QUADRILATERALS

Introduction

A quadrilateral is a 2-dimensional closed polygon with exactly four straight sides, four vertices (corners), and four interior angles. The word comes from Latin: "quadri" meaning four, and "latus" meaning side. From the screen you are reading on to the walls of your room, quadrilaterals are everywhere in the built environment.

1. Basic Properties of Quadrilaterals

• Angles: A quadrilateral has four interior angles.
• Angle Sum Property: The sum of the interior angles of ANY quadrilateral is always $360^\circ$.
  • Proof: If you draw a diagonal line connecting two opposite corners, you split the quadrilateral into two triangles. Since each triangle has $180^\circ$, two triangles have $180^\circ + 180^\circ = 360^\circ$.
• Diagonals: A quadrilateral has exactly two diagonals (line segments connecting opposite non-adjacent vertices).

MCQs - Basic Properties

1. A quadrilateral is a polygon with exactly how many sides?
   a) 3
   b) 4
   c) 5
   d) 6
   Answer: b
2. What is the sum of the interior angles of a quadrilateral?
   a) $180^\circ$
   b) $270^\circ$
   c) $360^\circ$
   d) $540^\circ$
   Answer: c
3. How many diagonals can be drawn inside a quadrilateral?
   a) 1
   b) 2
   c) 3
   d) 4
   Answer: b
4. Three angles of a quadrilateral are $60^\circ$, $100^\circ$, and $120^\circ$. What is the measure of the fourth angle?
   a) $60^\circ$
   b) $80^\circ$
   c) $90^\circ$
   d) $100^\circ$
   Answer: b ($360 - (60+100+120) = 80$)
5. A quadrilateral is split into two triangles by drawing a:
   a) Median
   b) Altitude
   c) Diagonal
   d) Perpendicular bisector
   Answer: c
6. If the angles of a quadrilateral are in the ratio 1:2:3:4, what is the measure of the largest angle?
   a) $36^\circ$
   b) $72^\circ$
   c) $108^\circ$
   d) $144^\circ$
   Answer: d
7. The word "quadrilateral" comes from Latin words meaning:
   a) Four sides
   b) Four angles
   c) Four points
   d) Four lines
   Answer: a
8. How many vertices does a quadrilateral have?
   a) 2
   b) 3
   c) 4
   d) 5
   Answer: c
9. In quadrilateral ABCD, if angle A = $90^\circ$, angle B = $90^\circ$, and angle C = $90^\circ$, what is the measure of angle D?
   a) $45^\circ$
   b) $60^\circ$
   c) $90^\circ$
   d) $120^\circ$
   Answer: c
10. A regular quadrilateral is more commonly known as a:
    a) Rectangle
    b) Rhombus
    c) Square
    d) Trapezoid
    Answer: c

2. Convex vs. Concave Quadrilaterals

Just like other polygons, quadrilaterals can be classified by their shape:
* Convex Quadrilateral: All four interior angles are less than $180^\circ$. Both diagonals lie completely inside the shape. (These are the common shapes like squares, rectangles, and trapezoids).
* Concave Quadrilateral (Dart/Arrowhead): One interior angle is greater than $180^\circ$ (a reflex angle). The shape looks like it has been "dented" inwards. One of the diagonals lies outside the shape!

MCQs - Convex vs Concave

1. A quadrilateral where all interior angles are less than $180^\circ$ is called a:
   a) Concave quadrilateral
   b) Complex quadrilateral
   c) Convex quadrilateral
   d) Reflex quadrilateral
   Answer: c
2. A concave quadrilateral is sometimes informally called a(n):
   a) Box
   b) Dart or Arrowhead
   c) Diamond
   d) Ellipse
   Answer: b
3. In a concave quadrilateral, how many interior angles are reflex angles (greater than $180^\circ$)?
   a) Exactly one
   b) Exactly two
   c) Exactly three
   d) All four
   Answer: a (If it had two, it wouldn't close with only 4 sides)
4. Where do the diagonals of a convex quadrilateral lie?
   a) Both lie outside the figure
   b) One inside, one outside
   c) Both lie entirely inside the figure
   d) They lie exactly on the edges
   Answer: c
5. Where do the diagonals of a concave quadrilateral lie?
   a) Both lie outside
   b) One lies inside, and one lies outside the figure
   c) Both lie inside
   d) They do not have diagonals
   Answer: b
6. Which of the following shapes is ALWAYS a convex quadrilateral?
   a) Dart
   b) Arrowhead
   c) Rectangle
   d) Star
   Answer: c
7. Can a convex quadrilateral have a reflex angle?
   a) Yes, exactly one
   b) Yes, up to two
   c) No, it cannot have any reflex angles
   d) Only if it is irregular
   Answer: c
8. In a concave quadrilateral, what is true about the angle that is greater than $180^\circ$?
   a) It points "outwards" from the center
   b) It points "inwards" towards the center
   c) It must be exactly $360^\circ$
   d) It must be adjacent to an acute angle
   Answer: b
9. If a quadrilateral has four angles measuring $45^\circ, 45^\circ, 200^\circ,$ and $70^\circ$, what type of quadrilateral is it?
   a) Convex
   b) Concave
   c) Regular
   d) Impossible to exist
   Answer: b
10. Why is a standard square considered a convex quadrilateral?
    a) Its diagonals are equal
    b) All its interior angles are $90^\circ$, which is less than $180^\circ$
    c) It has four equal sides
    d) Its diagonals intersect at right angles
    Answer: b

Fun Facts about Quadrilaterals!

• The Family Tree: Quadrilaterals have a very strict hierarchy or "family tree." A square is a rectangle, and a rectangle is a parallelogram, and a parallelogram is a quadrilateral! (We'll explore this in the next section).
• Tessellation Champions: Any quadrilateral—even a totally irregular, weird-looking convex one—can be used to tessellate (tile) a flat plane without leaving any gaps!
• Not Structurally Sound: Unlike triangles, which are rigid, quadrilaterals are inherently unstable. If you build a square frame out of wood and push on the corner, it will collapse into a rhombus. This is why builders always put a diagonal cross-brace (creating two triangles) to stabilize square walls!