Question

choose all answers that describe the polygon ( i j k l ) if ( overline{i j} cong overline{j k} cong overline{k l} cong overline{l i}, m angle i=95^{circ}, m angle j=85^{circ} ), ( m angle k=95^{circ} ), and ( m angle l=85^{circ} ).

Explanation:

Step1: Determine it's a quadrilateral

A polygon with four sides is a quadrilateral. Here, \(IJKL\) has four sides (\(\overline{IJ},\overline{JK},\overline{KL},\overline{LI}\)), so it is a quadrilateral.

Step2: Check for parallelogram properties

In a parallelogram, opposite angles are equal. Here, \(\angle I = 95^{\circ}\) and \(\angle K=95^{\circ}\), \(\angle J = 85^{\circ}\) and \(\angle L = 85^{\circ}\). Also, in a quadrilateral, if the sum of adjacent angles is \(180^{\circ}\) (\(95^{\circ}+85^{\circ}=180^{\circ}\)), and opposite sides are equal (\(\overline{IJ}\cong\overline{JK}\cong\overline{KL}\cong\overline{LI}\)), it satisfies parallelogram properties.

Step3: Check for rhombus properties

A rhombus is a parallelogram with all sides equal. Since \(\overline{IJ}\cong\overline{JK}\cong\overline{KL}\cong\overline{LI}\) (all - sides equal) and it is a parallelogram (from Step 2), it is a rhombus.

Step4: Eliminate other options

• Rectangle: A rectangle has all angles equal to \(90^{\circ}\). Here, angles are \(95^{\circ}\) and \(85^{\circ}\), so not a rectangle.
• Square: A square is a rectangle and a rhombus (all angles \(90^{\circ}\) and all sides equal). Since angles are not \(90^{\circ}\), not a square.
• Trapezoid: A trapezoid has only one pair of parallel sides. But we have a parallelogram (two pairs of parallel sides), so not a trapezoid.

Answer:

• Quadrilateral
• Parallelogram
• Rhombus