Question

if the measure of \\( \angle b a d \\) is \\( 28 ^ { \circ } \\) and the measure of \\( \angle b d c \\) is \\( 112 ^ { \circ } \\), what is the measure of \\( \angle a c d \\)? enter deg after any value that is in degrees. \\( m \angle a c d = \\)

Explanation:

Step1: Prove $\triangle ABD\cong\triangle ACD$

By SSS (Side - Side - Side) congruence criterion (two pairs of equal sides and common side \(AD\)), \(\triangle ABD\cong\triangle ACD\). So, \(\angle BAD=\angle CAD = 28^{\circ}\).

Step2: Use the property of the exterior angle of a triangle

We know that \(\angle BDC\) is the exterior angle of \(\triangle ADC\). According to the exterior - angle theorem (\(\angle BDC=\angle CAD+\angle ACD\)).
Substitute \(\angle BDC = 112^{\circ}\) and \(\angle CAD=28^{\circ}\) into the formula: \(112^{\circ}=28^{\circ}+\angle ACD\).

Step3: Solve for \(\angle ACD\)

Rearrange the equation: \(\angle ACD=112^{\circ}- 28^{\circ}\).

Answer:

\(84\mathrm{deg}\)