Find the measurement of the first and third angles if the ratio of the three consecutive angles of a cyclic quadrilateral is 1:2:3
x+3x=180
4x =180
x =180÷4
x =45
First angle = 45
Third angle = 3×45
= 135
Let the first three angles be
x , 2x, and 3x
recall that in any cyclic quad , the opposite angles add up to 180°
So, wouldn't x and 3x be opposite angles ?
so ......
Let's assume the measurement of the first angle is x degrees.
Since the ratio of consecutive angles in a cyclic quadrilateral is 1:2:3, the second angle will be 2x degrees and the third angle will be 3x degrees.
The sum of the interior angles of a cyclic quadrilateral is always 360 degrees. Therefore, we can write the equation:
x + 2x + 3x = 360
Combining like terms, we get:
6x = 360
Dividing both sides by 6, we find:
x = 60
So, the measurement of the first angle is 60 degrees.
The second angle will be:
2x = 2 * 60 = 120 degrees.
And the third angle will be:
3x = 3 * 60 = 180 degrees.
Therefore, the measurement of the first angle is 60 degrees, and the third angle is 180 degrees.
To find the measurement of the first and third angles in a cyclic quadrilateral when the ratio of the three consecutive angles is 1:2:3, follow these steps:
1. Let's assume that the three consecutive angles have measurements x, 2x, and 3x.
2. In a cyclic quadrilateral, the sum of the opposite angles is 180 degrees. Therefore, the sum of the first and third angles (x + 3x) should equal 180 degrees.
3. Combining like terms, we have 4x = 180 degrees.
4. Divide both sides of the equation by 4: x = 45 degrees.
5. Now that we know x, we can find the measurements of the first and third angles:
- The first angle = x = 45 degrees.
- The third angle = 3x = 3 * 45 = 135 degrees.
So, the measurement of the first angle is 45 degrees, and the measurement of the third angle is 135 degrees.