In parallelogram ABCD, `angleA=3cm` times `angleB`. Find all the angles of the parallelogram. In the same parallelogram, if AB=5x-7 and CD=3x+1, find the length of CD.

Answer

To solve the problem step by step, we will follow the given information about the parallelogram ABCD, where angle A is three times angle B. We will also find the length of side CD based on the expressions provided.

### Step 1: Set up the equations for angles
Given that angle A is three times angle B, we can express this relationship mathematically:
- Let angle B = x
- Then angle A = 3x

### Step 2: Use the properties of a parallelogram
In a parallelogram, opposite angles are equal, and the sum of all angles is 360 degrees. Therefore:
- Angle C = angle A = 3x
- Angle D = angle B = x

The equation for the sum of the angles in the parallelogram is:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} = 360^\circ \]
Substituting the values we have:
\[ 3x + x + 3x + x = 360^\circ \]

### Step 3: Simplify the equation
Combine like terms:
\[ 8x = 360^\circ \]

### Step 4: Solve for x
To find the value of x, divide both sides by 8:
\[ x = \frac{360^\circ}{8} = 45^\circ \]

### Step 5: Find angles A, B, C, and D
Now that we have x, we can find the angles:
- Angle B = x = 45°
- Angle A = 3x = 3(45°) = 135°
- Angle C = angle A = 135°
- Angle D = angle B = 45°

Thus, the angles of the parallelogram are:
- Angle A = 135°
- Angle B = 45°
- Angle C = 135°
- Angle D = 45°

### Step 6: Set up the equation for the sides
We are given:
- AB = 5x - 7
- CD = 3x + 1

Since AB = CD in a parallelogram, we can set up the equation:
\[ 5x - 7 = 3x + 1 \]

### Step 7: Solve for x
Rearranging the equation gives:
\[ 5x - 3x = 1 + 7 \]
\[ 2x = 8 \]
\[ x = 4 \]

### Step 8: Find the length of CD
Now substitute x back into the expression for CD:
\[ CD = 3x + 1 = 3(4) + 1 = 12 + 1 = 13 \]

### Final Answer
The angles of the parallelogram ABCD are:
- Angle A = 135°
- Angle B = 45°
- Angle C = 135°
- Angle D = 45°

The length of CD is:
- CD = 13 units