Question

completing proofs involving linear pairs
given: ( mangle elg = 124^circ )
prove: ( x = 28 )

statements	reasons
2. ( mangle eld = 2x )	2. given
3. ( angle elg ) and ( angle eld ) are a linear pair.	3. definition of a linear pair
4. ( mangle eld + mangle elg = 180 )	4.
5. ( 2x + 124 = 180 )	5. substitution
6.	6. subtraction property
7. ( x = 28 )	7. division property

complete the steps in the two - column proof.
options for step 4 reason: addition property of equality, angle addition postulate, definition of congruence, definition of vertical angles

Explanation:

Step-by-Step Explanation for Step 4 (Reason):

Step 1: Recall Linear Pair Postulate

A linear pair of angles is supplementary, meaning their measures add up to \(180^\circ\). So if \(\angle ELG\) and \(\angle ELD\) are a linear pair, then \(m\angle ELD + m\angle ELG = 180^\circ\). This is the angle addition postulate for supplementary angles (linear pair postulate).

Step 2: Identify the Reason for Step 4

In step 4, the statement is \(m\angle ELD + m\angle ELG = 180\), which follows from the definition of a linear pair (step 3) and the angle addition postulate (which states that if two angles form a linear pair, their measures sum to \(180^\circ\)). So the reason for step 4 is the angle addition postulate.

Step-by-Step Explanation for Step 6 (Statement):

Step 1: Start with the Equation from Step 5

From step 5, we have \(2x + 124 = 180\).

Step 2: Apply Subtraction Property

Using the subtraction property of equality (subtract 124 from both sides), we get \(2x + 124 - 124 = 180 - 124\), which simplifies to \(2x = 56\). So the statement for step 6 is \(2x = 56\).

Final Answers for the Blanks:

• Step 4 (Reason): angle addition postulate
• Step 6 (Statement): \(2x = 56\)

(For the two - column proof blanks:

• The reason for step 4 is "angle addition postulate".
• The statement for step 6 is \(2x = 56\))

Answer:

Step-by-Step Explanation for Step 4 (Reason):

Step 1: Recall Linear Pair Postulate

A linear pair of angles is supplementary, meaning their measures add up to \(180^\circ\). So if \(\angle ELG\) and \(\angle ELD\) are a linear pair, then \(m\angle ELD + m\angle ELG = 180^\circ\). This is the angle addition postulate for supplementary angles (linear pair postulate).

Step 2: Identify the Reason for Step 4

In step 4, the statement is \(m\angle ELD + m\angle ELG = 180\), which follows from the definition of a linear pair (step 3) and the angle addition postulate (which states that if two angles form a linear pair, their measures sum to \(180^\circ\)). So the reason for step 4 is the angle addition postulate.

Step-by-Step Explanation for Step 6 (Statement):

Step 1: Start with the Equation from Step 5

From step 5, we have \(2x + 124 = 180\).

Step 2: Apply Subtraction Property

Using the subtraction property of equality (subtract 124 from both sides), we get \(2x + 124 - 124 = 180 - 124\), which simplifies to \(2x = 56\). So the statement for step 6 is \(2x = 56\).

Final Answers for the Blanks:

• Step 4 (Reason): angle addition postulate
• Step 6 (Statement): \(2x = 56\)

(For the two - column proof blanks:

• The reason for step 4 is "angle addition postulate".
• The statement for step 6 is \(2x = 56\))