Question

given: hijk is a rectangle; hikl is a parallelogram. prove: \\( \triangle hjl \\) is isosceles. 1. hijk is a rectangle. 1. given 2. \\( \overline{ik} \cong \overline{hj} \\) 2. click here to insert 3. hikl is a parallelogram. 3. given 4. click here to insert 4. click here to insert

Explanation:

Step1: Properties of rectangle

In a rectangle, the diagonals are congruent. Since \(HIJK\) is a rectangle, by the property of rectangles (diagonals of a rectangle are congruent), we have \(\overline{IK}\cong\overline{HJ}\).

Step2: Properties of parallelogram

In a parallelogram, the opposite sides are congruent. Since \(HIKL\) is a parallelogram, by the property of parallelograms (opposite sides of a parallelogram are congruent), we have \(\overline{IK}\cong\overline{HL}\).

Step3: Transitive property

By the transitive property of congruence (\(a\cong b\) and \(b\cong c\) implies \(a\cong c\)), since \(\overline{IK}\cong\overline{HJ}\) and \(\overline{IK}\cong\overline{HL}\), we get \(\overline{HJ}\cong\overline{HL}\).

Step4: Definition of isosceles triangle

A triangle is isosceles if at least two of its sides are congruent. In \(\triangle HJL\), since \(\overline{HJ}\cong\overline{HL}\), by the definition of an isosceles triangle, \(\triangle HJL\) is isosceles.

Answer:

1. Diagonals of a rectangle are congruent; 4. \(\overline{IK}\cong\overline{HL}\) (opposite sides of a parallelogram are congruent)