Can a quadrilateral be inscribed in a circle if degree measures of its three consecutive angles are: a) 700, 800 and 1900; b) 300, 900 and 1500; c) in the ratio 3 : 7 : 2?
Thank you in advance for your help.
opposite angles must be supplementary, So if they are in the ratio 3:2 then they are 108 and 72.
Not sure what all those huge measures refer to. They are certainly not angles of a cyclic quadrilateral.
To determine whether a quadrilateral can be inscribed in a circle, we need to consider the angle measures of the quadrilateral.
a) In the given case, the degree measures of the three consecutive angles are 700, 800, and 1900. To check if they can be angles of a quadrilateral inscribed in a circle, we need to calculate the sum of these angles.
700 + 800 + 1900 = 3400
In any quadrilateral, the sum of interior angles is always 360 degrees. However, in this case, the sum of the three given angles is 3400 degrees, which is greater than 360 degrees. Therefore, this set of angles cannot be the angles of a quadrilateral inscribed in a circle.
b) Next, let's consider the angles with degrees 300, 900, and 1500. As before, we find their sum:
300 + 900 + 1500 = 2700
Again, the sum of these angles is not equal to 360 degrees. Therefore, this set of angles also cannot be the angles of a quadrilateral inscribed in a circle.
c) Finally, let's examine the angles in the ratio 3:7:2. Let's assume the angles are 3x, 7x, and 2x, respectively. To find the sum of these angles, we add them up:
3x + 7x + 2x = 12x
For any quadrilateral inscribed in a circle, the sum of its interior angles will be 360 degrees. So, we set the sum equal to 360:
12x = 360
x = 360/12
x = 30
Now that we know x = 30, we can find the individual angle measures:
First angle: 3x = 3 * 30 = 90 degrees
Second angle: 7x = 7 * 30 = 210 degrees
Third angle: 2x = 2 * 30 = 60 degrees
Let's verify if their sum is indeed 360 degrees:
90 + 210 + 60 = 360
Since the sum is equal to 360 degrees, the angles with a ratio of 3:7:2 can be the angles of a quadrilateral inscribed in a circle.
In conclusion, for the given cases, only option c) - angles in the ratio 3:7:2 - can be the angles of a quadrilateral inscribed in a circle.