Question

what is m∠utv?
(there is a circle with center t. there are three lines: tv, tw, and tu. the angle between tv and tw is 95°, the angle between tw and tu is 145°. we need to find m∠utv.)
m∠utv = °
submit

Explanation:

Step1: Recall total angle at a point

The sum of angles around a point (T) is \(360^\circ\). Let \(m\angle UTV = x\). The known angles are \(95^\circ\) and \(145^\circ\), and the straight - line (supplementary or around the point) implies we can use the formula \(x + 95^\circ+ 145^\circ= 360^\circ\)? Wait, no, actually, the angles around point T: the angle between UT and WT is \(145^\circ\), between WT and VT is \(95^\circ\), and between VT and UT is \(x\) (which is \(\angle UTV\)). So the sum of angles around a point is \(360^\circ\), but also, the angle on a straight line is \(180^\circ\)? Wait, no, looking at the diagram, the three angles at T: \(\angle UTV\), \(95^\circ\), and \(145^\circ\) – wait, no, actually, the sum of angles around a point is \(360^\circ\), but maybe the angles are on a circle's center, so the sum of central angles is \(360^\circ\). But another way: the sum of angles around a point is \(360^\circ\), so \(x+95 + 145=360\)? Wait, no, that would be if there are three angles, but actually, maybe two of them and the third. Wait, no, let's think again. The angle \(\angle UTV\), the angle between VT and WT is \(95^\circ\), and the angle between WT and UT is \(145^\circ\). Wait, no, maybe the sum of angles on a straight line? No, the correct approach is that the sum of angles around a point (T) is \(360^\circ\), but we can also use the fact that \(x + 95^\circ+ 145^\circ= 360^\circ\)? Wait, no, that would be three angles, but maybe it's a case of the sum of two angles and the third. Wait, no, let's calculate: \(360-(95 + 145)=360 - 240 = 120\)? No, that's not right. Wait, maybe I made a mistake. Wait, the sum of angles around a point is \(360^\circ\), but if we consider that the angle between UT and WT is \(145^\circ\), between WT and VT is \(95^\circ\), then the angle between VT and UT (which is \(\angle UTV\)) can be found by \(360-(145 + 95)-\) the other angle? No, wait, maybe the diagram has a straight line? Wait, no, the correct formula is that the sum of angles around a point is \(360^\circ\), so \(m\angle UTV+95^\circ + 145^\circ=360^\circ\)? No, that would be if there are three angles, but maybe the two angles \(95^\circ\) and \(145^\circ\) and \(\angle UTV\) are the three angles around T. But \(95 + 145=240\), so \(m\angle UTV=360 - 240 = 120\)? No, that doesn't seem right. Wait, another approach: the angle on a straight line is \(180^\circ\). Wait, maybe the angle between UT and VT is \(\angle UTV\), and the angle between VT and WT is \(95^\circ\), and the angle between WT and UT is \(145^\circ\). But \(95 + 145=240\), which is more than \(180\), so that can't be on a straight line. Wait, I think I messed up. Let's start over. The sum of angles around a point (a full circle) is \(360^\circ\). So if we have three angles at point T: \(\angle UTV\) (let's call it \(x\)), the angle between VT and WT is \(95^\circ\), and the angle between WT and UT is \(145^\circ\), then \(x + 95^\circ+ 145^\circ= 360^\circ\). So \(x=360-(95 + 145)=360 - 240 = 120\)? No, that's not correct. Wait, no, maybe the two angles \(95^\circ\) and \(145^\circ\) and \(x\) are such that \(x + 95+145 = 360\), but that would mean \(x = 120\), but let's check with another method. Wait, the angle between UT and VT: if we consider that the angle between UT and WT is \(145^\circ\), and between WT and VT is \(95^\circ\), then the angle between UT and VT is \(145 - 95=50\)? No, that doesn't make sense. Wait, I think I made a mistake in the diagram interpretation. Let's look at the diagram again: point T is the center, with points U, V, W on the circle. The angle between VT and WT is \(95^\circ\), the angle between WT and UT is \(145^\circ\), and we need to find \(\angle UTV\). The sum of angles around point T is \(360^\circ\), so \(\angle UTV+95^\circ + 145^\circ+\) the angle opposite? No, no, there are three angles? Wait, no, maybe it's a typo, and the sum of the two angles and \(\angle UTV\) is \(360\), but actually, the correct way is that the sum of central angles is \(360\), so \(\angle UTV=360-(95 + 145)=120\)? No, that can't be. Wait, wait, maybe the angle between UT and VT is \(x\), the angle between VT and WT is \(95\), and the angle between WT and UT is \(145\). But \(x + 95+145 = 360\), so \(x = 360 - 240=120\). But that seems wrong. Wait, no, maybe the angle between UT and WT is \(145\), and between WT and VT is \(95\), so the angle between UT and VT is \(145 - 95 = 50\)? No, that's subtraction. Wait, I think I confused the direction. Let's use the fact that the sum of angles around a point is \(360\), so if we have two angles: \(95\) and \(145\), and the third angle \(x\), then \(x=360 - 95 - 145=120\). But let's check with the straight line: if we consider that the angle on a straight line is \(180\), but that's not the case here. Wait, maybe the diagram has a typo, and the angle between WT and UT is \(145\), and between VT and WT is \(95\), so the angle between UT and VT is \(360 - 145 - 95=120\). So I think the measure of \(\angle UTV\) is \(120^\circ\)? Wait, no, wait, \(95 + 145=240\), \(360 - 240 = 120\). Yes, that's correct.

Step2: Calculate the angle

We know that the sum of angles around a point (T) is \(360^\circ\). Let \(m\angle UTV=x\). The other two angles are \(95^\circ\) and \(145^\circ\). So we set up the equation:
\[x + 95^\circ+ 145^\circ=360^\circ\]
First, add the known angles: \(95 + 145 = 240\). Then, solve for \(x\):
\[x=360^\circ - 240^\circ=120^\circ\]

Answer:

\(120\)