Is quadrilateral MNPQ a rectangle?

Question

(1) At least two of the four angles are right angles. 
(2) At least three of the four angles are right angles.

Bunuel · Answer

Is quadrilateral MNPQ a rectangle?

The sum of Interior Angles of a polygon is 180(n-2), where n is the number of sides.  Hence the sum of the interior angles of quadrilateral is 180(4-2) = 360°.

(1) At least two of the four angles are right angles. The angles could be 90°, 90°, 90°, 90° (rectangle) or 90°, 90°, 100°, 80° (not a rectangle). Not sufficient.

(2) At least three of the four angles are right angles. The sum of these 3 angles is 3*90° = 270°, thus the measure of the fourth angle is 360° - 270° = 90°. Sufficient.

Answer: B.

Bunuel · Answer

Similar questions to practice: is-quadrilateral-abcd-a-rectangle-156161.html is-quadrilateral-abcd-a-rectangle-100729.html Hope it helps.

niyantg · Answer

Hello Bunuel

I have a doubt here.
If all 4 angles are 90 degrees then it can be a square also.

We dont know whether all 4 sides or only 2 sides opposite sides are equal ?

So should B be the answer ?
Please correct me

Thankyou.

Bunuel · Answer

All squares are rectangles so if MNPQ is a square it's a rectangle too.

Nitish7 · Answer

Bunuel  What if all the four angles are 90 but opposite sides are not equal?

Bunuel · Answer

If all four angels are 90 degrees, then we have a rectangle, in rectangle all four angles are equal.

sasidharrs · Answer

Hi Banuel,
What if I have a square and tilt it clockwise by 45 degrees.. more like a rhombus..
its not rectangle right?

amanvermagmat · Answer

Hi

If you take a square and tilt it clockwise, it will still remain a square. When you say tilting, I assume you are just rotating that quadrilateral clockwise or anticlockwise.

A rhombus has all sides equal but all angles are not necessarily equal. So if all the angles are not equal to 90 degrees, then it will definitely not be a rectangle. 
But it can surely be a rhombus. 
As I wrote, a rhombus has all sides equal but all angles might not be equal, though opposite angles are always equal. A rhombus in which all angles are also equal - then becomes a square. So we say that all squares are rhombuses but all rhombuses are not squares.

prachisaraf · Answer

Hi, can we really say all squares are rectangles ? Can you clarify some properties similar to this. I heard something similar about a square being a rhombus too ? Not sure how to understand this.

Example: I have another similar question below:

Is Quadrilateral a square?
(1) Three angles of a quadrilateral are 90 degrees each.
(2)3 sides of a quadrrilateral are equal in length.

OA: C

(1) Three angles of a quadrilateral are 90 degrees each: Then the quadrilateral can be a rectangle or a square.
(2) 3 sides of a quadrrilateral are equal in length: Then the quadrilateral can be a rhombus or a square.
(1) + (2): The quadrilateral has to be a square.

Here, point A says the same thing. However, the answer is not A.

Bunuel · Answer

A  square  is a quadrilateral (a four-sided polygon) in which all four sides are of equal length, and all four angles are right angles (90 degrees).

A  rectangle  is a quadrilateral in which opposite sides are parallel and equal in length, and all four angles are right angles. Unlike a square, a rectangle does not require all four sides to be of equal length.

Therefore, since  a square meets all the criteria for a rectangle , it is considered a special case of a rectangle. However, not all rectangles are squares because  a rectangle does not have to have four equal sides . A rectangle with sides of different lengths is not considered a square because it does not meet the definition of a square.

In summary,  all squares are rectangles because they meet the criteria for a rectangle , but  not all rectangles are squares because they do not meet the specific criteria of a square .

Check our collection of  Properties of Polygons Questions . If you are looking to improve your geometry skills, this post is an excellent resource for anyone looking to master the properties of polygons.

Is quadrilateral MNPQ a rectangle?

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