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 = 4 \\)
\\( m \angle a c d = 4 \\)
thats not it. try again.

Explanation:

Step1: Prove triangles congruent

Since the two sides are equal (marked in the figure), triangles \(ABD\) and \(ACD\) are congruent (by SSS - Side - Side - Side congruence criterion as \(AB = AC\), \(BD=CD\), \(AD = AD\)). So, \(\angle BAD=\angle CAD = 28^{\circ}\).

Step2: Use triangle angle - sum property

In \(\triangle BDC\), \(BD = CD\), so \(\angle DBC=\angle DCB\). Using the angle - sum property of a triangle (\(\angle DBC+\angle DCB+\angle BDC = 180^{\circ}\)), and \(\angle BDC = 112^{\circ}\), we get \(2\angle DCB=180^{\circ}- 112^{\circ}=68^{\circ}\), so \(\angle DCB = 34^{\circ}\).

Step3: Find \(\angle ACD\)

\(\angle ACD=\angle ACB-\angle DCB\). First, in \(\triangle ABC\), \(AB = AC\), \(\angle BAC=\angle BAD+\angle CAD=28^{\circ}+28^{\circ} = 56^{\circ}\). Then \(\angle ACB=\frac{180^{\circ}-\angle BAC}{2}=\frac{180 - 56}{2}=62^{\circ}\). So \(\angle ACD=62^{\circ}-34^{\circ}=28^{\circ}\).

Answer:

\(28\mathrm{deg}\)