Given that triangles WYX and EFG are similar (△WYX ∼ △EFG) with a scale factor of 2, we can determine the measures of the angles based on the properties of similar triangles.
In similar triangles, corresponding angles are equal. This means:
- ∠W corresponds to ∠E
- ∠Y corresponds to ∠F
- ∠X corresponds to ∠G
From the information provided, we know:
- ∠E = 50°
- ∠X = 65°
We need to find ∠W. Since the triangles are isosceles, we can determine the measures of the remaining angles.
In triangle EFG:
- The angles in a triangle sum to 180°. So, we can find ∠F: \[ \text{Let } \angle F = x \] Then, \[ 50° + x + \text{(the angle at G)} = 180° \]
The angle at G can be found since triangle EFG is isosceles, implying: \[ \text{(the angle at G)} = \angle F \] So: \[ 50° + x + x = 180° \] or: \[ 50° + 2x = 180° \] \[ 2x = 130° \] \[ x = 65° \] Thus, ∠F = 65°.
Since ∠F corresponds to ∠Y, then ∠Y = 65° (as given), and since triangle WYX is also isosceles, the angles at W and X must be equal because they are the non-base angles. Thus:
To find ∠W: \[ \text{Let } \angle W = \angle X = 65° \]
However, we also need to find the third angle ∠Y which is similar across both triangles: Since we know the respective angles’ relationships in triangle EFG, use the property of isosceles triangles where two angles at bases equal the other non-base angle.
Thus: \[ \angle W + \angle X + \angle Y = 180° \] So: \[ \angle W + 65° + 65° = 180° \]
Solving gives: \[ \angle W + 130° = 180° \] \[ \angle W = 50° \]
However, the original question queries relation of angle at W given all data, under similarity and given isosceles context: To answer correctly as per angle correspondence:
The measure of \(\angle W\) thus equals \(\angle E\) based on similarity.
The possible responses provided lead us to conclude: \(\angle W\) matches \(\angle E = 50°\).
Therefore, final assertion: ∠W = 50° (is your necessary logical final query)
Response: 50°.
1 answer