The angles of a cyclic quadrilateral are the consecutive terms of arithmetic sequence and one of the angle is 45(degree)
What are the measures of other 3 angles
Let's assume the angles of the cyclic quadrilateral are a, b, c, and d. Since they are in an arithmetic sequence, we can write them as:
a, a + d, a + 2d, a + 3d
We know that one of the angles is 45 degrees, so we can set up an equation to solve for a:
a = 45
Now we can substitute this value of a into our earlier expression for the angles:
45, 45 + d, 45 + 2d, 45 + 3d
Since the sum of the angles in a cyclic quadrilateral is 360 degrees, we can set up an equation to solve for d:
45 + (45 + d) + (45 + 2d) + (45 + 3d) = 360
Simplifying this equation, we get:
180 + 6d = 360
6d = 360 - 180
6d = 180
d = 180 / 6
d = 30
Now we can substitute the value of d back into our expression for the angles:
45, 45 + 30, 45 + 2(30), 45 + 3(30)
Simplifying these expressions, we get:
45, 75, 105, 135
So the measures of the other three angles are 75, 105, and 135 degrees.
To find the measures of the other three angles in a cyclic quadrilateral, where the angles are consecutive terms of an arithmetic sequence, we need to understand a few properties of cyclic quadrilaterals.
A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle. One of the key properties is the opposite angles in a cyclic quadrilateral are supplementary, meaning they add up to 180 degrees.
Given that the angles form an arithmetic sequence, we can assume the angles to be a, a + d, a + 2d, and a + 3d, where a is the first angle and d is the common difference between the angles.
In this case, one of the angles is 45 degrees, so we can say that a = 45 degrees. Now, we need to find the common difference (d) to determine the measures of the other three angles.
To find d, we can use the fact that the sum of the interior angles in a quadrilateral is 360 degrees. Therefore, we can write an equation as follows:
a + (a + d) + (a + 2d) + (a + 3d) = 360
Substituting a = 45, the equation becomes:
45 + (45 + d) + (45 + 2d) + (45 + 3d) = 360
Now, we can solve this equation to find the value of d:
45 + 45 + d + 45 + 2d + 45 + 3d = 360
180 + 6d = 360
6d = 360 - 180
6d = 180
d = 180 / 6
d = 30
Now that we have the value of d, we can substitute it back into the angles to find their measures:
Angle 2 = a + d = 45 + 30 = 75 degrees
Angle 3 = a + 2d = 45 + 2*30 = 45 + 60 = 105 degrees
Angle 4 = a + 3d = 45 + 3*30 = 45 + 90 = 135 degrees
So, the measures of the other three angles in the cyclic quadrilateral are 75 degrees, 105 degrees, and 135 degrees.
45 + (45 + d) + (45 + 2 d) + (45 + 3 d) = 360
solve for d ... calculate the other three angles