10 Triangle 1 has an angle that measures \( 42^{\circ} \) and an angle that measures \( 57^{\circ} \). Triangle 2 has an angle that measures \( 42^{\circ} \) and an angle that measures \( a^{\circ} \), where \( a \neq 57^{\circ} \). Based on the information, Jennifer claims that triangle 1 and triangle 2 cannot be similar. What value of a, in degrees, will refute Jennifer's claim?

To determine the value of \( a \) that would support the similarity between Triangle 1 and Triangle 2, let's first find the missing angle in Triangle 1. The sum of the angles in a triangle is always \( 180^{\circ} \). So, we can find the third angle in Triangle 1 as follows: \[ 180^{\circ} - (42^{\circ} + 57^{\circ}) = 180^{\circ} - 99^{\circ} = 81^{\circ} \] Now, for Triangle 2, we already have one angle as \( 42^{\circ} \) and the new angle \( a \). To maintain similarity, the third angle in Triangle 2 must also be \( 81^{\circ} \). Thus, we can find \( a \): \[ 180^{\circ} - (42^{\circ} + a) = 81^{\circ} \] \[ 180^{\circ} - 42^{\circ} - 81^{\circ} = a \] \[ a = 57^{\circ} \] However, since we need \( a \) to be different from \( 57^{\circ} \) and also make the triangles similar, we can create a similar triangle by changing the angle. To refute Jennifer's claim, let \( a \) be equal to \( 81^{\circ} \). Now we have the angles as \( 42^{\circ} \), \( 81^{\circ} \) in Triangle 2. Since both triangles share angles \( 42^{\circ} \) and \( 81^{\circ} \), they are indeed similar. Hence, the angle \( a \) must be: \[ a = 81^{\circ} \]