Question

1. if ( \triangle abc cong \triangle def ), ( ab = 8 ), ( bc = 19 ), ( ac = 14 ), ( ef = 4x - 1 ), and ( de = y - 6 ), find the values of ( x ) and ( y ).
2. if ( \triangle zmk cong \triangle apy ), ( mangle m = 112^{circ} ), ( mangle y = 41^{circ} ), ( mangle k = (13x - 37)^{circ} ), and ( mangle a = (2y + 7)^{circ} ), find the values of ( x ) and ( y ).
3. if ( \triangle bts cong \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 \(\triangle ABC\cong\triangle DEF\), then \(BC = EF\) and \(AB=DE\).
For \(BC = EF\):
We have the equation \(19 = 4x - 1\).
Add \(1\) to both sides of the equation: \(19+1=4x-1 + 1\), which simplifies to \(20 = 4x\).
Divide both sides by \(4\): \(\frac{20}{4}=\frac{4x}{4}\), so \(x = 5\).

Step2: Solve for \(y\) using \(AB = DE\)

We have the equation \(8=y - 6\).
Add \(6\) to both sides of the equation: \(8 + 6=y-6+6\), so \(y=14\).

Answer:

\(x = 5\) and \(y = 14\)