In right triangle $JKL$, angle $J$ measures 60 degrees and angle $K$ measures 30 degrees. When drawn, the angle bisectors of angles $J$ and $K$ intersect at a point $M$. What is the measure of obtuse angle $JMK$?

[asy]
import geometry;
import olympiad;
unitsize(0.8inch);
dotfactor = 3;
defaultpen(linewidth(1pt)+fontsize(10pt));

pair J,K,L,M,U,V;

J = (0,0);
K = (1,2);
L = (1,0);

draw(J--K--L--cycle);
draw(rightanglemark(J,L,K,5));

label("$J$",J,W);
label("$K$",K,N);
label("$L$",L,E);

U = (1,2/3);
V = (2/(2+sqrt(3)),0);

draw(J--U);
draw(K--V);

M = intersectionpoint(J--U,K--V);
dot("M",M,NW);
[/asy]
Since $JM$ bisects $\angle J$, we know that the measure of $\angle KJM$ is $60/2 = 30$ degrees. Similarly, since $MK$ bisects $\angle K$, we know that the measure of $\angle JKM$ is $30/2 = 15$ degrees. Finally, since the sum of the measures of the angles of a triangle always equals $180$ degrees, we know that the sum of the measures of $\angle JKM$, $\angle KJM$, and $\angle JMK$ equals $180$ degrees. Thus, the measure of $\angle JMK = 180 - 30 - 15 = \boxed{135}$ degrees.
135
Level 2
Geometry
Problem: In right triangle $JKL$, angle $J$ measures 60 degrees and angle $K$ measures 30 degrees. When drawn, the angle bisectors of angles $J$ and $K$ intersect at a point $M$. What is the measure of obtuse angle $JMK$?

[asy]
import geometry;
import olympiad;
unitsize(0.8inch);
dotfactor = 3;
defaultpen(linewidth(1pt)+fontsize(10pt));

pair J,K,L,M,U,V;

J = (0,0);
K = (1,2);
L = (1,0);

draw(J--K--L--cycle);
draw(rightanglemark(J,L,K,5));

label("$J$",J,W);
label("$K$",K,N);
label("$L$",L,E);

U = (1,2/3);
V = (2/(2+sqrt(3)),0);

draw(J--U);
draw(K--V);

M = intersectionpoint(J--U,K--V);
dot("M",M,NW);
[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the measure of obtuse angle JMK, we first note the angles in triangle JKL. Angle J is 60 degrees and angle K is 30 degrees. The third angle, angle L, can be calculated using the triangle sum theorem, which states that the sum of the angles in a triangle is 180 degrees. Therefore, angle L is 90 degrees, since 180 - 60 - 30 = 90.

Next, we consider the angle bisectors of angles J and K. The angle bisector of angle J divides it into two equal angles of 30 degrees each. Similarly, the angle bisector of angle K divides it into two equal angles of 15 degrees each. 

At point M, where the angle bisectors intersect, we can analyze the angles formed. The angle JMK consists of the angle from the bisector of angle J and the angle from the bisector of angle K. Thus, angle JMK can be expressed as the sum of the angles from the two bisectors: 

Angle JMK = angle from bisector of J + angle from bisector of K = 30 degrees + 15 degrees = 45 degrees.

However, we need to find the obtuse angle JMK. The full angle around point M is 360 degrees. Therefore, the obtuse angle JMK is given by:

Obtuse angle JMK = 360 degrees - 45 degrees = 315 degrees.

Since we are interested in the angle between the two lines extending from J and K through M, we can also consider the supplementary angle to angle JMK. The supplementary angle to 45 degrees is:

Supplementary angle = 180 degrees - 45 degrees = 135 degrees.

Thus, the measure of obtuse angle JMK is 135 degrees.<|END_OF_LATENT|> The final answer is $135$.