In `triangle ABC, D` is point on BC such that `angle BAD =(1)/(2) angle ADC, angle BAC=84^(@) and angle C=42^(@)`. What is the measure of `angle ADB` ?

Options

A. `102^(@)`
B. `68^(@)`
C. `94^(@)`
D. `72^(@)`

Answer

To solve the problem step by step, let's follow the reasoning laid out in the video transcript.

### Step 1: Understand the given information
We have triangle ABC with:
- \( \angle BAC = 84^\circ \)
- \( \angle C = 42^\circ \)
- \( D \) is a point on \( BC \) such that \( \angle BAD = \frac{1}{2} \angle ADC \)

### Step 2: Find \( \angle ABC \)
Using the property that the sum of angles in a triangle is \( 180^\circ \):
\[ \angle ABC + \angle BAC + \angle C = 180^\circ \]
Substituting the known angles:
\[ \angle ABC + 84^\circ + 42^\circ = 180^\circ \]
\[ \angle ABC + 126^\circ = 180^\circ \]
\[ \angle ABC = 180^\circ - 126^\circ = 54^\circ \]

### Step 3: Set up the angles involving point D
Let \( \angle ADC = 2x \). According to the problem:
\[ \angle BAD = x \]
Since \( \angle BAD = \frac{1}{2} \angle ADC \), we have:
\[ \angle BAD = x \quad \text{and} \quad \angle ADC = 2x \]

### Step 4: Use the exterior angle theorem in triangle ADB
In triangle ADB, the exterior angle \( \angle ADC \) is equal to the sum of the two opposite interior angles:
\[ \angle ADC = \angle ADB + \angle ABD \]
Substituting the known values:
\[ 2x = \angle ADB + 54^\circ \]

### Step 5: Solve for \( x \)
From the equation \( 2x = \angle ADB + 54^\circ \), we can express \( \angle ADB \) as:
\[ \angle ADB = 2x - 54^\circ \]

### Step 6: Find the value of \( x \)
Using the triangle ADB's interior angles, we know:
\[ \angle ADB + \angle BAD + \angle ABD = 180^\circ \]
Substituting the known angles:
\[ (2x - 54^\circ) + x + 54^\circ = 180^\circ \]
Simplifying:
\[ 3x = 180^\circ \]
\[ x = 60^\circ \]

### Step 7: Find \( \angle ADB \)
Now substituting \( x \) back into the expression for \( \angle ADB \):
\[ \angle ADB = 2(60^\circ) - 54^\circ \]
\[ \angle ADB = 120^\circ - 54^\circ = 66^\circ \]

### Step 8: Find the final angle
Now we have:
\[ \angle ADB + 54^\circ + 54^\circ = 180^\circ \]
This gives:
\[ \angle ADB + 108^\circ = 180^\circ \]
\[ \angle ADB = 180^\circ - 108^\circ = 72^\circ \]

Thus, the measure of \( \angle ADB \) is \( 72^\circ \).

### Final Answer
\[ \angle ADB = 72^\circ \]