Derek wrote the following paragraph proof for the vertical angles theorem:

the sum of angle 1 and angle 4 and the sum of angle 3 and angle 4 are each equal to 180 degrees by the definition of supplementary angles. the sum of angle 1 and angle 4 is equal to the sum of angle 3 and angle 4 by the transitive property of equality. angle 1 is equal to angle 3

which phrase completes the proof?

a. by construction using a straightedge
b. by the definition of a perpendicular bisector
c. by the subtraction property of equality
d. by the vertical angles theroem

Option C is correct.

By the Subtraction Property of Equality.

Explanation:

Supplementary Angles: Two angles are Supplementary if the sum of the  measure of angles is 180 degree.

Given:

   {By definition of supplementary angle}                                              .......[1]

Transitive Property of equality:

If a=b and

b = c,

then a = c

By the transitive property,

[1} ⇒            ......[2]

Subtraction property of equality states that you subtract the same number from both sides of an equation.

Subtract from both the sides in [2];

Simplify:

    {By subtraction property of equality}

Therefore, the only phrase which completes the proof is; by the subtraction property of equality

step-by-step explanation:

you divide the denominator by the numerator on both sides and if the quotients are equal then the fractions are equivalent