3.) m < 7 = 155°, m < 8 = 25°
4.) m < 5 = 30°
  m < 6 = 30°
  m < 7  = 60°
  m < 8  = 60°
Step-by-step explanation:
3.) By definition, angles that do not share a common side are called nonadjacent angles. Two nonadjacent angles formed by two intersecting lines are called vertical angles.
Given that < PQT + < TQR = 180°Then it also means that the sum of m < 7 and m < 8 will also equal 180°.  Also, < PQT ≅ < SQR because they are vertical angles, therefore, their measurements must also be congruent.  Similarly, < PQS ≅ < TQR because they are vertical angles, and their measurements must also be congruent. Â
m < 7 = 5x + 5
m < 8 = x - 5
m < 7 + m < 8 = 180°
Substitute the values of m < 7 and m < 8 into the equation:
5x + 5 + x - 5 = 180°
6x + 0 = 180°
6x  = 180°
Divide 6 on both sides of the equation to solve for x:
x = 30°
Plug in x = 30° to find the value of m< 7 and m< 8:
m < 7 = 5x + 5 = 5(30) + 5 = 150 + 5 = 155°
m < 8 = x - 5 = 30 - 5 = 25°
4.) This problem is an example of angles on a straight line. By definition, the sum of angles on a straight line is equal to 180°.
Therefore, the measurements of the following angles add up to 180°: Â
 < UVX + < XVY + < YVZ + <ZVW = 180°   m < 5 + m < 6 + m < 7 + m < 8 = 180°
m < 5 = 5x
m < 6 = 4x + 6
m < 7 = 10x
m < 8 = 12x - 12
Substitute the values of each measurement onto the following equation: Â
5x + 4x + 6 + 10x + 12x - 12 = 180°
Combine like terms: Â
31x - 6 = 180°
Add 6 on both sides of the equation:
31x - 6 + 6 = 180° + 6
31x = 186
Solve for x:
x = 6
Plug in x = 6° to find the values of m < 5,  m <  6,  m < 7, and  m < 8:
5(6) + 4(6) + 6 + 10(6) + 12(6) - 12 = 180°
180° = 180°
Therefore:
m < 5 = 5(6) = 30°
m < 6 = 4(6) + 6 = 30°
m < 7 = 10(6) = 60°
m < 8 = 12(6) - 12 = 60°
