Answer: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. m < 7 = 5x + 5m < 8 = x - 5m < 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:[tex]\frac{6x}{6} = \frac{180}{6}[/tex]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°: < UVX + < XVY + < YVZ + <ZVW = 180° m < 5 + m < 6 + m < 7 + m < 8 = 180°m < 5 = 5xm < 6 = 4x + 6m < 7 = 10xm < 8 = 12x - 12Substitute the values of each measurement onto the following equation: 5x + 4x + 6 + 10x + 12x - 12 = 180°Combine like terms: 31x - 6 = 180°Add 6 on both sides of the equation:31x - 6 + 6 = 180° + 631x = 186Solve for x:[tex]\frac{31x}{31} = \frac{186}{31}[/tex]x = 6Plug 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°