Three consecutive angles of a cyclic quadrilateral are in the ratio 1 : 4 : 5. What is the measure of the fourth angle (in degrees)?

Introduction / Context:
This problem is about angle properties of a cyclic quadrilateral, which is a quadrilateral whose vertices all lie on a circle. In such a quadrilateral, opposite angles are supplementary, meaning they add up to 180 degrees. Using the given ratio of three consecutive angles, we can find the actual angles and then determine the fourth angle. This kind of question checks knowledge of both ratio manipulation and special properties of cyclic quadrilaterals.

Given Data / Assumptions:
• The quadrilateral is cyclic, so opposite interior angles sum to 180 degrees.
• Three consecutive angles are in the ratio 1 : 4 : 5.
• Let the three consecutive angles be A, B, and C, with A : B : C = 1 : 4 : 5.
• The fourth angle is D, which we must find in degrees.

Concept / Approach:
Assign a variable x to represent the common ratio unit. Then the three consecutive angles can be written as x, 4x, and 5x. In a quadrilateral, the sum of all four interior angles is 360 degrees. In a cyclic quadrilateral, each pair of opposite angles also sums to 180 degrees. If we take A, B, C, D in order, then A and C are opposite angles. We can write equations using both the 360 degree sum and the 180 degree opposite angle property to solve for x and then find D.

Step-by-Step Solution:
Step 1: Let A = x, B = 4x, and C = 5x according to the given ratio 1 : 4 : 5. Step 2: Since the quadrilateral is cyclic, opposite angles sum to 180 degrees. Angles A and C are opposite, so A + C = 180 degrees. Step 3: Substitute A = x and C = 5x to get x + 5x = 180. Step 4: Simplify to 6x = 180, so x = 180 / 6 = 30 degrees. Step 5: Now A = x = 30 degrees, B = 4x = 120 degrees, and C = 5x = 150 degrees. Step 6: The sum of all interior angles is 360 degrees, so A + B + C + D = 360. Step 7: Substitute values: 30 + 120 + 150 + D = 360, so 300 + D = 360. Step 8: Solve for D: D = 360 - 300 = 60 degrees.

Verification / Alternative check:
We can verify the opposite angle property using the found values. A is 30 degrees and C is 150 degrees; their sum is 30 + 150 = 180 degrees, satisfying the cyclic quadrilateral condition. B is 120 degrees and D is 60 degrees, and their sum is also 120 + 60 = 180 degrees, confirming that they form the other pair of opposite angles. The total of all angles is 30 + 120 + 150 + 60 = 360 degrees, which matches the general quadrilateral angle sum. This consistent set of checks confirms that D = 60 degrees is correct.

Why Other Options Are Wrong:
120 degrees would make B + D = 120 + 120 = 240 degrees, which violates the 180 degree opposite angle condition for a cyclic quadrilateral.
30 degrees would force two angles to be equal and break the given 1 : 4 : 5 ratio structure when the full quadrilateral sum is considered.
80 and 100 degrees also fail either the total angle sum of 360 degrees or the requirement that opposite angles in a cyclic quadrilateral must be supplementary. They do not fit the consistent set of values that we derived.

Common Pitfalls:
A common mistake is to use only the 360 degree sum and ignore the special cyclic property, which can lead to incorrect systems of equations. Others may misidentify which angles are opposite or may assign the ratio incorrectly to nonconsecutive angles. Miscalculating the ratio sum or arithmetic errors when solving for x can also occur. Carefully naming the angles in order and explicitly using both the 360 degree and 180 degree rules helps avoid these traps.

Final Answer:
The measure of the fourth angle of the cyclic quadrilateral is 60°.