Question

1. if ( \triangle abccong\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 zmkcong\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 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 \(\triangle ABC\cong\triangle DEF\), 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 \(4\): \(x=\frac{20}{4}=5\).
For \(AB = DE\):
We have the equation \(8=y - 6\).
Add \(6\) to both sides of the equation: \(8+6=y-6 + 6\), which simplifies to \(y = 14\).

Answer:

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