Question

finally, find the measure of angle udw. angle udw measures degrees.

Explanation:

Step1: Recall the sum of angles around a point

The sum of all angles around a point \( D \) is \( 360^\circ \). We know three of the angles: \( 85^\circ \) (between \( U \) and \( L \)), \( 95^\circ \) (between \( L \) and \( T \)), and \( 85^\circ \) (between \( T \) and the other line). Let the measure of \( \angle UDW \) be \( x \). So, \( 85^\circ + 95^\circ + 85^\circ + x = 360^\circ \)? Wait, no, actually, looking at the diagram, the angles around point \( D \) should also consider that a straight line is \( 180^\circ \), but maybe a better way: the angles adjacent to the yellow angle ( \( \angle UDW \)) and the given angles. Wait, actually, the sum of angles around a point is \( 360^\circ \), but also, the angles on a straight line sum to \( 180^\circ \). Wait, looking at the diagram, the lines \( U - T \) and the other line (let's say \( W - \) the left line) intersect at \( D \). So the angles around \( D \): \( \angle UDL = 85^\circ \), \( \angle LDLT = 95^\circ \), \( \angle TDW = 85^\circ \), and \( \angle WDU \) (which is \( \angle UDW \)) is what we need. Wait, actually, the sum of angles around a point is \( 360^\circ \), so \( 85^\circ + 95^\circ + 85^\circ + \angle UDW = 360^\circ \)? Wait, no, that would be if all four angles are around the point. Wait, maybe the lines are two intersecting lines, so vertical angles and linear pairs. Wait, another approach: the sum of angles on one side of a straight line is \( 180^\circ \). Wait, the line \( U - T \) is a straight line? No, \( U \) is up, \( T \) is down, so \( U - D - T \) is a straight line, so \( \angle UDL + \angle LDLT + \angle TDW + \angle WDU = 360^\circ \)? No, that's around the point. Wait, actually, the angles given: \( 85^\circ \) (between \( U \) and \( L \)), \( 95^\circ \) (between \( L \) and \( T \)), \( 85^\circ \) (between \( T \) and \( W \)), so the remaining angle \( \angle UDW \) should satisfy \( 85 + 95 + 85 + x = 360 \). Let's calculate that: \( 85 + 95 = 180 \), \( 180 + 85 = 265 \), so \( x = 360 - 265 = 95 \)? Wait, no, that doesn't seem right. Wait, maybe the lines are perpendicular? No, the angles given are \( 85 \), \( 95 \), \( 85 \). Wait, another way: the sum of angles around a point is \( 360^\circ \), so if we have three angles: \( 85^\circ \), \( 95^\circ \), \( 85^\circ \), then the fourth angle is \( 360 - (85 + 95 + 85) = 360 - 265 = 95 \)? Wait, but maybe the diagram is such that \( \angle UDW \) is supplementary to one of the angles? Wait, no, let's check again. Wait, the problem is to find \( \angle UDW \). Let's list all angles around \( D \):

- \( \angle UDL = 85^\circ \)
- \( \angle LDLT = 95^\circ \)
- \( \angle TDW = 85^\circ \)
- \( \angle WDU = \angle UDW \) (let's call this \( x \))

So, \( 85 + 95 + 85 + x = 360 \)

Calculating the sum of the known angles: \( 85 + 95 = 180 \), \( 180 + 85 = 265 \)

Then, \( x = 360 - 265 = 95 \)? Wait, no, that can't be. Wait, maybe the lines \( U - T \) and \( L - W \) are intersecting, so the sum of angles on a straight line (e.g., \( U - D - T \)) is \( 180^\circ \). Wait, \( \angle UDL + \angle LDLT + \angle TDW = 85 + 95 + 85 = 265 \), which is more than \( 180 \), so that's not a straight line. Wait, maybe the diagram is two intersecting lines, so \( U - T \) and \( L - W \) intersect at \( D \), forming four angles. Then, the vertical angles should be equal. Wait, \( \angle UDL = 85^\circ \), so the vertical angle to \( \angle TDW \) should be equal? But \( \angle TDW \) is \( 85^\circ \), so that's vertical to \( \angle UDL \)? No, vertical angles are opposite each other. So if \( U - T \) and \( L - W \) intersect at \( D \), then \( \angle UDL \) and \( \angle TDW \) are vertical angles, so they should be equal. But here \( \angle UDL = 85^\circ \) and \( \angle TDW = 85^\circ \), so that's correct. Then \( \angle LDLT = 95^\circ \) and \( \angle UDW \) are vertical angles, so they should be equal. Therefore, \( \angle UDW = 95^\circ \). Ah, that makes sense. So vertical angles are equal, so since \( \angle LDLT = 95^\circ \), then \( \angle UDW = 95^\circ \).

Step2: Confirm using vertical angles

When two lines intersect, vertical angles are equal. Here, lines \( U - T \) and \( L - W \) intersect at \( D \). So \( \angle LDLT \) (which is \( 95^\circ \)) and \( \angle UDW \) are vertical angles, so they are equal. Therefore, \( \angle UDW = 95^\circ \).

Answer:

\( 95 \)