In the given figure , ` angle ACE = 43^(@) and angle CAF = 62^(@) `, find the values of a,b and c.

Answer

To solve the problem, we need to find the values of angles A, B, and C given the angles ACE and CAF. Let's break it down step by step.

### Step 1: Identify Given Angles
We are given:
- Angle ACE = 43°
- Angle CAF = 62°

### Step 2: Find Angle AEC
In triangle ACE, the sum of the angles is 180°. Therefore, we can write the equation:
\[ \text{Angle ACE} + \text{Angle CAF} + \text{Angle AEC} = 180° \]

Substituting the known values:
\[ 43° + 62° + \text{Angle AEC} = 180° \]

Calculating:
\[ \text{Angle AEC} = 180° - 43° - 62° \]
\[ \text{Angle AEC} = 180° - 105° \]
\[ \text{Angle AEC} = 75° \]

### Step 3: Find Angle ABD
In cyclic quadrilateral ABDE, we know that the sum of opposite angles is 180°. Thus:
\[ \text{Angle ABD} + \text{Angle AEC} = 180° \]

Substituting the value of Angle AEC:
\[ \text{Angle ABD} + 75° = 180° \]

Calculating:
\[ \text{Angle ABD} = 180° - 75° \]
\[ \text{Angle ABD} = 105° \]

So, we have:
\[ A = 105° \]

### Step 4: Find Angle B in Triangle ABF
In triangle ABF, the sum of the angles is also 180°. Therefore:
\[ \text{Angle ABF} + \text{Angle BAF} + \text{Angle AFB} = 180° \]

Letting:
- Angle ABF = A (which we found to be 105°)
- Angle BAF = 62° (given)
- Angle AFB = B (unknown)

We can write:
\[ 105° + 62° + B = 180° \]

Calculating:
\[ B = 180° - 105° - 62° \]
\[ B = 180° - 167° \]
\[ B = 13° \]

So, we have:
\[ B = 13° \]

### Step 5: Find Angle C in Triangle ADF
In triangle ADF, the sum of the angles is again 180°. Thus:
\[ \text{Angle ADF} + \text{Angle DEF} + \text{Angle EFD} = 180° \]

Letting:
- Angle ADF = C (unknown)
- Angle DEF = 180° - 75° = 105° (since it is a linear pair with Angle AEC)
- Angle EFD = B = 13° (found earlier)

We can write:
\[ C + 105° + 13° = 180° \]

Calculating:
\[ C + 118° = 180° \]
\[ C = 180° - 118° \]
\[ C = 62° \]

So, we have:
\[ C = 62° \]

### Final Values
- A = 105°
- B = 13°
- C = 62°