Question

your turn

1. according to the theorems, label the rectangles appropriately.

conditions for rectangles
if box, then the parallelogram is a rectangle.
if box, then the parallelogram is a rectangle.

use the given information to determine whether the quadrilateral is necessarily a rectangle. explain your reasoning.

1. given: \\(ef \cong gh, fg \cong he, fh \cong ge\\)
2. given: \\(m\angle feg = 45^{\circ}, m\angle geh = 50^{\circ}\\)

Explanation:

Identify conditions for rectangles

Using the Properties of Rectangles knowledge point

• A parallelogram is a rectangle if it has one right angle.
• A parallelogram is a rectangle if its diagonals are congruent.

Analyze Question 2

Using the Properties of Rectangles knowledge point

• Given: \(EF \cong GH\), \(FG \cong HE\), and \(FH \cong GE\).
• Since opposite sides are congruent (\(EF \cong GH\) and \(FG \cong HE\)), \(EFGH\) is a parallelogram.
• Since the diagonals are congruent (\(FH \cong GE\)), the parallelogram is a rectangle.
• Therefore, \(EFGH\) is necessarily a rectangle.

Analyze Question 3

Using the Properties of Rectangles knowledge point

• Given: \(m\angle FEG = 45^\circ\) and \(m\angle GEH = 50^\circ\).
• The measure of angle \(FEH\) is \(m\angle FEH = m\angle FEG + m\angle GEH = 45^\circ + 50^\circ = 95^\circ\).
• Since \(m\angle FEH = 95^\circ

eq 90^\circ\), the quadrilateral does not have a right angle.

• Therefore, \(EFGH\) is not necessarily a rectangle.

Answer:

Question 1

• First Condition: If a parallelogram has one right angle, then the parallelogram is a rectangle.
• Second Condition: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Question 2

• Conclusion: Yes, the quadrilateral is necessarily a rectangle.
• Explanation: The given conditions \(EF \cong GH\) and \(FG \cong HE\) establish that \(EFGH\) is a parallelogram because its opposite sides are congruent. The third condition, \(FH \cong GE\), states that the diagonals are congruent. A parallelogram with congruent diagonals is a rectangle.

Question 3

• Conclusion: No, the quadrilateral is not necessarily a rectangle.
• Explanation: The measure of the interior angle \(\angle FEH\) is the sum of \(m\angle FEG\) and \(m\angle GEH\), which is \(45^\circ + 50^\circ = 95^\circ\). Since a rectangle must have four \(90^\circ\) angles, a quadrilateral with an interior angle of \(95^\circ\) cannot be a rectangle.