Question

completing proofs involving linear pairs

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. ( \boldsymbol{cdot} )
5. ( 2x + 124 = 180 )	5. substitution
6. ( \boldsymbol{cdot} )	6. subtraction property
7. ( x = 28 )	7. division property

given: ( mangle elg = 124^circ )
prove: ( x = 28 )

diagram: collinear points d, l, g; line through l with h; rays le, lf forming ( angle eld = (2x)^circ )

complete the steps in the two - column proof.

options:

• addition property of equality
• angle addition postulate
• definition of congruence
• definition of vertical angles

Explanation:

Step 4 Reason:

Linear pairs are supplementary (sum to \(180^\circ\)), so the reason is "linear pair postulate" (or recognizing that linear pairs are supplementary, which is a postulate about adjacent angles forming a straight line summing to \(180^\circ\)). From the options, the closest related concept here is the idea that linear pairs have a sum of \(180^\circ\), but among the given options, the "angle addition postulate" isn't correct here. Wait, actually, the correct reason for step 4 is that linear pairs are supplementary (their measures sum to \(180^\circ\)), which is a theorem based on the definition of a linear pair (adjacent angles forming a straight line). But from the options, maybe a misstep—wait, no, the options given: "angle addition postulate" is for adding angles, but here it's about linear pairs. Wait, maybe the problem's options have a typo, but actually, the correct reason for step 4 is that if two angles form a linear pair, then they are supplementary (sum to \(180^\circ\)), which is a postulate. But among the options, the "angle addition postulate" is not it. Wait, maybe the user's options are misrepresented, but let's proceed with the proof.

Step 6 Statement:

Starting from \(2x + 124 = 180\), using the subtraction property of equality (subtract 124 from both sides), we get \(2x = 180 - 124\), so \(2x = 56\).

Step 4 Reason:

The reason for \(m\angle ELD + m\angle ELG = 180\) is that linear pairs are supplementary (their measures sum to \(180^\circ\)), which is a postulate (sometimes called the linear pair postulate). But from the given options, the closest might be a misoption, but actually, the correct reason is "linear pair postulate" (not in the options? Wait, the options are: addition property, angle addition, definition of congruence, vertical angles. Wait, maybe the problem intended "angle addition postulate" is incorrect, but perhaps a mistake. Alternatively, maybe the user made a typo, but let's focus on step 6.

Step 6:

From \(2x + 124 = 180\), subtract 124 from both sides: \(2x + 124 - 124 = 180 - 124\), so \(2x = 56\).

Answer:

Step 4 Reason: (Correct concept: linear pairs are supplementary, but from options, maybe a mistake—however, proceeding with step 6)
Step 6 Statement: \(2x = 56\)