Question

problem 23
given: hijk is a rectangle; hikl is a parallelogram.
prove: △hjl is isosceles.

1. hijk is a rectangle.
2. \\( \overline { ik } \cong \overline { hj } \\)
3. hikl is a parallelogram.
4. given
5. click here to insert
6. given
7. click here to insert
8. 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, 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 of congruence

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:

The reason for statement 2 is "Diagonals of a rectangle are congruent". The statement 4 is "\(\overline{IK}\cong\overline{HL}\)" and the reason for statement 4 is "Opposite sides of a parallelogram are congruent".