Quadrilaterals — Questions and Answers
Welcome to HSLC Guru. This page gives the complete ASSEB (Assam State School Education Board) Class 8 General Mathematics Chapter 3 Quadrilaterals guide — a clear explanation of the concepts, fully worked answers to every question of Exercise 3.1 and Exercise 3.2, the textbook worked examples, labelled figures and extra practice questions.
Summary
This chapter studies polygons and, in particular, quadrilaterals. A simple, closed plane figure made up of a finite number of line segments is a polygon. According to the number of sides (or vertices) a polygon is named a triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10) and so on.
A line segment joining two non-consecutive vertices is a diagonal. A polygon in which every interior angle is less than $180^\circ$ and all diagonals lie inside is a convex polygon; if at least one interior angle is greater than $180^\circ$ it is a concave polygon. The sum of the interior angles of an n-sided polygon is $(2n-4)\times 90^\circ$, and the sum of the exterior angles of any polygon is always $360^\circ$.
Depending on their sides and angles, special quadrilaterals arise — the trapezium, parallelogram, rectangle, rhombus, square and kite. Their definitions, properties and mutual relationships (the family of quadrilaterals) are the heart of this chapter.
Summary: This ASSEB Class 8 General Mathematics Chapter 3 (Quadrilaterals) guide explains polygons and quadrilaterals, interior and exterior angles, the angle-sum formula (2n − 4) × 90°, the constant exterior-angle sum of 360°, and the special quadrilaterals — trapezium, parallelogram, rectangle, rhombus, square and kite — with full worked answers to Exercise 3.1 and Exercise 3.2, labelled figures and extra practice questions.
Textbook Questions and Answers
Important Formulae and Ideas
- Sum of interior angles of an n-sided polygon $= (2n-4)\times 90^\circ$.
- Each interior angle of a regular polygon $= \dfrac{(2n-4)\times 90^\circ}{n}$.
- Sum of exterior angles of any polygon $= 360^\circ$; each exterior angle of a regular polygon $= \dfrac{360^\circ}{n}$.
- Number of diagonals of an n-sided polygon $= \dfrac{n(n-3)}{2}$.
- Interior angle $+$ exterior angle $= 180^\circ$ (at one vertex).
The figure above shows the family of quadrilaterals — every square is also a rectangle, a rhombus, a parallelogram and a trapezium.
Worked Examples (from the textbook)
Example 1. Find the sum of the interior angles of a regular polygon of 12 sides.
Answer: $n=12$; sum $=(2\times 12-4)\times 90^\circ=20\times 90^\circ=1800^\circ$.
Example 2. The sum of the interior angles of a regular polygon of 15 sides is $2340^\circ$. Find each interior angle.
Answer: Each interior angle $=\dfrac{2340^\circ}{15}=156^\circ$.
Example 3. Find the number of sides of a regular polygon whose each interior angle is $160^\circ$.
Answer: $\dfrac{(2n-4)}{n}\times 90^\circ=160^\circ \Rightarrow 180n-360=160n \Rightarrow 20n=360 \Rightarrow n=18$. Number of sides $=18$.
Example 4. Find the number of sides of a regular polygon whose each exterior angle is $36^\circ$.
Answer: Number of sides $=\dfrac{360^\circ}{36^\circ}=10$.
Example 5. Is a regular polygon possible whose each exterior angle is (i) $24^\circ$ (ii) $22^\circ$ (iii) $40^\circ$?
Answer: (i) $\dfrac{360}{24}=15$ — a whole number, so possible (15 sides). (ii) $\dfrac{360}{22}=16.36\ldots$ — not a whole number, so not possible. (iii) $\dfrac{360}{40}=9$ — possible (9 sides).
Example 6. Is a regular polygon possible whose each interior angle is (i) $24^\circ$ (ii) $70^\circ$ (iii) $135^\circ$?
Answer: (i) exterior $=156^\circ$; $\dfrac{360}{156}$ is not a whole number — not possible. (ii) exterior $=110^\circ$; $\dfrac{360}{110}$ is not a whole number — not possible. (iii) exterior $=45^\circ$; $\dfrac{360}{45}=8$ — possible (8 sides).
Example 7. (i) What is the smallest possible interior angle of a regular polygon? (ii) What is the largest possible exterior angle?
Answer: The polygon with the fewest sides is a triangle, and a regular triangle (equilateral) has each interior angle $60^\circ$. Hence (i) smallest interior angle $=60^\circ$; (ii) largest exterior angle $=180^\circ-60^\circ=120^\circ$.
Example 8. Find the unknown angles in the following parallelograms.
Answer: (i) $80^\circ+b=180^\circ \Rightarrow b=100^\circ$; $c=80^\circ$; $d=b=100^\circ$. (ii) $a+25^\circ+110^\circ=180^\circ \Rightarrow a=45^\circ$; $25^\circ+45^\circ+c=180^\circ \Rightarrow c=110^\circ$; $b=25^\circ$. (iii) $a+130^\circ=180^\circ \Rightarrow a=50^\circ$; $b=130^\circ$; $d=130^\circ$ (corresponding angle).
Example 9. The two adjacent angles of a parallelogram are in the ratio $5:4$. Find each angle.
Answer: $5k+4k=180^\circ \Rightarrow 9k=180^\circ \Rightarrow k=20^\circ$. The angles are $100^\circ, 80^\circ, 100^\circ, 80^\circ$.
Example 10. The diagonals of a parallelogram meet at O. If $OD=15, OA=14, OB=3b, OC=2a$, find $a$ and $b$.
Answer: The diagonals of a parallelogram bisect each other. So $OB=OD \Rightarrow 3b=15 \Rightarrow b=5$; $OC=OA \Rightarrow 2a=14 \Rightarrow a=7$.
Example 11. Is it possible to draw a parallelogram with (i) $AB=CD=6$ cm, $BC=AD=4$ cm (ii) $\angle B=80^\circ, \angle C=90^\circ$ (iii) $\angle A=60^\circ, \angle C=70^\circ$?
Answer: (i) Possible — opposite sides are equal. (ii) Not possible — adjacent angles must be supplementary ($80^\circ+90^\circ\neq 180^\circ$). (iii) Not possible — opposite angles must be equal ($60^\circ\neq 70^\circ$).
Example 12. Name the quadrilaterals whose diagonals (i) bisect each other (ii) are perpendicular bisectors of each other (iii) are equal.
Answer: (i) parallelogram, rectangle, rhombus, square. (ii) rhombus, square. (iii) rectangle, square.
Exercise 3.1
1. Draw the following polygons — (i) convex hexagon (ii) concave heptagon (iii) concave pentagon.
Answer: In a convex hexagon every angle is less than $180^\circ$ and all diagonals lie inside. In a concave heptagon and a concave pentagon at least one angle is greater than $180^\circ$ (one vertex is pushed inward). Sample figures are shown below.
2. Draw the following convex polygons. Mark the diagonals in each case and find the total number of diagonals — (i) regular hexagon (ii) octagon (iii) nonagon (iv) decagon.
Answer: Number of diagonals $=\dfrac{n(n-3)}{2}$. (i) hexagon: $\dfrac{6\times 3}{2}=9$. (ii) octagon: $\dfrac{8\times 5}{2}=20$. (iii) nonagon: $\dfrac{9\times 6}{2}=27$. (iv) decagon: $\dfrac{10\times 7}{2}=35$.
3. Draw the following regular polygons. Find the sum of the angles and the measure of each angle — (i) regular hexagon (ii) regular nonagon (iii) regular polygon of 12 sides.
Answer: (i) hexagon: sum $=(2\times 6-4)\times 90^\circ=720^\circ$; each angle $=\dfrac{720^\circ}{6}=120^\circ$. (ii) nonagon: sum $=(2\times 9-4)\times 90^\circ=1260^\circ$; each angle $=\dfrac{1260^\circ}{9}=140^\circ$. (iii) 12 sides: sum $=(2\times 12-4)\times 90^\circ=1800^\circ$; each angle $=\dfrac{1800^\circ}{12}=150^\circ$.
4. Find the measures of the angles $a, b$ in the following figures.
Answer: (i) This is a pentagon; sum of interior angles $=540^\circ$. So $a=540^\circ-(115^\circ+95^\circ+105^\circ+120^\circ)=540^\circ-435^\circ=105^\circ$.
(ii) All five sides are marked equal, so we treat this as a regular pentagon; each interior angle $=108^\circ$ and each exterior angle $=72^\circ$. Hence $a=108^\circ$ (interior angle at the bottom vertex) and $b=72^\circ$ (exterior angle).
(iii) This is a quadrilateral; sum of interior angles $=360^\circ$. So $a=360^\circ-(60^\circ+80^\circ+70^\circ)=150^\circ$; and $b$ is the exterior angle of $80^\circ$, so $b=180^\circ-80^\circ=100^\circ$.
(iv) This is a hexagon with exterior angles marked. The exterior angle of $130^\circ$ is $b=180^\circ-130^\circ=50^\circ$. The exterior angles add up to $360^\circ$, so $55^\circ+60^\circ+a+70^\circ+65^\circ+b=360^\circ \Rightarrow 300^\circ+a=360^\circ \Rightarrow a=60^\circ$. Hence $a=60^\circ, b=50^\circ$.
(v) This is a pentagon with two right angles ($90^\circ$). At the vertex with $70^\circ$ exterior, the interior angle $b=180^\circ-70^\circ=110^\circ$; at the vertex with $80^\circ$ exterior, the interior angle $=100^\circ$. The interior angles of a pentagon add up to $540^\circ$, so $90^\circ+110^\circ+90^\circ+100^\circ+a=540^\circ \Rightarrow a=150^\circ$. Hence $a=150^\circ, b=110^\circ$.
5. Find the number of sides of a regular polygon if one exterior angle measures $30^\circ$.
Answer: Number of sides $=\dfrac{360^\circ}{30^\circ}=12$.
6. Find the measure of each exterior angle of a regular polygon of 20 sides.
Answer: Each exterior angle $=\dfrac{360^\circ}{20}=18^\circ$.
7. The interior angle of a regular polygon is given below. Find the number of sides — (i) $120^\circ$ (ii) $144^\circ$ (iii) $156^\circ$ (iv) $135^\circ$ (v) $165^\circ$.
Answer: Number of sides $=\dfrac{360^\circ}{180^\circ-\text{interior angle}}$. (i) $\dfrac{360}{60}=6$. (ii) $\dfrac{360}{36}=10$. (iii) $\dfrac{360}{24}=15$. (iv) $\dfrac{360}{45}=8$. (v) $\dfrac{360}{15}=24$.
8. The number of sides of polygons is given below. Find the sum of the interior angles — (i) $12$ (ii) $14$ (iii) $20$ (iv) $24$ (v) $25$.
Answer: Sum $=(2n-4)\times 90^\circ$. (i) $(24-4)\times 90^\circ=1800^\circ$. (ii) $(28-4)\times 90^\circ=2160^\circ$. (iii) $(40-4)\times 90^\circ=3240^\circ$. (iv) $(48-4)\times 90^\circ=3960^\circ$. (v) $(50-4)\times 90^\circ=4140^\circ$.
9. State with reasons whether the following statements are true or false.
Answer: (i) “Each exterior angle of a regular polygon cannot be $25^\circ$” — True, because $\dfrac{360}{25}=14.4$ is not a whole number. (ii) “Each interior angle can be $1^\circ$” — False (a $1^\circ$ interior angle gives a $179^\circ$ exterior angle, and $\dfrac{360}{179}$ is not a whole number). (iii) “The maximum exterior angle is $90^\circ$” — False, the maximum exterior angle is $120^\circ$ (equilateral triangle). (iv) “The maximum interior angle is $180^\circ$” — False, an interior angle of a convex polygon is always less than $180^\circ$. (v) “The minimum interior angle is $60^\circ$” — True (equilateral triangle).
Exercise 3.2
1. Find the unknown quantity (side/angle) of the following quadrilaterals. The arrow mark (→) on opposite sides means the pair of sides is parallel.
Answer: (i) This is a general quadrilateral (no parallel marks); sum of angles $=360^\circ$. $a=360^\circ-(110^\circ+80^\circ+75^\circ)=95^\circ$.
(ii) This is a parallelogram; the angle adjacent to $105^\circ$ is $a=180^\circ-105^\circ=75^\circ$; the opposite angles give $b=105^\circ$ and $c=a=75^\circ$.
(iii) This is a rhombus (all sides equal) with one diagonal drawn. Since the interior angle $\angle B=110^\circ$, the opposite $\angle D=110^\circ$ and $\angle A=\angle C=70^\circ$. With the diagonal, $\angle DCA=40^\circ$; so $b=\angle ACB=70^\circ-40^\circ=30^\circ$. Also $AB\parallel DC$ gives $\angle CAB=\angle DCA=40^\circ$ (alternate angles), so $a=\angle DAC=70^\circ-40^\circ=30^\circ$. Hence $a=30^\circ, b=30^\circ$.
(iv) This is a parallelogram; opposite sides are equal. $2b+1=9 \Rightarrow b=4$; $4a=24 \Rightarrow a=6$.
(v) This is a rhombus with both diagonals drawn (they cut at right angles). The diagonals bisect each other, so $b=8$ (half of the horizontal diagonal) and $c=15$ (half of the vertical diagonal). The side $a=\sqrt{8^2+15^2}=\sqrt{289}=17$. Hence $a=17, b=8, c=15$.
(vi) This is a parallelogram; the angle adjacent to $75^\circ$ is $a=180^\circ-75^\circ=105^\circ$; the opposite angles give $b=75^\circ$ and $c=105^\circ$.
(vii) This is a rectangle with diagonal $AC=10$. The diagonals of a rectangle are equal and bisect each other, so $BD=10$ and $OA=OB=OC=OD=5$. Hence $x=OD=5, y=OB=5$.
(viii) This is a square with side $AB=6$. All sides are equal, so $x=y=z=6$. Hence $x+y+z=18$.
(ix) This is a rhombus with side $AB=7$. All sides are equal, so $x=y=z=7$. Hence $x+y+z=21$.
(x) This is a parallelogram; adjacent angles are supplementary. $(3x-2)+(2x-3)=180 \Rightarrow 5x-5=180 \Rightarrow x=37$. So the angles are $3x-2=109^\circ$ and $2x-3=71^\circ$, i.e. $\angle A=71^\circ, \angle B=109^\circ, \angle C=71^\circ, \angle D=109^\circ$.
(xi) This is a parallelogram (with $AD=DC$ marked) and diagonal $AC$; $\angle D=68^\circ$. In $\triangle ADC$, $AD=DC$, so $\angle DAC=\angle DCA=\dfrac{180^\circ-68^\circ}{2}=56^\circ$; hence $y=\angle DCA=56^\circ$. Also $AB\parallel DC$ gives $x=\angle CAB=\angle DCA=56^\circ$ (alternate angles). Hence $x=56^\circ, y=56^\circ$.
(xii) This is a pentagon with exterior angles marked. The exterior angles add up to $360^\circ$, so $90^\circ+90^\circ+60^\circ+40^\circ+x=360^\circ \Rightarrow 280^\circ+x=360^\circ \Rightarrow x=80^\circ$.
2. Two adjacent angles of a parallelogram are in the ratio $4:5$. Find the angles.
Answer: $4k+5k=180^\circ \Rightarrow 9k=180^\circ \Rightarrow k=20^\circ$. The adjacent angles are $80^\circ$ and $100^\circ$; hence all four angles are $80^\circ, 100^\circ, 80^\circ, 100^\circ$.
3. Two adjacent angles of a parallelogram are such that one is $30^\circ$ more than the other. Find each angle.
Answer: Let the angles be $\theta$ and $\theta+30^\circ$. Then $\theta+(\theta+30^\circ)=180^\circ \Rightarrow 2\theta=150^\circ \Rightarrow \theta=75^\circ$. The angles are $75^\circ, 105^\circ, 75^\circ, 105^\circ$.
4. $AC$ and $BD$ are diagonals of parallelogram $ABCD$. $a=37^\circ$ and $\angle ABC=120^\circ$. Find $b$.
Answer: The diagonal $BD$ splits $\angle ADC$ into $a$ and $b$, so $\angle ADC=a+b$. Opposite angles of a parallelogram are equal, so $\angle ADC=\angle ABC=120^\circ$. Hence $37^\circ+b=120^\circ \Rightarrow b=83^\circ$.
5. In rectangle $ABCD$, diagonal $AC=6x-2$ and $BD=4x+2$. Find $x$ and the length of $AC$.
Answer: Diagonals of a rectangle are equal: $6x-2=4x+2 \Rightarrow 2x=4 \Rightarrow x=2$. $AC=6(2)-2=10$.
6. The diagonals of a rectangle meet at O. $AC=3x+1$ and $BD=8x-24$. Find $x$, $AO$ and $BO$.
Answer: Diagonals equal: $3x+1=8x-24 \Rightarrow 5x=25 \Rightarrow x=5$. $AC=BD=16$; $AO=BO=\dfrac{16}{2}=8$.
7. In square $ABCD$, $\angle A=4x+30^\circ$. Find $x$.
Answer: Each angle of a square is $90^\circ$; $4x+30^\circ=90^\circ \Rightarrow 4x=60^\circ \Rightarrow x=15$.
8. In rectangle $ABCD$, $AB=2x+5, BC=20, AD=3x+5$. Find the sum of the four sides.
Answer: Opposite sides equal: $AD=BC \Rightarrow 3x+5=20 \Rightarrow x=5$. $AB=2(5)+5=15$. Perimeter $=2(AB+BC)=2(15+20)=70$.
9. In parallelogram $ABCD$, $\angle B=2x+10^\circ$ and $\angle D=3x-13^\circ$. Find all angles.
Answer: Opposite angles equal: $2x+10^\circ=3x-13^\circ \Rightarrow x=23$. $\angle B=\angle D=2(23)+10^\circ=56^\circ$; $\angle A=\angle C=180^\circ-56^\circ=124^\circ$. Hence $\angle A=124^\circ, \angle B=56^\circ, \angle C=124^\circ, \angle D=56^\circ$.
10. If the sum of the four sides of a square is $36$ cm, find the length of each side.
Answer: Side $=\dfrac{36}{4}=9$ cm.
11. In rhombus $ABCD$, $AB=3x+4$ and $BC=2x+7$. Find $DC$ and $AD$.
Answer: All sides of a rhombus are equal: $3x+4=2x+7 \Rightarrow x=3$. $AB=3(3)+4=13$. Hence $DC=AD=13$.
12. The diagonals $AC$ and $BD$ of parallelogram $ABCD$ are bisected at the point E. If $AE=10x$, find $AC$ when $x=3$.
Answer: E is the midpoint of $AC$, so $AE=\dfrac{1}{2}AC$. $AE=10x=10\times 3=30$; hence $AC=2\times 30=60$.
13. Perpendiculars $AE$ and $CF$ are drawn on the diagonal $BD$ of parallelogram $ABCD$. Show that $\triangle AED \cong \triangle CFB$.
Answer: In $\triangle AED$ and $\triangle CFB$ — $\angle AED=\angle CFB=90^\circ$ (perpendiculars); $AD=CB$ (opposite sides of a parallelogram); and since $AD\parallel CB$ with $BD$ a transversal, $\angle ADE=\angle CBF$ (alternate angles). Hence by the AAS congruence criterion, $\triangle AED \cong \triangle CFB$. (Proved)
14. The sides $AB$ and $CD$ of quadrilateral $ABCD$ are extended to $P$ and $Q$ respectively. Prove that $\angle ADQ+\angle CBP=\angle A+\angle C$.
Answer: Since $ABP$ is a straight line, $\angle CBP=180^\circ-\angle B$; and since $CDQ$ is a straight line, $\angle ADQ=180^\circ-\angle D$. Adding, $\angle ADQ+\angle CBP=360^\circ-(\angle B+\angle D)$. In a quadrilateral $\angle A+\angle B+\angle C+\angle D=360^\circ$, so $\angle B+\angle D=360^\circ-(\angle A+\angle C)$. Therefore $\angle ADQ+\angle CBP=360^\circ-[360^\circ-(\angle A+\angle C)]=\angle A+\angle C$. (Proved)
15. The bisectors of $\angle A$ and $\angle B$ of parallelogram $ABCD$ meet at $O$. Find $\angle AOB$.
Answer: In a parallelogram $\angle A+\angle B=180^\circ$ (co-interior angles). In $\triangle AOB$, $\angle OAB=\dfrac{\angle A}{2}$ and $\angle OBA=\dfrac{\angle B}{2}$. So $\angle AOB=180^\circ-\left(\dfrac{\angle A}{2}+\dfrac{\angle B}{2}\right)=180^\circ-\dfrac{180^\circ}{2}=90^\circ$.
16. Find the angles of a quadrilateral if its four angles are in the ratio $3:4:5:6$.
Answer: $3k+4k+5k+6k=360^\circ \Rightarrow 18k=360^\circ \Rightarrow k=20^\circ$. The angles are $60^\circ, 80^\circ, 100^\circ, 120^\circ$.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. The sum of the exterior angles of any polygon is — (a) $180^\circ$ (b) $360^\circ$ (c) $540^\circ$ (d) $720^\circ$.
Answer: (b) $360^\circ$.
2. The sum of the interior angles of an $n$-sided polygon is — (a) $n\times 180^\circ$ (b) $(2n-4)\times 90^\circ$ (c) $360^\circ$ (d) $\dfrac{360^\circ}{n}$.
Answer: (b) $(2n-4)\times 90^\circ$.
3. Each interior angle of a regular hexagon is — (a) $108^\circ$ (b) $120^\circ$ (c) $135^\circ$ (d) $144^\circ$.
Answer: (b) $120^\circ$.
4. The quadrilateral whose diagonals are perpendicular bisectors of each other (but not equal) is — (a) rectangle (b) rhombus (c) trapezium (d) kite.
Answer: (b) rhombus.
5. The diagonals of a rectangle are — (a) equal (b) perpendicular (c) unequal (d) do not bisect each other.
Answer: (a) equal (and they bisect each other).
6. Two adjacent angles of a parallelogram are — (a) equal (b) supplementary (c) complementary (d) right angles.
Answer: (b) supplementary (sum $180^\circ$).
7. The number of diagonals of a pentagon is — (a) $2$ (b) $5$ (c) $9$ (d) $10$.
Answer: (b) $5$ ($\dfrac{5\times 2}{2}=5$).
8. If each exterior angle of a regular polygon is $45^\circ$, the number of sides is — (a) $6$ (b) $8$ (c) $9$ (d) $10$.
Answer: (b) $8$ ($\dfrac{360}{45}=8$).
9. A square is a — (a) parallelogram (b) rectangle (c) rhombus (d) all of the above.
Answer: (d) all of the above.
10. The sum of the interior angles of a triangle is — (a) $90^\circ$ (b) $180^\circ$ (c) $270^\circ$ (d) $360^\circ$.
Answer: (b) $180^\circ$.
Fill in the Blanks
1. The sum of the exterior angles of an $n$-sided polygon is ______ .
Answer: $360^\circ$.
2. A quadrilateral with one pair of opposite sides parallel is called a ______ .
Answer: trapezium.
3. Each angle of a square is ______ .
Answer: $90^\circ$.
4. Each interior angle of a regular decagon is ______ .
Answer: $144^\circ$ ($\dfrac{(2\times 10-4)\times 90^\circ}{10}=144^\circ$).
5. The diagonals of a parallelogram ______ each other.
Answer: bisect.
True or False
1. Every square is a rhombus.
Answer: True.
2. Every rhombus is a square.
Answer: False.
3. Both pairs of opposite sides of a trapezium are parallel.
Answer: False (only one pair is parallel).
4. The diagonals of a rectangle are equal in length.
Answer: True.
5. All diagonals of a convex polygon lie inside it.
Answer: True.
Short Answer Questions
1. What is the difference between a regular polygon and an irregular polygon?
Answer: A regular polygon has all sides of equal length and all angles of equal measure; an irregular polygon has unequal sides or unequal angles.
2. Write one difference between a rhombus and a square.
Answer: Both have all sides equal; but each angle of a square is $90^\circ$, whereas a rhombus need not have right angles. (Also, the diagonals of a square are equal, while those of a rhombus are generally unequal.)
3. Find the number of sides of a regular polygon whose each interior angle is $108^\circ$.
Answer: Exterior angle $=180^\circ-108^\circ=72^\circ$; number of sides $=\dfrac{360^\circ}{72^\circ}=5$ (pentagon).
4. Three angles of a quadrilateral are $70^\circ, 80^\circ, 100^\circ$. Find the fourth angle.
Answer: Fourth angle $=360^\circ-(70^\circ+80^\circ+100^\circ)=110^\circ$.
Key Terms
| Term | Meaning |
|---|---|
| Polygon | A simple, closed plane figure made of a finite number of line segments |
| Quadrilateral | A polygon with four sides |
| Interior angle | An angle inside the polygon at a vertex |
| Exterior angle | The angle formed outside when a side is extended |
| Diagonal | A line segment joining two non-consecutive vertices |
| Convex polygon | Every interior angle is less than $180^\circ$ |
| Concave polygon | At least one interior angle is greater than $180^\circ$ |
| Regular polygon | All sides and all angles are equal |
| Parallelogram | A quadrilateral with both pairs of opposite sides parallel |
| Trapezium | A quadrilateral with one pair of opposite sides parallel |
| Rectangle | A parallelogram with all angles right angles |
| Rhombus | A parallelogram with all sides equal |
| Square | A quadrilateral with all sides and all angles equal |
| Kite | A quadrilateral with two pairs of adjacent sides equal |
| Congruent | Having the same shape and size |