Question

a. solve for x =
show all work:
if m\angle a=(5x - 7)^{\circ}, find x, m\angle a and m\angle e.

b. find m\angle a and m\angle e.
show all work:

Explanation:

Step1: Determine the relationship between the angles

Assume \(\triangle PRE\) and \(\triangle STA\) are congruent. So \(\angle P=\angle A=(5x - 7)^{\circ}\), and in \(\triangle PRE\), using the angle - sum property of a triangle (\(\angle R+\angle P+\angle E = 180^{\circ}\)). Also, if \(\triangle PRE\cong\triangle STA\), then \(\angle S = 50^{\circ}=\angle E\) (corresponding angles of congruent triangles).
In \(\triangle PRE\), \(\angle R = 82^{\circ}\), \(\angle E=50^{\circ}\). By the angle - sum property of a triangle (\(\angle R+\angle P+\angle E=180^{\circ}\)), we substitute \(\angle P=(5x - 7)^{\circ}\)

$$82+(5x - 7)+50 = 180$$

Step2: Solve the equation for \(x\)

Simplify the left - hand side of the equation:

$$82 + 5x-7+50=180$$

$$125 + 5x=180$$

Subtract 125 from both sides:

$$5x=180 - 125$$

$$5x=55$$

Divide both sides by 5:

$$x = 11$$

Step3: Find \(m\angle A\)

Substitute \(x = 11\) into \(m\angle A=(5x - 7)^{\circ}\)

$$m\angle A=5\times11-7$$

$$m\angle A=55 - 7$$

$$m\angle A = 48^{\circ}$$

Answer:

a. \(x = 11\)
b. \(m\angle A=48^{\circ}\), \(m\angle E = 50^{\circ}\)