Question

Using the vertical angle image from problem 1, let ∠2 = 4x-4 and ∠4 = x+50.

A. What is the value of x?

B. What is the measure of ∠2?

C. Find the measure of ∠3.

Answer 1

A. x = 18

B. m∠2 = 68°

C. m∠2 = 112°

Explanation:

Given:

• 
  `\angle 2=4x-4`

• 
  `\angle4=x+50`

Part A

Vertical Angles Theorem

When two straight lines intersect, the opposite vertical angles are congruent.

Therefore:

`\implies \angle 2 = \angle 4`

`\implies 4x-4= x+50`

`\implies 4x-4-x= x+50-x`

`\implies 3x-4= 50`

`\implies 3x-4+4= 50+4`

`\implies 3x=54`

`\implies 3x/ 3=54 / 3`

`\implies x=18`

Part B

To find the measure of ∠2, substitute the found value of x into the given expression for ∠2:

`\implies m\angle 2 =(4x-4)^(\circ)`

`\implies m\angle 2 =[4(18)-4]^(\circ)`

`\implies m\angle 2 =(72-4)^(\circ)`

`\implies m\angle 2 =68^(\circ)`

Part C

Angles on a straight line sum to 180°:

`\implies m\angle 2 +m\angle 3=180^(\circ)`

`\implies 68 ^(\circ)+m\angle 3=180^(\circ)`

`\implies m\angle 3=180^(\circ)-68 ^(\circ)`

`\implies m\angle 3=112 ^(\circ)`

Answer 2

• A) x = 18, B) ∠2 = 68°, C) ∠3 = 112°

Explanation:

According to vertical angles theorem, when two lines intersect the opposite angles are equal and adjacent angles are supplementary.

Given

• ∠2 = 4x-4 and ∠4 = x+50

A) Find the value of x:

• 4x - 4 = x + 50
• 4x - x = 50 + 4
• 3x = 54
• x = 54/3
• x = 18

B) Find the measure of angle 2:

• ∠2 = 4*18 - 4 = 72 - 4 = 68

C) Angles 2 and 3 are supplementary so their sum is 180°.

• ∠2 + ∠3 = 180
• 68 + ∠3 = 180
• ∠3 = 180 - 68
• ∠3 = 112°