Question

1. given: the diagram below.

Explanation:

Step1: Identify Similar Triangles

The triangles appear similar (vertical angles, right angles? Wait, the right angles? Wait, the sides: 27, 15 and 117, 65? Wait, check ratios. $\frac{27}{117} = \frac{15}{65}$? Simplify: $\frac{27}{117} = \frac{3}{13}$, $\frac{15}{65} = \frac{3}{13}$. So similar by SAS (vertical angles). So the angle $a$: since the triangles are similar, and the angle at the intersection is vertical, but wait, maybe it's a right angle? Wait, no, the right angles? Wait, the sides: 15, 27 and 65, 117. Wait, maybe the triangles are right triangles? Wait, 15-27-? Wait, no, 15² + 27²? No, 15²=225, 27²=729, sum=954. 65²=4225, 117²=13689, sum=17914. Not equal. Wait, but the ratios: 27/117 = 3/13, 15/65=3/13. So the included angle (vertical angle) is equal, so triangles are similar by SAS similarity. Therefore, the corresponding angles: the angle with side 15 and 27, and the angle with side 65 and 117. Wait, but the angle $a$: if the triangles are similar, and one of the angles is 90 degrees? Wait, maybe the right angles. Wait, the right angles (the little square) on both triangles? So they are right triangles, similar by SAS (right angle, vertical angle, and ratio of sides). So angle $a$ is 90 degrees? Wait, no, wait, the right angle is at N and I? So angle at N and I are right angles. Then vertical angles at the intersection. So the triangles are similar (right angle, vertical angle), so angle $a$ is equal to the right angle? Wait, no, wait, the right angle is at N and I, so angle at N is 90, angle at I is 90, vertical angles at the intersection, so the triangles are similar (AA similarity: right angle and vertical angle). Therefore, angle $a$ is 90 degrees? Wait, no, wait, maybe I misread. Wait, the diagram: N and I have right angles (the little square). So triangle AN (right angle at N) and triangle IS (right angle at I). Vertical angles at the intersection, so angles are equal. So triangles are similar (AA). Therefore, angle $a$ is equal to the right angle? Wait, no, angle $a$ is at the intersection, between 65 and the vertical line? Wait, maybe the angle $a$ is 90 degrees? Wait, no, let's check the ratios again. Wait, 27/117 = 3/13, 15/65=3/13. So the sides adjacent to the vertical angle are in ratio 3/13, so triangles are similar by SAS. Therefore, the angle opposite? Wait, no, the right angles: if angle at N is 90, angle at I is 90, then angle $a$ is equal to 90 degrees? Wait, maybe the problem is to find angle $a$, and since the triangles are similar and right-angled, angle $a$ is 90 degrees. Wait, but maybe I made a mistake. Wait, let's re-express:

Given two triangles with sides 27, 15 and 117, 65, and vertical angles. So $\frac{27}{117} = \frac{15}{65} = \frac{3}{13}$, so SAS similarity (vertical angle is common). Therefore, the triangles are similar, so the corresponding angles: the angle at N (right angle) corresponds to angle at I (right angle), so angle $a$ is equal to 90 degrees? Wait, no, angle $a$ is the vertical angle? Wait, no, the angle $a$ is between the side of length 65 and the transversal. Wait, maybe the triangles are right triangles, so angle $a$ is 90 degrees. Wait, but let's check the sum of angles. If the triangles are right-angled, then angle $a$ is 90 degrees. Alternatively, maybe the angle $a$ is calculated via linear pair? Wait, no, the diagram shows a right angle at N and I. So angle $a$ is 90 degrees. Wait, but maybe the problem is to find angle $a$, and since the triangles are similar and right-angled, angle $a$ is 90 degrees. Wait, maybe I'm overcomplicating. Let's check the ratios again: 27/117 = 3/13, 15/65 = 3/13. So the triangles are similar, so the included angle (vertical angle) is equal, and the other angles: since they are right triangles (right angles at N and I), then angle $a$ is 90 degrees.

Step2: Confirm Right Angle

Since the triangles have right angles (marked by the square) and are similar by SAS, the angle $a$ is equal to the right angle, so $a = 90^\circ$.

Answer:

$\boxed{90^\circ}$