Question

12 if (\triangle btscong\triangle ghd), (bs = 25), (ts = 14), (bt = 31), (gd = 4x - 11), (mangle s = 56^{circ}), (mangle b = 21^{circ}), and (mangle h=(7y + 5)^{circ}), find the values of (x) and (y).

Explanation:

Step1: Use the property of congruent triangles (corresponding sides are equal)

Since \(\triangle BTS\cong\triangle GHD\), then \(GD = BT\).
We know that \(GD = 4x-11\) and \(BT = 31\). So, \(4x-11=31\).

Step2: Solve the equation for \(x\)

Add \(11\) to both sides of the equation \(4x - 11=31\):
\(4x-11 + 11=31+11\), which simplifies to \(4x=42\).
Divide both sides by \(4\): \(x=\frac{42}{4}=\frac{21}{2}=10.5\).

Step3: Use the property of congruent triangles (corresponding angles are equal)

Since \(\triangle BTS\cong\triangle GHD\), then \(\angle H=\angle B\).
We know that \(m\angle H=(7y + 5)^{\circ}\) and \(m\angle B = 21^{\circ}\). So, \(7y+5 = 21\).

Step4: Solve the equation for \(y\)

Subtract \(5\) from both sides of the equation \(7y+5=21\):
\(7y+5 - 5=21-5\), which simplifies to \(7y=16\).
Divide both sides by \(7\): \(y=\frac{16}{7}\approx2.29\).

Answer:

\(x = 10.5\) and \(y=\frac{16}{7}\)