In `Delta ABC, AB = AC.` The poins D and E are on BC. Such that AD = AE. If` /_ACE = 48^(@)` and `/_DAE = 32^(@)`, then measure of `/_BAD` is

Options

A. `26^(@)`
B. `22^(@)`
C. `24^(@)`
D. `29^(@)`

Answer

To solve the problem step by step, we will use the properties of triangles and the information given in the question.

### Step 1: Understand the triangle properties
We have triangle ABC where AB = AC. This means that triangle ABC is isosceles, and the angles opposite to equal sides are also equal.

### Step 2: Identify the angles
Given:
- Angle ACE = 48°
- Angle DAE = 32°

Since AB = AC, we can say that angle ABC = angle ACB. Let's denote these angles as x. Therefore, we have:
- Angle ABC = angle ACB = x

### Step 3: Use the triangle angle sum property
In triangle ABC, the sum of the angles is 180°. Therefore, we can write:
\[ x + x + 48° = 180° \]
\[ 2x + 48° = 180° \]
Subtracting 48° from both sides gives:
\[ 2x = 132° \]
Dividing by 2:
\[ x = 66° \]
Thus, angle ABC = angle ACB = 66°.

### Step 4: Analyze triangle ADE
In triangle ADE, we know that AD = AE (given). Therefore, angles ADE and AED are equal. Let's denote these angles as y.

Using the triangle angle sum property in triangle ADE:
\[ y + y + 32° = 180° \]
\[ 2y + 32° = 180° \]
Subtracting 32° from both sides gives:
\[ 2y = 148° \]
Dividing by 2:
\[ y = 74° \]
Thus, angle ADE = angle AED = 74°.

### Step 5: Use the exterior angle theorem in triangle ABD
In triangle ABD, angle DAE is an exterior angle to triangle ABD. According to the exterior angle theorem:
\[ \text{Angle DAE} = \text{Angle ABD} + \text{Angle BAD} \]
Substituting the known values:
\[ 32° = 66° + \text{Angle BAD} \]
Rearranging gives:
\[ \text{Angle BAD} = 32° - 66° \]
\[ \text{Angle BAD} = -34° \]
This is incorrect since angles cannot be negative. Let's correct this by using the right exterior angle:
\[ \text{Angle DAE} = \text{Angle ABD} + \text{Angle BAD} \]
We have:
\[ 74° = 66° + \text{Angle BAD} \]
So:
\[ \text{Angle BAD} = 74° - 66° \]
\[ \text{Angle BAD} = 8° \]

### Final Answer
Thus, the measure of angle BAD is **8°**.