Question

1. in this diagram, ( mangle1 = 4x + 30 ) and ( mangle3 = 2x + 48 ). write a step - by - step explanation for an absent classmate showing how to find ( mangle2 ).
2. construct viable arguments. ( angle rst ) is adjacent to ( angle tsu ), ( mangle rst=40^{circ} ), and ( mangle tsu = 50^{circ} ). explain how you can use the angle addition postulate to show that ( angle rsu ) is a right angle.

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. So \(m\angle1 = m\angle3\)

$$4x + 30=2x + 48$$

Step2: Solve for \(x\)

Subtract \(2x\) from both sides:

$$4x-2x + 30=2x-2x + 48$$

$$2x+30 = 48$$

Subtract 30 from both sides:

$$2x+30 - 30=48 - 30$$

$$2x=18$$

Divide both sides by 2:

$$x = 9$$

Step3: Find \(m\angle1\)

Substitute \(x = 9\) into \(m\angle1=4x + 30\)

$$m\angle1=4\times9+30=36 + 30=66^{\circ}$$

Step4: Use linear - pair property

\(\angle1\) and \(\angle2\) form a linear pair. So \(m\angle1+m\angle2 = 180^{\circ}\)

$$m\angle2=180 - m\angle1$$

Substitute \(m\angle1 = 66^{\circ}\)

$$m\angle2=180-66 = 114^{\circ}$$

Answer:

\(m\angle2 = 114^{\circ}\)