Question

Please help me with both questions.

Answer 1

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 + 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:

$\frac{6x}{6} = \frac{180}{6}$

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 = 5x

m < 6 = 4x + 6

m < 7 = 10x

m < 8 = 12x - 12

Substitute 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° + 6

31x = 186

Solve for x:

$\frac{31x}{31} = \frac{186}{31}$

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°