Hello, and welcome to your mathematics lesson. Today, we are diving into rectilinear figures. We will explore polygons, with special focus on quadrilaterals — parallelograms, rectangles, rhombuses, squares, and trapeziums. By the end of this lesson, you will understand the properties that make each shape unique, and how to apply powerful theorems to solve problems.
Let us begin with the word rectilinear. It simply means forming straight lines. A rectilinear figure is any plane shape bounded entirely by straight line segments. When we have a closed figure with at least three sides, we call it a polygon.
Polygons are named by their number of sides. Three sides make a triangle, four make a quadrilateral, five a pentagon, six a hexagon, seven a heptagon, and eight an octagon. There are more, of course, but these are the most common.
Now, let us distinguish between two types of polygons. A convex polygon has every interior angle less than one hundred eighty degrees. Imagine a shape where all corners point outward — that is convex. A concave polygon, however, has at least one interior angle greater than one hundred eighty degrees, creating a dent inward. Unless stated otherwise, when we say polygon, we mean a convex polygon.
Here comes a fundamental result. The sum of interior angles of a polygon with n sides equals (2n – 4) right angles, or (2n – 4) × 90°.
Why does this work? Imagine drawing all possible diagonals from one vertex of your polygon. This divides the polygon into (n – 2) triangles. Since each triangle has angles summing to one hundred eighty degrees, the total becomes (n – 2) × 180°, which simplifies to our formula.
Now for exterior angles. If you extend each side of a polygon in order, either clockwise or anticlockwise, the sum of these exterior angles is always four right angles — that is, three hundred sixty degrees. This remarkable fact holds true for every polygon, regardless of how many sides it has.
Let us see this in action. Suppose the sum of interior angles of a polygon is five times the sum of its exterior angles. We know exterior angles sum to three hundred sixty degrees, so interior angles sum to five times that: one thousand eight hundred degrees. Using our formula: (2n – 4) × 90 = 1800. Solving: 2n – 4 = 20, so n = 12. The polygon has twelve sides — a dodecagon.
A regular polygon has all sides equal and all angles equal. This means all interior angles are equal, and all exterior angles are equal too.
For a regular polygon with n sides: Each interior angle equals (2n – 4) × 90° / n. Each exterior angle equals 360° / n. At every vertex, interior angle plus exterior angle equals one hundred eighty degrees.
Here is a useful relationship. If each exterior angle of a regular polygon is x°, then the number of sides is 360/x. More sides means larger interior angles and smaller exterior angles.
Consider this: each interior angle of a regular polygon is one hundred sixty degrees. Then each exterior angle is twenty degrees, giving eighteen sides. If another polygon has two-thirds as many sides, that is twelve sides. Its exterior angle becomes thirty degrees, and its interior angle becomes one hundred fifty degrees.
Now we turn to quadrilaterals — polygons with exactly four sides. A quadrilateral has four vertices, four sides, and two diagonals joining opposite vertices. The sum of all four interior angles is always three hundred sixty degrees.
Let us meet the special quadrilaterals, each with distinct properties.
First, the trapezium. A trapezium has exactly one pair of opposite sides parallel. The other pair is not parallel. If the non-parallel sides happen to be equal, we call it an isosceles trapezium. In an isosceles trapezium, base angles are equal, diagonals are equal, and if diagonals intersect at point O, then OA = OB and OC = OD.
Next, the parallelogram — a quadrilateral with both pairs of opposite sides parallel. Here are its key properties. Opposite sides are equal: AB = DC and AD = BC. Opposite angles are equal: ∠A = ∠C and ∠B = ∠D. Consecutive angles are supplementary — they add to one hundred eighty degrees. Diagonals bisect each other: OA = OC and OB = OD. Each diagonal divides the parallelogram into two congruent triangles. And diagonals create four triangles of equal area.
A rectangle is a parallelogram with all angles equal to ninety degrees. Since it is a parallelogram, all parallelogram properties apply. Additionally, diagonals are equal in length: AC = BD.
A rhombus is a parallelogram with all four sides equal. Its diagonals bisect each other at ninety degrees — they are perpendicular. Each diagonal also bisects the angles at the vertices it connects.
A square combines the best of both: it is a parallelogram with all sides equal and all angles ninety degrees. Its diagonals are equal, bisect each other, and meet at right angles. Each diagonal bisects the vertex angles, creating forty-five degree angles with the sides.
Let us summarize the diagonal properties in a clear hierarchy. All parallelograms have diagonals that bisect each other. Rectangles add equal diagonals. Rhombuses add perpendicular diagonals that bisect vertex angles. Squares have all these properties: equal, perpendicular, bisecting each other, and bisecting vertex angles.
Now we prove why these properties hold.
Theorem: In a parallelogram, opposite sides are equal. Given parallelogram ABCD, draw diagonal AC. Triangles ABC and CDA have: angle BAC equals angle DCA as alternate angles, angle BCA equals angle DAC as alternate angles, and AC is common. By angle-side-angle congruence, the triangles are congruent. Therefore, corresponding parts give us AB = DC and AD = BC.
Theorem: In a parallelogram, opposite angles are equal. Using diagonal BD, triangles ABD and CDB are congruent by angle-side-angle. Corresponding angles give ∠A = ∠C and ∠B = ∠D.
Theorem: If one pair of opposite sides of a quadrilateral are equal and parallel, it is a parallelogram. Given AB = DC and AB ∥ DC, join AC. Triangles ABC and CDA are congruent by side-angle-side: AB = DC, alternate angles ∠BAC = ∠DCA are equal, and AC is common. This gives ∠BCA = ∠DAC, which are alternate angles, so AD ∥ BC. Both pairs of opposite sides are parallel — hence, a parallelogram.
Theorem: Diagonals of a parallelogram bisect each other. In parallelogram ABCD, diagonals intersect at O. Triangles AOB and COD have: AB = DC as opposite sides, ∠OAB = ∠OCD as alternate angles, and ∠OBA = ∠ODC as alternate angles. By angle-side-angle, the triangles are congruent. Therefore, OA = OC and OB = OD.
Theorem: In a rhombus, diagonals meet at right angles. Since all sides of a rhombus are equal, AB = BC. Diagonals bisect each other, so OA = OC. With OB common, triangles AOB and COB are congruent by side-side-side. Thus, ∠AOB = ∠COB. These angles form a linear pair summing to one hundred eighty degrees, so each must be ninety degrees.
Theorem: In a rectangle, diagonals are equal. Consider triangles DAB and CBA. We have AD = BC as opposite sides, AB is common, and both angles at A and B are 90°. By side-angle-side, the triangles are congruent, giving AC = BD.
Theorem: In a square, diagonals are equal and perpendicular. A square is both a rectangle and a rhombus. From rectangle properties, diagonals are equal. From rhombus properties, diagonals are perpendicular. Hence both properties hold.
Let us apply these theorems to solve problems.
In a parallelogram ABCD, if AB = 3x – 1 and DC = 2(x + 1), setting these opposite sides equal gives x = 3. Using alternate angles and the angle sum property of triangles, we can find unknown angles step by step.
In a rhombus ABCD, if AB = 3x + 2 and AD = 4x – 4, setting adjacent sides equal gives x = 6. With ∠DAB = 60°, consecutive angle ABC is 120°. Diagonal BD bisects this angle, so ∠ABD = 60°. Triangle ABD has angles 60° each, making it equilateral, so BD = AB, giving y = 21.
For an isosceles trapezium with AB ∥ DC and AD = BC, draw a line through C parallel to DA, meeting AB produced at E. Since AECD is a parallelogram, AD = CE, and with AD = BC, we get CE = BC, making ∠CBE = ∠E. Triangles ABC and BAD are congruent, proving diagonals are equal. Using supplementary angles and parallel lines, ∠A = ∠ABC and similarly ∠D = ∠C.
Here is a beautiful result: when two parallel lines are cut by a transversal, and all four interior angles at the two intersection points are bisected, the four bisectors form a rectangle. Each angle of this quadrilateral proves to be ninety degrees through careful angle chasing using the properties of parallel lines and angle bisectors.
Another elegant problem: in parallelogram ABCD, if E is the midpoint of AB and DE bisects angle D, then BC = BE, CE bisects angle C, and ∠DEC = 90°. This follows from angle relationships and the fact that consecutive angles in a parallelogram are supplementary.
Let us recap the essential takeaways from this chapter.
First, the sum of interior angles of an n-sided polygon is (2n – 4) right angles, while exterior angles always sum to four right angles, or three hundred sixty degrees.
Second, a regular polygon has equal sides and equal angles, with each exterior angle equal to 360°/n.
Third, quadrilaterals include trapeziums, parallelograms, rectangles, rhombuses, and squares — each with progressively more specialized properties.
Fourth, in a parallelogram, opposite sides and angles are equal, consecutive angles are supplementary, and diagonals bisect each other.
Fifth, rectangles add equal diagonals to parallelogram properties. Rhombuses add perpendicular diagonals that bisect vertex angles.
Sixth, squares possess all these properties: equal sides, right angles, equal diagonals, perpendicular diagonals, and diagonal angle bisection.
Remember, geometry is about seeing relationships and building logical arguments step by step. Every theorem we proved today connects to others, creating a beautiful network of mathematical truth. Practice visualizing these shapes, drawing their diagonals, and tracking how angles and sides relate.
Thank you for your attention. Keep exploring, keep questioning, and enjoy the elegance of geometry. Until next time, happy learning!