → SAS.

For option 3: $m\angle STU = 37^{\circ}$ and $m\angle VTU = 37^{\circ}$. We know $TU$ is common, but we need two sides. We know $TV = 20$, but we don't know $ST$ (unless we assume, but from the diagram $ST$ is not marked as 20). Wait, the diagram shows $TV = 20$ ft. If we don't know $ST$, then option 3 is not sufficient.

For option 4: $ST = 20$ ft (so $ST = TV$ since $TV = 20$ ft), $UV = 14$ ft (so $SU = UV$ since $SU = 14$ ft), and $m\angle UST=98^{\circ}$. In $\triangle STU$, we have sides $ST = 20$, $SU = 14$ and included angle $\angle UST = 98^{\circ}$. In $\triangle VTU$, sides $TV = 20$, $UV = 14$ and included angle (we can calculate it as $180-(m\angle TUV + m\angle VTU)$. But since $ST = TV$, $SU = UV$ and we can show that the included angles are equal (because the sum of angles in a triangle is $180^{\circ}$). So, this also satisfies SAS.

Wait, the correct options are:
- $UV = 14$ ft and $m\angle TUV = 45^{\circ}$
- $m\angle STU = 37^{\circ}$ and $m\angle VTU = 37^{\circ}$ → No, wait, no. Wait, the SAS requires two sides and the included angle. Let's start over.

We want to prove $\triangle STU\cong\triangle VTU$ by SAS. So we need $ST = VT$, $SU = VU$ and $\angle STU=\angle VTU$ (or $SU = VU$, $TU = TU$ and $\angle STU=\angle VTU$, or $ST = VT$, $TU = TU$ and $\angle STU=\angle VTU$).

From the diagram:
- $TV = 20$ ft, $SU = 14$ ft, $\angle STU = 45^{\circ}$, $\angle VTU$ is adjacent to $98^{\circ}$ angle.

1. **Option 1: $UV = 14$ ft and $m\angle TUV = 45^{\circ}$**
   - $SU = 14$ ft, $UV = 14$ ft ⇒ $SU = UV$
   - $TU$ is common to both triangles.
   - $\angle STU = 45^{\circ}$, $m\angle TUV = 45^{\circ}$. Also, $\angle STU$ and $\angle VTU$: Since $m\angle TUV = 45^{\circ}$ and in $\triangle VTU$, if we consider the angles, $\angle VTU$ can be related. But more directly, $SU = UV$, $TU = TU$, and $\angle STU=\angle VTU$ (because $m\angle TUV = 45^{\circ}$ and the angle at $T$ for $\triangle STU$ is $45^{\circ}$). So, this satisfies SAS.
2. **Option 2: $TU = 26$ ft**
   - Just knowing $TU$ does not give us the other side and the included angle. So, this is not sufficient.
3. **Option 3: $m\angle STU = 37^{\circ}$ and $m\angle VTU = 37^{\circ}$**
   - We know $TU$ is common, but we need two sides. We know $TV = 20$ ft, but we don't know $ST$ (unless we assume $ST = 20$ ft, but it's not marked in the diagram). So, this is not sufficient for SAS.
4. **Option 4: $ST = 20$ ft, $UV = 14$ ft, and $m\angle UST = 98^{\circ}$**
   - $ST = 20$ ft, $TV = 20$ ft (from diagram) ⇒ $ST = TV$
   - $SU = 14$ ft, $UV = 14$ ft (given $UV = 14$ ft) ⇒ $SU = UV$
   - In $\triangle STU$, we have sides $ST = 20$, $SU = 14$ and included angle $\angle UST = 98^{\circ}$. In $\triangle VTU$, sides $TV = 20$, $UV = 14$ and the included angle (we can calculate it as $180-(m\angle TUV + m\angle VTU)$. But since $ST = TV$, $SU = UV$ and the sum of angles in a triangle is $180^{\circ}$, the included angles will be equal. So, this satisfies SAS.
5. **Option 5: $m\angle UST = 98^{\circ}$ and $m\angle TUV = 45^{\circ}$**
   - This gives us two non - included angles. We don't have the required sides for SAS. So, this is not sufficient.

So the correct options are:
- $UV = 14$ ft and $m\angle TUV = 45^{\circ}$
- $ST = 20$ ft, $UV = 14$ ft, and $m\angle UST = 98^{\circ}$

# Answer:
- $UV = 14$ ft and $m\angle TUV = 45^{\circ}$
- $ST = 20$ ft, $UV = 14$ ft, and $m\angle UST = 98^{\circ}$

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