Three angles of a quadrilateral are in the ratio 1:2:3. The sum of the least and the greatest of these angles is 200°. Find all the four angles of the quadrilateral.
x + 2x + 3x + y = 360
y + 6x = 360
x + 3x = 200
x = 50
so
y + 300 = 360
y = 60
50, 100, 150, 60
Let's denote the three angles of the quadrilateral as x, 2x, and 3x since they are in the ratio 1:2:3.
According to the information given, the sum of the least (x) and the greatest (3x) angles is 200°.
So, x + 3x = 200°
Combining like terms, we get 4x = 200°
Dividing both sides by 4, we find x = 50°.
Now that we know the value of x, we can find the three angles of the quadrilateral:
- The first angle is x = 50°.
- The second angle is 2x = 2 * 50° = 100°.
- The third angle is 3x = 3 * 50° = 150°.
Therefore, the four angles of the quadrilateral are:
- Angle 1: 50°
- Angle 2: 100°
- Angle 3: 150°
- Angle 4: The remaining angle, which can be found by subtracting the sum of the other three angles from 360° (since the sum of all angles in a quadrilateral is 360°). So, angle 4 = 360° - (50° + 100° + 150°). Solving this equation gives us the final angle: angle 4 = 60°.
Hence, the four angles of the quadrilateral are 50°, 100°, 150°, and 60°.
To find the angles of the quadrilateral, we can start by assigning variables to the three angles in the ratio of 1:2:3. Let's call the smallest angle "x", the second largest angle "2x", and the largest angle "3x".
According to the question, the sum of the smallest and the largest of these angles is 200°. So we can create an equation:
x + 3x = 200
Simplifying the equation, we have:
4x = 200
Next, we solve for "x" by dividing both sides of the equation by 4:
x = 200 / 4
x = 50
Now we know that the smallest angle (x) is 50°. Plugging this value back into the ratios, we can find the other angles:
The second largest angle = 2x = 2 * 50 = 100°
The largest angle = 3x = 3 * 50 = 150°
To find the remaining angle, we know that the sum of all four angles of a quadrilateral is 360°. Therefore, we subtract the sum of the three known angles from 360:
Sum of all four angles = 360°
Sum of the known angles = x + 2x + 3x = 6x = 6 * 50 = 300°
Remaining angle = Sum of all four angles - Sum of the known angles
Remaining angle = 360° - 300° = 60°
Therefore, the four angles of the quadrilateral are 50°, 100°, 150°, and 60°.
x+3x=200
x=50 deg
360=sum of three + fourth
360=x+2x+3x + fourth
360=6*50+fourth
fourth= ..... degrees