Question

complete the proof by choosing the correct
eason\.
given ( mangle ejg = mangle hjf )
prove ( mangle 1 = mangle 3 )

statement	reason
2. ( mangle hjf = mangle 2 + mangle 3 )	reason?
3. ( mangle ejg = mangle hjf )	reason?
4. ( mangle 1 + mangle 2 = mangle 2 + mangle 3 )	reason?
5. ( mangle 1 = mangle 3 )	reason?

Explanation:

To solve this proof, we analyze each step's reason:

Step 1: \( m\angle EJG = m\angle 1 + m\angle 2 \)

This is the Angle Addition Postulate, which states that if a point lies in the interior of an angle, the sum of the two smaller angles formed equals the measure of the larger angle.

Step 2: \( m\angle HJF = m\angle 2 + m\angle 3 \)

Similarly, this uses the Angle Addition Postulate (same logic as Step 1: a point splits \( \angle HJF \) into \( \angle 2 \) and \( \angle 3 \)).

Step 3: \( m\angle EJG = m\angle HJF \)

This is given in the problem statement (the initial “Given” condition: \( m\angle EJG = m\angle HJF \)).

Step 4: \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \)

This is the Substitution Property of Equality. Since \( m\angle EJG = m\angle 1 + m\angle 2 \) (Step 1) and \( m\angle EJG = m\angle HJF \) (Step 3), we substitute \( m\angle EJG \) in Step 3 with \( m\angle 1 + m\angle 2 \), and \( m\angle HJF \) with \( m\angle 2 + m\angle 3 \) (Step 2).

Step 5: \( m\angle 1 = m\angle 3 \)

This is the Subtraction Property of Equality. Subtract \( m\angle 2 \) from both sides of \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \), so \( m\angle 1 + m\angle 2 - m\angle 2 = m\angle 2 + m\angle 3 - m\angle 2 \), simplifying to \( m\angle 1 = m\angle 3 \).

Final Answer (for each step’s reason):

1. Angle Addition Postulate
2. Angle Addition Postulate
3. Given
4. Substitution Property of Equality
5. Subtraction Property of Equality

Answer:

To solve this proof, we analyze each step's reason:

Step 1: \( m\angle EJG = m\angle 1 + m\angle 2 \)

This is the Angle Addition Postulate, which states that if a point lies in the interior of an angle, the sum of the two smaller angles formed equals the measure of the larger angle.

Step 2: \( m\angle HJF = m\angle 2 + m\angle 3 \)

Similarly, this uses the Angle Addition Postulate (same logic as Step 1: a point splits \( \angle HJF \) into \( \angle 2 \) and \( \angle 3 \)).

Step 3: \( m\angle EJG = m\angle HJF \)

This is given in the problem statement (the initial “Given” condition: \( m\angle EJG = m\angle HJF \)).

Step 4: \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \)

This is the Substitution Property of Equality. Since \( m\angle EJG = m\angle 1 + m\angle 2 \) (Step 1) and \( m\angle EJG = m\angle HJF \) (Step 3), we substitute \( m\angle EJG \) in Step 3 with \( m\angle 1 + m\angle 2 \), and \( m\angle HJF \) with \( m\angle 2 + m\angle 3 \) (Step 2).

Step 5: \( m\angle 1 = m\angle 3 \)

This is the Subtraction Property of Equality. Subtract \( m\angle 2 \) from both sides of \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \), so \( m\angle 1 + m\angle 2 - m\angle 2 = m\angle 2 + m\angle 3 - m\angle 2 \), simplifying to \( m\angle 1 = m\angle 3 \).

Final Answer (for each step’s reason):

1. Angle Addition Postulate
2. Angle Addition Postulate
3. Given
4. Substitution Property of Equality
5. Subtraction Property of Equality