Question

from this given statement, select the definition, property, postulate, or theorem that justifies the prove statement. given: ∠abc and ∠ghj are complementary; ∠ghj and ∠rtu are complementary. prove: ∠abc ≅ ∠rtu after you pick your answer press go. a b c d e if two angles are a linear pair, then they are supplementary.

Explanation:

1. Recall the definition of complementary angles: Two angles are complementary if their measures sum to $90^\circ$. So, $m\angle ABC + m\angle GHJ = 90^\circ$ and $m\angle GHJ + m\angle RTU = 90^\circ$.
2. Use the transitive property of equality: Since both sums equal $90^\circ$, $m\angle ABC + m\angle GHJ = m\angle GHJ + m\angle RTU$.
3. Subtract $m\angle GHJ$ from both sides: This gives $m\angle ABC = m\angle RTU$.
4. Apply the definition of congruent angles: Angles with equal measures are congruent, so $\angle ABC \cong \angle RTU$. The theorem justifying this is the Congruent Complements Theorem (if two angles are complementary to the same angle, they are congruent). The given option A (about linear pairs and supplementary angles) does not apply here.

Answer:

(Note: The provided option A is incorrect for this proof. The correct theorem is the "Congruent Complements Theorem" which states that if two angles are complementary to the same angle (or to congruent angles), then they are congruent to each other. Since the given options are not fully shown, but based on the problem, the reasoning is as follows:

Given $\angle ABC$ and $\angle GHJ$ are complementary, so $m\angle ABC + m\angle GHJ = 90^\circ$. Also, $\angle GHJ$ and $\angle RTU$ are complementary, so $m\angle GHJ + m\angle RTU = 90^\circ$. By the transitive property of equality, $m\angle ABC + m\angle GHJ = m\angle GHJ + m\angle RTU$. Subtracting $m\angle GHJ$ from both sides gives $m\angle ABC = m\angle RTU$, so $\angle ABC \cong \angle RTU$ by the definition of congruent angles (equal measures). The theorem justifying this is the Congruent Complements Theorem.)

(If we assume the options include the Congruent Complements Theorem, the correct option would be the one stating that if two angles are complementary to the same angle, they are congruent. Since the given option A is about linear pairs and supplementary angles, it's not the right choice. But as the full options aren't provided, we explain the correct reasoning.)