Question

1. triangle abc is isosceles with ab = ac. if ab = 3x + 4 and ac = 5x - 2, find x.
2. triangle def is isosceles with de = df. if de = 7x - 9 and df = 4x, find x.
3. triangle ghi has base angles ∠g and ∠h congruent. if m∠g = 4x + 10 and m∠h = 6x - 10, find x.
4. triangle jkl is isosceles with jl = kl. if jl = 2x + 9 and kl = 4x - 3, find x.

Explanation:

Step1: Set up the equation

Since \(AB = AC\) in isosceles \(\triangle ABC\), we set \(3x + 4=5x - 2\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(4 = 2x-2\).
Add \(2\) to both sides: \(6 = 2x\).
Divide both sides by \(2\): \(x = 3\).

Step1: Set up the equation

Since \(DE = DF\) in isosceles \(\triangle DEF\), we set \(7x - 9=4x\).

Step2: Solve for \(x\)

Subtract \(4x\) from both sides: \(3x-9 = 0\).
Add \(9\) to both sides: \(3x=9\).
Divide both sides by \(3\): \(x = 3\).

Step1: Set up the equation

Since \(\angle G\cong\angle H\) in \(\triangle GHI\), we set \(4x + 10=6x - 10\).

Step2: Solve for \(x\)

Subtract \(4x\) from both sides: \(10 = 2x-10\).
Add \(10\) to both sides: \(20 = 2x\).
Divide both sides by \(2\): \(x = 10\).

Step1: Set up the equation

Since \(JL = KL\) in isosceles \(\triangle JKL\), we set \(2x + 9=4x - 3\).

Step2: Solve for \(x\)

Subtract \(2x\) from both sides: \(9 = 2x-3\).
Add \(3\) to both sides: \(12 = 2x\).
Divide both sides by \(2\): \(x = 6\).

Answer:

For problem 11: \(x = 3\)
For problem 12: \(x = 3\)
For problem 13: \(x = 10\)
For problem 14: \(x = 6\)