Question

understand how angles formed by intersecting lines are related. find unknown angle measures.

Explanation:

Step1: Use the property of a straight - line angle

The sum of angles on a straight line is \(180^{\circ}\). Let the unknown angle be \(x\). We know that \(51^{\circ}+x + 51^{\circ}+129^{\circ}=360^{\circ}\) (sum of angles around a point is \(360^{\circ}\)), or we can use the straight - line property. If we consider the straight line with angles \(129^{\circ}\) and \(x\) (they are adjacent angles on a straight line).
Another way: Since the sum of adjacent angles forming a straight line is \(180^{\circ}\). We can also note that the angle adjacent to \(129^{\circ}\) (let's call it \(y\)) is \(y = 180^{\circ}-129^{\circ}=51^{\circ}\) (linear pair). And the unknown angle \(x\) and \(129^{\circ}\) are vertical angles. Wait, no. Let's use the sum of angles around a point formula \(A + B+C + D=360^{\circ}\). Let \(A = 51^{\circ}\), \(B=x\), \(C = 51^{\circ}\), \(D = 129^{\circ}\). Then \(51+x + 51+129=360\).
Simplify the left - hand side: \(51+51+129+x=(51 + 51)+129+x=102+129+x=231+x\).
So, \(231+x=360\).

Step2: Solve for \(x\)

Subtract 231 from both sides of the equation \(231+x=360\).
\(x=360 - 231\).

Answer:

\(129^{\circ}\)