Question

Show by contrast that if in a quadrilateral there are no obtuse angles, that is, angles greater than 90 degrees, then said quadrilateral is a rectangle.

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Gene

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1. Let the four angles of the quadrilateral be $A$, $B$, $C$, and $D$. Since there are no obtuse angles, each angle $A$, $B$, $C$, and $D$ is less than or equal to 90 degrees.

A \leq 90^\circ

B \leq 90^\circ

C \leq 90^\circ

D \leq 90^\circ

2. The sum of the angles in a quadrilateral is 360 degrees:

A + B + C + D = 360^\circ

3. To satisfy the condition that each angle is less than or equal to 90 degrees while their sum equals 360 degrees, they all must be exactly 90 degrees:

A = 90^\circ

B = 90^\circ

C = 90^\circ

D = 90^\circ

4. A quadrilateral with four right angles is a rectangle. Therefore, the quadrilateral must be a rectangle.

Therefore, a quadrilateral with no obtuse angles must be a rectangle.

The answer shows the solution from start to end, citing it is a rectangle.

2. The sum of the angles in a quadrilateral is 360 degrees:

3. To satisfy the condition that each angle is less than or equal to 90 degrees while their sum equals 360 degrees, they all must be exactly 90 degrees:

4. A quadrilateral with four right angles is a rectangle. Therefore, the quadrilateral must be a rectangle.

Therefore, a quadrilateral with no obtuse angles must be a rectangle.

The answer shows the solution from start to end, citing it is a rectangle.

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