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Prove that a quadrilateral whose diagonals are congruent and bisect each other is a rectangle.

a. Let [tex]AC[/tex] and [tex]BD[/tex] be two line segments that bisect each other at [tex]E[/tex], with [tex]AC \cong BD[/tex]. Prove that [tex]ABCD[/tex] is a rectangle.

- Because [tex]AC[/tex] and [tex]BD[/tex] bisect each other, [tex]m\angle ABC = m\angle DCB =[/tex] (Simplify your answer). Therefore, [tex]ABCD[/tex] is a rectangle.

b. Explain how to use part (a) and only a compass and straightedge to construct any rectangle.

- Choose the correct answer below:
A. Construct two congruent line segments that perpendicularly bisect each other. The endpoints of these line segments will be the midpoints of the sides of a rectangle.
B. Construct two congruent line segments that perpendicularly bisect each other. The endpoints of these line segments will be the vertices of a rectangle.
C. Construct two congruent line segments that bisect each other. The endpoints of these line segments will be the midpoints of the sides of a rectangle.
D. Construct two congruent line segments that bisect each other. The endpoints of these line segments will be the vertices of a rectangle.

c. Construct another rectangle not congruent to the rectangle in part (b) but whose diagonals are congruent to the diagonals of the rectangle in part (b). Why are the rectangles not congruent?

- Choose the correct answer below:
A. If the diagonals are not the same length, the corresponding sides of the two rectangles will not be congruent.
B. If the diagonals are not the same length and if the angles formed by the diagonals are not congruent, the corresponding sides of the two rectangles will not be congruent.
C. If the angles formed by the diagonals are not congruent, the corresponding sides of the two rectangles will not be congruent.
D. If the angles formed by the diagonals are not right angles, the corresponding sides of the two rectangles will not be congruent.

Answer :

a. To prove that a quadrilateral with congruent diagonals that bisect each other is a rectangle, we can use the properties of diagonals in a parallelogram.

Given quadrilateral ABCD with diagonals AC and BD that bisect each other at point E, and AC ≅ BD.

Now, let's consider triangles ABE and CDE.

By the Side-Side-Side (SSS) congruence criterion:

- AE ≅ CE (given)

- BE ≅ DE (given)

- AB ≅ CD (opposite sides of a parallelogram are congruent)

Therefore, by SSS, triangles ABE and CDE are congruent.

By the Corresponding Parts of Congruent Triangles (CPCTC), we can conclude that ∠AEB ≅ ∠CED and ∠ABE ≅ ∠CDE.

Since corresponding angles in congruent triangles are congruent, we have:

m∠ABC = m∠DCB (corresponding angles are congruent)

m∠ABE = m∠CDE (corresponding angles are congruent)

From the above, we can deduce that m∠ABC = m∠DCB = m∠ABE = m∠CDE.

Since the opposite angles are congruent, we can conclude that ABCD is a rectangle.

b. Using a compass and straightedge, we can construct any rectangle based on the properties established in part (a):

1. Draw a line segment AB.

2. Bisect AB using a compass to find the midpoint M.

3. Using the same compass width, draw two circles centered at A and B with radius AM or BM.

4. The intersection points of the two circles will give us points C and D.

5. Connect points C and D to complete the rectangle ABCD.

c. To construct another rectangle not congruent to the one in part (b) but with congruent diagonals, we can follow these steps:

1. Draw a line segment AB.

2. Bisect AB using a compass to find the midpoint M.

3. Draw a line segment perpendicular to AB at M.

4. Extend the line segment from M in both directions to intersect AB at points C and D.

5. Connect points C and D to form a rectangle.

The rectangles in parts (b) and (c) will have congruent diagonals but different side lengths and angles, making them non-congruent.

learn more about congruent triangles from :

https://brainly.com/question/29789999

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