Examkin

Question

Difficulty: MediumPolygon Angles and Properties

An irregular convex hexagon has two interior angles that are right angles. The remaining four interior angles have measures in the ratio 4:5:5:64:5:5:6. What is the measure, in degrees, of the largest interior angle of this hexagon?

Answer: 162 degrees

Answer

The correct answer is 162162 degrees. The sum of the interior angles of a hexagon is 720∘720^\circ. Subtracting the two right angles (180∘180^\circ total) leaves a remaining sum of 540∘540^\circ. The ratio of the remaining four angles is 4:5:5:64:5:5:6, which can be represented as 4x,5x,5x,6x4x, 5x, 5x, 6x, summing to 20x=54020x = 540. Solving for the multiplier gives x=27x = 27. The largest angle is 6(27)=162∘6(27) = 162^\circ, which is also greater than the two 90∘90^\circ angles.
The sum of the interior angles of a hexagon is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ. Subtracting the two right angles (180∘180^\circ) gives a remaining sum of 540∘540^\circ for the other four angles. Let these four angles be 4x,5x,5x,4x, 5x, 5x, and 6x6x. Their sum is 20x=54020x = 540, which solves to x=27x = 27. The largest angle is 6x=6(27)=162∘6x = 6(27) = 162^\circ, which is also larger than the two 90∘90^\circ angles.

Step-by-Step Solution

1
Calculate the sum of all interior angles of a convex hexagon.
The sum is (6−2)×180∘=720∘(6-2) \times 180^\circ = 720^\circ.
The sum of the interior angles of any convex nn-gon is given by (n−2)×180∘(n-2) \times 180^\circ.
2
Subtract the sum of the two right angles from the total sum.
The remaining sum is 720∘−180∘=540∘720^\circ - 180^\circ = 540^\circ.
Two right angles contribute 90∘+90∘=180∘90^\circ + 90^\circ = 180^\circ to the total.
3
Set up a linear equation representing the ratio of the remaining four angles.
The equation is 4x+5x+5x+6x=5404x + 5x + 5x + 6x = 540, which simplifies to 20x=54020x = 540, yielding x=27x = 27.
The angles are proportional to the parts of the ratio, and their sum must equal the remaining 540∘540^\circ.
4
Calculate the largest angle from the ratio and compare with the right angles.
The largest angle is 6×27=162∘6 \times 27 = 162^\circ.
The largest term in the ratio is 66, and the resulting angle 162∘162^\circ is larger than both 90∘90^\circ and the other calculated angles (108∘108^\circ and 135∘135^\circ).

Key Concept

The sum of the interior angles of a convex polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ, and ratio relationships can be solved using algebraic multipliers.
Rate this question

Topics

Polygon Angles and Properties

More questions from this topic