In `DeltaABC`, D is a point on side BC such that `angleADC=2angleBAD`. If `angleA=80^(@) and angleC=38^(@)`, then what is the measure of `angleADB`?

Options

A. `56^(@)`
B. `62^(@)`
C. `58^(@)`
D. `52^(@)`

Answer

To solve the problem, let's break it down step by step:

### Step 1: Understand the Triangle and Given Angles
We have triangle ABC where:
- Angle A = 80°
- Angle C = 38°

We need to find Angle ADB, and we know that D is a point on side BC such that angle ADC = 2 * angle BAD.

### Step 2: Calculate Angle B
Using the triangle angle sum property:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180° \]
Substituting the known values:
\[ 80° + \text{Angle B} + 38° = 180° \]
\[ \text{Angle B} = 180° - 80° - 38° = 62° \]

### Step 3: Set Up the Relationship Between Angles
Let:
- Angle BAD = θ
- Therefore, Angle ADC = 2θ

According to the exterior angle theorem:
\[ \text{Angle ADC} = \text{Angle BAD} + \text{Angle ABD} \]
Substituting the known values:
\[ 2θ = θ + 62° \]

### Step 4: Solve for θ
Rearranging the equation:
\[ 2θ - θ = 62° \]
\[ θ = 62° \]

### Step 5: Find Angle ADB
In triangle ABD, the sum of angles is:
\[ \text{Angle ABD} + \text{Angle BAD} + \text{Angle ADB} = 180° \]
Substituting the known values:
\[ 62° + 62° + \text{Angle ADB} = 180° \]
\[ \text{Angle ADB} = 180° - 124° = 56° \]

### Final Answer
Thus, the measure of Angle ADB is:
\[ \text{Angle ADB} = 56° \]