Question

if ( \triangle bat cong \triangle dog ), and ( mangle b = 14^{circ} ), ( mangle g = 29^{circ} ), and ( mangle o=(10x + 7)^{circ} ), find ( x ) and ( mangle o ).
( x=)
( mangle o=)

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle BAT\cong\triangle DOG\), corresponding angles are equal. So \(\angle B=\angle D = 14^{\circ}\), \(\angle A=\angle O\), \(\angle T=\angle G=29^{\circ}\).

Step2: Use the angle - sum property of a triangle

In a triangle, the sum of interior angles is \(180^{\circ}\). For \(\triangle DOG\), \(m\angle D+m\angle O+m\angle G = 180^{\circ}\).
Substitute \(m\angle D = 14^{\circ}\) and \(m\angle G=29^{\circ}\) into the equation: \(14+(10x + 7)+29=180\).

Step3: Simplify the equation

Combine like terms: \(14 + 7+29+10x=180\), \(50+10x=180\).
Subtract \(50\) from both sides: \(10x=180 - 50\), \(10x=130\).

Step4: Solve for \(x\)

Divide both sides by \(10\): \(x=\frac{130}{10}=13\).

Step5: Find \(m\angle O\)

Substitute \(x = 13\) into \(m\angle O=(10x + 7)^{\circ}\). \(m\angle O=(10\times13 + 7)^{\circ}=(130+7)^{\circ}=137^{\circ}\).

Answer:

\(x = 13\)
\(m\angle O=137\)