Question

if (mangle adb = mangle bdc) and (cdperp ae), which is a valid conclusion?
i (overrightarrow{db}) bisects (angle adc)
ii (angle adb) and (angle adf) are supplementary angles
iii (angle adb) and (angle bdc) are complementary angles
o i only
o i and ii
o i and iii
o i, ii, and iii

Explanation:

Step1: Recall angle - bisector definition

Since \(m\angle ADB=m\angle BDC\), by the definition of an angle - bisector (a ray that divides an angle into two congruent angles), \(\overrightarrow{DB}\) bisects \(\angle ADC\). So statement I is true.

Step2: Analyze supplementary angles

\(\angle ADE = 180^{\circ}\) because \(\overrightarrow{AE}\) is a straight - line. \(\angle ADB+\angle BDE=\angle ADE = 180^{\circ}\), and \(\angle BDE=\angle ADF\) (vertical angles are congruent). So \(\angle ADB+\angle ADF = 180^{\circ}\), and \(\angle ADB\) and \(\angle ADF\) are supplementary angles. Statement II is true.

Step3: Analyze complementary angles

We know that \(CD\perp AE\), so \(\angle CDE = 90^{\circ}\), and \(m\angle ADB=m\angle BDC\), but there is no information to suggest that \(m\angle ADB + m\angle BDC=90^{\circ}\). In fact, \(\angle ADB+\angle BDC\) is not necessarily \(90^{\circ}\), so \(\angle ADB\) and \(\angle BDC\) are not complementary angles. Statement III is false.

Answer:

I and II only