In a `Delta ABC, (AB)/(AC) = (BD)/(DC) , /_B = 70^@ and /_C = 50^@`, then `/_BAD`?

Options

A. `60^@`
B. ` 20^@`
C. `30^@`
D. `50^@`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Identify the given information
We have a triangle ABC where:
- \( \frac{AB}{AC} = \frac{BD}{DC} \)
- \( \angle B = 70^\circ \)
- \( \angle C = 50^\circ \)

### Step 2: Find angle A
To find angle A, we use the fact that the sum of angles in a triangle is \( 180^\circ \):
\[ \angle A = 180^\circ - \angle B - \angle C \]
Substituting the known values:
\[ \angle A = 180^\circ - 70^\circ - 50^\circ \]
\[ \angle A = 180^\circ - 120^\circ = 60^\circ \]

### Step 3: Use the property of the angle bisector
Since \( \frac{AB}{AC} = \frac{BD}{DC} \), we can conclude that AD is the angle bisector of \( \angle BAC \). This means that:
\[ \angle BAD = \angle CAD \]

### Step 4: Calculate angle BAD
Since \( \angle A = 60^\circ \) and AD is the angle bisector, we can find \( \angle BAD \) as follows:
\[ \angle BAD = \frac{\angle A}{2} = \frac{60^\circ}{2} = 30^\circ \]

### Final Answer
Thus, the measure of angle \( \angle BAD \) is:
\[ \angle BAD = 30^\circ \]

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