ABCD is a cyclic quadrilateral in which BC is paralleld to AD, angle `ADC=110^(@)` and angle `BAC=50^(@)`. Find angle DAC and angle DCA.

Answer

To solve the problem, we will follow these steps:

### Step 1: Draw the cyclic quadrilateral
We start by drawing a circle and labeling the points A, B, C, and D such that ABCD forms a cyclic quadrilateral. We know that BC is parallel to AD.

### Step 2: Identify given angles
From the problem, we have:
- Angle ADC = 110°
- Angle BAC = 50°

### Step 3: Use the property of cyclic quadrilaterals
In a cyclic quadrilateral, opposite angles are supplementary. Therefore, we can find angle ABC:
\[ \text{Angle ABC} = 180° - \text{Angle ADC} = 180° - 110° = 70° \]

### Step 4: Use the angle sum property in triangle ABC
In triangle ABC, we can apply the angle sum property:
\[ \text{Angle BAC} + \text{Angle ABC} + \text{Angle BCA} = 180° \]
Substituting the known values:
\[ 50° + 70° + \text{Angle BCA} = 180° \]
\[ \text{Angle BCA} = 180° - 120° = 60° \]

### Step 5: Use the property of parallel lines
Since BC is parallel to AD, angle BCA is equal to angle CAD (alternate interior angles):
\[ \text{Angle DAC} = \text{Angle BCA} = 60° \]

### Step 6: Find angle DCA
Now, we need to find angle DCA in triangle ACD. We know:
- Angle ADC = 110°
- Angle DAC = 60°

Using the angle sum property in triangle ACD:
\[ \text{Angle DAC} + \text{Angle DCA} + \text{Angle ADC} = 180° \]
Substituting the known values:
\[ 60° + \text{Angle DCA} + 110° = 180° \]
\[ \text{Angle DCA} = 180° - 170° = 10° \]

### Final Answers
- Angle DAC = 60°
- Angle DCA = 10°

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