If, in the quadrilateral JKLM, the measures of the angles J, K, L, and M are in the ratio $ 2 : 3 : 5 : 6 $, t

Question

If, in the quadrilateral JKLM, the measures of the angles J, K, L, and M are in the ratio $ 2 : 3 : 5 : 6 $, then the quadrilateral is a:

Examoogle · Accepted Answer

The correct answer is Option C: trapezium. Let angles be $2x$, $3x$, $5x$, and $6x$. Their sum is $2x + 3x + 5x + 6x = 16x = 360^\circ$, giving $x = 22.5^\circ$. The angles are $45^\circ, 67.5^\circ, 112.5^\circ, 135^\circ$. Since no pair of adjacent or opposite sides/angles satisfies parallel or kite properties (such as equal adjacent sides or parallel opposite sides), check standard quadrilateral classifications. Specifically, opposite angles do not sum to $180^\circ$ and are not equal. Wait, let us check trapezium: a trapezium requires at least one pair of parallel sides, which corresponds to consecutive angles summing to $180^\circ$ ($45^\circ + 135^\circ = 180^\circ$ and $67.5^\circ + 112.5^\circ = 180^\circ$). Therefore, adjacent angles on two sides sum to $180^\circ$, meaning two sides are parallel!

Step-by-Step Solution & Concepts:
Shortcut Trick: Let angles be $2x$, $3x$, $5x$, and $6x$. Their sum is $2x + 3x + 5x + 6x = 16x = 360^\circ$, giving $x = 22.5^\circ$. The angles are $45^\circ, 67.5^\circ, 112.5^\circ, 135^\circ$. Since no pair of adjacent or opposite sides/angles satisfies parallel or kite properties (such as equal adjacent sides or parallel opposite sides), check standard quadrilateral classifications. Specifically, opposite angles do not sum to $180^\circ$ and are not equal. Wait, let us check trapezium: a trapezium requires at least one pair of parallel sides, which corresponds to consecutive angles summing to $180^\circ$ ($45^\circ + 135^\circ = 180^\circ$ and $67.5^\circ + 112.5^\circ = 180^\circ$). Therefore, adjacent angles on two sides sum to $180^\circ$, meaning two sides are parallel!

Formula Used: $$	ext{Sum of interior angles of a quadrilateral} = 360^\circ$$

Given: Ratio of angles J, K, L, and M of quadrilateral JKLM is 2 : 3 : 5 : 6

Steps: Step 1: Set up the angle ratio: Let the measures of angles $J$, $K$, $L$, and $M$ be $2x$, $3x$, $5x$, and $6x$ respectively.. Step 2: Apply sum of angles in a quadrilateral: The sum of all interior angles of a quadrilateral is always $360^\circ$. Therefore, set up the sum equation and solve for $x$.. Step 3: Calculate individual angles: Substitute $x = 22.5^\circ$ into each angle expression: Angle $J = 2(22.5^\circ) = 45^\circ$, Angle $K = 3(22.5^\circ) = 67.5^\circ$, Angle $L = 5(22.5^\circ) = 112.5^\circ$, Angle $M = 6(22.5^\circ) = 135^\circ$.. Step 4: Identify the quadrilateral type: Notice that $\angle J + \angle M = 45^\circ + 135^\circ = 180^\circ$ and $\angle K + \angle L = 67.5^\circ + 112.5^\circ = 180^\circ$. Since consecutive interior angles on two sides are supplementary, the opposite sides are parallel, which defines a trapezium.

Answer: C is correct because the sum of consecutive interior angles equals $180^\circ$, which confirms that the quadrilateral is a trapezium.

Exam Tip: In any polygon problems involving angle ratios, always check if consecutive interior angles sum to $180^\circ$ to immediately identify parallel lines and trapeziums.

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Discussion (0)

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Discussion (0)

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Related Questions

Discussion (0)

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Related Questions

Discussion (0)