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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Answer to a math 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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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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