Question

bell ringer: angle theorems

1. determine the theorem you could use to solve for $\angle x$. explain your reasoning.
2. determine the theorem you could use to solve for $\angle y$. explain your reasoning.

theorems
triangle angle sum theorem
$\angle 1+\angle 2+\angle 3 = 180^{\circ}$
exterior angle theorem
$\angle 1+\angle 2+ = \angle 4$
supplementary angles
$\angle 3+\angle 4 = 180^{\circ}$

Explanation:

1. For \(\angle x\), we can use the Exterior Angle Theorem. The exterior angle (\(\angle x\)) is equal to the sum of the two non - adjacent interior angles. In this case, if we consider the right - angled triangle formed (with a \(30^{\circ}\) angle and a right angle (\(90^{\circ}\))), the exterior angle \(\angle x\) is related to the non - adjacent interior angles.
2. For \(\angle y\), we can use the Triangle Angle Sum Theorem. In the right - angled triangle (with angles \(y\), \(30^{\circ}\), and \(90^{\circ}\)), the sum of the interior angles of a triangle is \(180^{\circ}\). So, \(y + 30^{\circ}+90^{\circ}=180^{\circ}\).

Answer:

1. Theorem: Exterior Angle Theorem. Reasoning: \(\angle x\) is an exterior angle of a right - angled triangle with non - adjacent interior angles \(30^{\circ}\) and \(90^{\circ}\).
2. Theorem: Triangle Angle Sum Theorem. Reasoning: \(\angle y\) is an interior angle of a right - angled triangle where the sum of interior angles (\(y + 30^{\circ}+90^{\circ}\)) must equal \(180^{\circ}\).