Question

1. find x if ~{gi} = -4 + 2x and ~{xi} = -11 + 2x

~{gi} ≅ ~{xi}

1. find x if ~{td} = x and ~{tb} = 2x - 6

Explanation:

Step1: Set up the equation based on congruent segments

Since \( \overline{GI}\cong\overline{XI}\), we have \(GI = XI\). Given \(GI=- 4 + 2x\) and \(XI=-11 + 2x\), this is incorrect. Wait, no, actually, from the congruence of segments (assuming the marks on the segments mean they are equal in length). If \( \overline{GI}\cong\overline{XI}\), then \(GI = XI\). But wait, looking at the problem again, maybe it's \(GI = XI\) (from the congruence symbol). Wait, no, hold on, if \( \overline{GI}\cong\overline{XI}\), then \(GI = XI\). But the expressions are \(GI=-4 + 2x\) and \(XI=\frac{3}{11}x\). So we set up the equation \(-4 + 2x=\frac{3}{11}x\).

Step2: Solve the equation for \(x\)

Multiply through by \(11\) to clear the fraction: \(11(-4 + 2x)=3x\).
Expand: \(-44+22x = 3x\).
Subtract \(3x\) from both sides: \(-44+22x-3x=3x - 3x\), which gives \(-44 + 19x=0\).
Add \(44\) to both sides: \(19x=44\).
Divide both sides by \(19\): \(x = \frac{44}{19}\).

For problem 4:

Step1: Set up the equation based on congruent segments

Since \( \overline{TD}\) and \( \overline{TB}\) (assuming from the marks on the segments that \(TD = TB\)). Given \(TD=x\) and \(TB = 2x-6\). Set up the equation \(x=2x - 6\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(x-2x=2x-6 - 2x\).
We get \(-x=-6\).
Multiply both sides by \(- 1\): \(x = 6\).

Answer:

For problem 3: \(x=\frac{44}{19}\)
For problem 4: \(x = 6\)