Question

prove: ( x = 4 )

convert the flow chart proof into a paragraph proof.

it is given that ( angle abe ) and ( angle dbc ) are vertical angles. by the (\boxed{quad}) ( angle abe ) is congruent to ( angle dbc ). by the definition of congruence, the
then, by the substitution property of equality,
gives ( x = 4 ).
congruent and supplementary angles theorem
congruent complements theorem
linear pair theorem
vertical angles theorem

Explanation:

First, identify the correct theorem to justify that vertical angles are congruent, which is the vertical angles theorem. Then, complete the paragraph by filling in the missing statements based on the flow chart logic: state the congruence leads to equal measures, substitute the angle expressions, and apply the subtraction property of equality to solve for $x$.

Answer:

It is given that $\angle ABE$ and $\angle DBC$ are vertical angles. By the vertical angles theorem, $\angle ABE$ is congruent to $\angle DBC$. By the definition of congruence, the measures of $\angle ABE$ and $\angle DBC$ are equal, so $m\angle ABE = m\angle DBC$. Then, by the substitution property of equality, substitute the given angle expressions: $2x + 6 = x + 10$. Applying the subtraction property of equality (subtract $x$ and 6 from both sides) gives $x = 4$.