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A short class with Sarah and her students. The other ways are under More.
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A quadrilateral is a polygon with four sides. Is a shape with three straight sides and one curved side a quadrilateral?
No, all four sides have to be straight line segments.
What about four segments that cross in the middle, like a bow-tie?
That's not allowed either. The sides can only meet at their endpoints.
Right. So a quadrilateral is closed, flat, has four straight sides and no crossings.
Then a square, a kite and even a lopsided four-sided shape all count.
If I call a quadrilateral ABCD, which vertices are joined by sides?
A to B, B to C, C to D and D back to A, in order around the shape.
So which segments are the diagonals?
AC and BD, because they join opposite vertices.
How many diagonals does every quadrilateral have?
Two. From each vertex there's only one vertex that isn't next to it.
Draw diagonal AC. What shapes do you get?
Two triangles, ABC and ACD.
And the angles of each triangle add up to?
180°. So the quadrilateral's angles add up to 360°.
Three angles are 85°, 95° and 110°. Find the fourth.
They add up to 290°, so the fourth is 70°.
And if three angles were 30°, 40° and 30°?
The fourth would be 260°. That's reflex, so the quadrilateral is concave.
a quadrilateral is a closed plane figure with four straight sides that meet only at their endpoints.
in ABCD the vertices go in order, the sides are AB, BC, CD and DA, and the diagonals are AC and BD.
convex has all angles less than 180°; concave has one reflex angle and one diagonal outside.
a diagonal splits a quadrilateral into two triangles, so the interior angles add up to 360°.
subtract the known angles from 360°, or set up an equation when the angles are given in terms of x.
Three angles are 85°, 95° and 110°: the fourth angle is 360° − 290° = 70°.
Angles in the ratio 1 : 2 : 3 : 4: 10x = 360°, so x = 36° and the angles are 36°, 72°, 108° and 144°.
Angles (2x + 10)°, (3x − 20)°, (x + 40)° and (4x − 30)°: 10x = 360, so x = 36 and the angles are 82°, 88°, 76° and 114°.
Three angles are 30°, 40° and 30°: the fourth is 260°, a reflex angle, so the quadrilateral is concave.
Four straight sides that close up tight, four corners joined, no sides that cross; their angles make a full turn right, three hundred sixty, nothing lost.
A farmer fences a field with four straight fence panels joined end to end. Walking round the field and turning at each corner, she turns through a full 360° by the time she is back at the gate.
"Two triangles make a quad" - one diagonal, two triangles, 2 × 180° = 360°.
F.S.C.S. - Four Straight sides, Closed, Sides meet only at ends.
Term
What is a quadrilateral?
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What do the interior angles of a quadrilateral add up to?
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What are the diagonals of quadrilateral ABCD?
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What makes a quadrilateral concave?
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Why do a quadrilateral's angles add up to 360°?
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A polygon with four straight sides, four vertices and four interior angles.
A closed plane figure made of straight line segments that meet only at their endpoints.
A corner of a polygon, where two sides meet; the plural is vertices.
A line segment joining two opposite (non-adjacent) vertices; a quadrilateral has two.
A quadrilateral whose interior angles are all less than 180°.
A quadrilateral with one interior angle greater than 180°.
An angle greater than 180° and less than 360°.
The interior angles of any quadrilateral add up to 360°.
A quadrilateral is a polygon with four straight sides, four vertices and four interior angles, and its interior angles always add up to 360°. This lesson defines the quadrilateral precisely, names its parts and uses the angle sum to find missing angles.
A quadrilateral is a closed plane figure made of four straight line segments that meet only at their endpoints, giving four vertices and four interior angles. Quadrilateral ABCD is named by listing its vertices in order around the shape; its sides are AB, BC, CD and DA, and its two diagonals are AC and BD. A quadrilateral can be convex, with every interior angle less than 180°, or concave, with one reflex angle. Because a diagonal inside the shape splits any quadrilateral into two triangles, the interior angles always sum to 2 × 180° = 360°. This lesson uses that fact to find missing angles, solve angle problems with algebra and test whether a set of angles can belong to a quadrilateral.
The word quadrilateral comes from Latin: "quadri" means four and "latus" means side. This lesson pins down exactly what counts as a quadrilateral before later lessons sort them into types and study their theorems.
Definition. A quadrilateral is a polygon with four sides. In detail:
A shape with a curved edge is not a quadrilateral, and neither is a "bow-tie" made of four segments that cross each other.
Naming the parts. In quadrilateral ABCD the vertices are listed in order around the shape, so A is joined to B, B to C, C to D and D back to A.
Convex and concave. In a convex quadrilateral every interior angle is less than 180° and both diagonals lie inside the shape. In a concave (re-entrant) quadrilateral one interior angle is a reflex angle, greater than 180°, and one diagonal lies outside the shape. An arrowhead, or dart, is concave.
The angle sum is 360°. Draw a diagonal that lies inside the quadrilateral, for example AC. It splits ABCD into triangles ABC and ACD, and the angles of each triangle add up to 180°. Together the six triangle angles make up exactly the four angles of the quadrilateral, so:
∠A + ∠B + ∠C + ∠D = 180° + 180° = 360°
This works for every quadrilateral, convex or concave, as long as you use a diagonal inside the shape. For a convex quadrilateral, the exterior angles (one at each vertex) also add up to 360°.
Worked examples.
Looking ahead. Squares, rectangles, rhombuses, parallelograms, trapeziums and kites are all quadrilaterals: each one is a quadrilateral with extra conditions on its sides, angles or parallel lines. The next lessons describe these types and the theorems about their sides, angles and diagonals.