From the image, what is the value of angle LNBA?

In the bottom diagram, we have points A, N, B, M on a circle. We are given that AB-diameter and LNBA = 73 degrees. We need to find LNMB.

If AB is a diameter, then the angle subtended by the diameter at any point on the circumference is 90 degrees. So, angle ANB = 90 degrees and angle AMB = 90 degrees.

We are given LNBA = 73 degrees. This is an inscribed angle subtending arc NA.

Therefore, the measure of arc NA = 2 * LNBA = 2 * 73 degrees = 146 degrees.

The central angle subtending arc NA is angle NOA. So, angle NOA = 146 degrees.

Since angle ANB = 90 degrees, we have angle AND + angle DNB = 90 degrees. (Assuming D is some point on AB or circumference).

In right triangle ANB, angle NAB + angle NBA = 90 degrees.

angle NAB + 73 degrees = 90 degrees.

angle NAB = 90 - 73 = 17 degrees.

Angle NAB subtends arc NB. So, arc NB = 2 * angle NAB = 2 * 17 degrees = 34 degrees.

The central angle subtending arc NB is angle NOB. So, angle NOB = 34 degrees.

We need to find LNMB. This is an inscribed angle subtending arc NB.

Therefore, LNMB = (1/2) * arc NB.

LNMB = (1/2) * 34 degrees = 17 degrees.

Let's verify using angles in triangle ANB.

Angle ANB = 90 degrees (subtended by diameter AB).

Angle NBA = 73 degrees (given).

Angle BAN = 180 - 90 - 73 = 17 degrees.

The angle LNMB is an inscribed angle subtending arc NB. The inscribed angle subtending the same arc NB is angle NAB.

Therefore, LNMB = angle NAB = 17 degrees.

Ответ: 17°