Question

if \\( \angle a c b \cong \angle a c d \\), \\( a b = 2 p \\), and \\( a d = p + 28 \\), what is the value of \\( p \\)?

Explanation:

Step1: Use the Angle - Bisector Theorem (HL Congruence)

Since \(\angle ACB\cong\angle ACD\), \(CA = CA\) (common side), and \(\angle ADC=\angle ABC = 90^{\circ}\) (right angles), by the Hypotenuse - Leg (HL) congruence criterion, \(\triangle ABC\cong\triangle ADC\).

Step2: Set up the equation

For congruent triangles \(\triangle ABC\) and \(\triangle ADC\), \(AB = AD\). Given \(AB = 2p\) and \(AD=p + 28\), we have the equation \(2p=p + 28\).

Step3: Solve the equation

Subtract \(p\) from both sides of the equation \(2p=p + 28\).
\(2p-p=p + 28-p\)
\(p=28\)

Answer:

\(28\)