Question

if \\( \overline { a b } \cong \overline { a d } \\), \\( m \angle a c b = v + 28 ^ { \circ } \\), and \\( m \angle a c d = 3 v \\), what is \\( m \angle a c b \\)?
\\( m \angle a c b = \square ^ { \circ } \\)

Explanation:

Step1: Use the Hypotenuse - Leg (HL) Congruence Theorem

Since \(\overline{AB}\cong\overline{AD}\) and \(CA = CA\) (common side), and \(\angle ABC=\angle ADC = 90^{\circ}\), then \(\triangle ABC\cong\triangle ADC\) by HL.

Step2: Set up the equation for angle measures

Because \(\triangle ABC\cong\triangle ADC\), \(\angle ACB=\angle ACD\). So \(v + 28=3v\).

Step3: Solve the equation for \(v\)

Subtract \(v\) from both sides: \(28=3v - v\), which simplifies to \(28 = 2v\). Then \(v=\frac{28}{2}=14\).

Step4: Find \(m\angle ACB\)

Substitute \(v = 14\) into \(m\angle ACB=v + 28\). So \(m\angle ACB=14 + 28\).

Answer:

\(42\)