One of the angles of a triangle is two-third angle of sum of adjacent angles of parallelogram. Remaining angles of the triangle are in ratio 5 : 7 respectively. What is the value of second largest angle of the triangle?

Options

A. `25^@`
B. `40^@`
C. `35^@`
D. Cannot be determined

Answer

To solve the problem step by step, we will follow the information provided in the question and the video transcript.

### Step 1: Understand the properties of a parallelogram
The sum of adjacent angles in a parallelogram is 180 degrees.

### Step 2: Calculate the angle of the triangle
We are given that one angle of the triangle is two-thirds of the sum of adjacent angles of the parallelogram. Therefore:
\[ \text{Angle of triangle} = \frac{2}{3} \times 180 = 120 \text{ degrees} \]

### Step 3: Find the sum of the remaining angles of the triangle
The sum of all angles in a triangle is 180 degrees. So, the sum of the remaining two angles is:
\[ \text{Remaining angles} = 180 - 120 = 60 \text{ degrees} \]

### Step 4: Set up the ratio of the remaining angles
Let the remaining angles be in the ratio of 5:7. We can denote these angles as \(5x\) and \(7x\).

### Step 5: Write the equation for the sum of the remaining angles
According to the ratio, we have:
\[ 5x + 7x = 60 \]
This simplifies to:
\[ 12x = 60 \]

### Step 6: Solve for \(x\)
To find \(x\), we divide both sides by 12:
\[ x = \frac{60}{12} = 5 \]

### Step 7: Calculate the remaining angles
Now we can find the values of the remaining angles:
- The second angle:
\[ 5x = 5 \times 5 = 25 \text{ degrees} \]
- The third angle:
\[ 7x = 7 \times 5 = 35 \text{ degrees} \]

### Step 8: Identify the angles of the triangle
Now we have all three angles of the triangle:
1. First angle: 120 degrees
2. Second angle: 25 degrees
3. Third angle: 35 degrees

### Step 9: Determine the second largest angle
To find the second largest angle, we can order the angles:
- 120 degrees (largest)
- 35 degrees (second largest)
- 25 degrees (smallest)

Thus, the second largest angle of the triangle is:
\[ \text{Second largest angle} = 35 \text{ degrees} \]

### Final Answer
The value of the second largest angle of the triangle is **35 degrees**.

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