Question

what is m∠hki?
(there is a circle with center k. there are three radii: kj, kh, and ki. the angle between kj and kh is 110°, and the angle between kj and ki is 110°. we need to find m∠hki.)

Explanation:

Step1: Recall the total degrees around a point

The sum of angles around a point (center of a circle, here point K) is \( 360^\circ \).

Step2: Calculate the remaining angle

We know two angles are \( 110^\circ \) each. Let \( x = m\angle HKI \). Then \( 110^\circ + 110^\circ + x = 360^\circ \). Simplify: \( 220^\circ + x = 360^\circ \). Subtract \( 220^\circ \) from both sides: \( x = 360^\circ - 220^\circ = 140^\circ \)? Wait, no, wait. Wait, looking at the diagram, maybe the two \( 110^\circ \) and \( \angle HKI \) are the three central angles? Wait, no, maybe I misread. Wait, the center is K, with two angles of \( 110^\circ \) (between J-K-I? Wait, no, the diagram: points J, H, I on the circle, K is center. So angles at K: \( \angle JK I = 110^\circ \), \( \angle J K H = 110^\circ \), and \( \angle H K I \) is what we need. Wait, no, the sum of central angles around K is \( 360^\circ \). So \( \angle JKI + \angle JKH + \angle HKI = 360^\circ \)? Wait, no, maybe the two angles are \( 110^\circ \) each, so \( 110 + 110 + \angle HKI = 360 \). So \( 220 + \angle HKI = 360 \), so \( \angle HKI = 360 - 220 = 140 \)? Wait, no, that can't be. Wait, maybe the diagram is different. Wait, maybe the two \( 110^\circ \) angles and \( \angle HKI \) are the three angles, but maybe I made a mistake. Wait, no, let's check again. Wait, the total around a point is \( 360^\circ \). So if two angles are \( 110^\circ \) each, then the third is \( 360 - 110 - 110 = 140 \)? Wait, but that seems large. Wait, maybe the diagram has two angles of \( 110^\circ \) and the third angle is \( \angle HKI \). So yes, \( 360 - 110 - 110 = 140 \). Wait, but maybe I misread the angles. Wait, the problem says "What is \( m\angle HKI \)?" with the diagram showing two \( 110^\circ \) angles at K. So sum of angles around K: \( 110 + 110 + m\angle HKI = 360 \). So \( m\angle HKI = 360 - 220 = 140 \). Wait, but let me confirm. The sum of central angles in a circle is \( 360^\circ \), so yes, that's the formula. So step 1: total degrees around a point is \( 360^\circ \). Step 2: subtract the two known angles from \( 360^\circ \). So \( 360 - 110 - 110 = 140 \). So \( m\angle HKI = 140^\circ \).

Answer:

\( 140 \)