Plane Geometry - PT education · PDF fileIC : PTtkmml09 (1) of (20) Point, line and plane...

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Point, line and plane

There are three basic concepts in geometry. These concepts are the “point”, “line” and “plane”.

Point

A fine dot, made by a sharp pencil on a sheet of paper, resembles a geometrical point very closely. The sharper the pencil, the closer is the dot to the concept of a geometrical point.

Plane

The surface of a smooth wall or the surface of a sheet of paper or the surface of a smooth blackboard are close examples of a plane. The surface of a blackboard is, however, limited in extent and so are the surfaces of a wall and a sheet of paper but the geometrical plane extends endlessly in all directions.

Lines

A line in geometry is always assumed to be a straight line. It extends infinitely far in both directions. It is determined if two points are known. It can be expressed in terms of the two points, wh ich are wr i t ten in capi ta l let ters . The fo l lowing l ine i s ca l l ed AB.

B A

Or, a line may be given one name with a small letter. The following line is named line k.

k

A line segment is a part of a line between two endpoints. It is named by its endpoints, for example, A and B.

B A AB is a line segment. It has a definite length.

If point P is on the line and at the same distance from A as well as from B, then P is the midpoint of segment AB. When we say AP = PB, we mean that the two line segments have the same length.

B A P

A part of a line with one endpoint is called a ray. AC is a ray of which A is an endpoint. The ray extends infinitely far in the direction away from the endpoint.

A C

Angles

An angle is formed by two rays with the same initial point. So an angle looks like the one in the figure below.

A B

C

Definition: An angle is the union of two noncollinear rays with a common initial point.

MATHS LECTURE # 09 Plane Geometry

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The two rays forming an angle are called the “arms” of the angle and the common initial point is called the “vertex” of the angle.

Notation: The angle formed by the two rays AB and AC, is denoted by the symbol ∠BAC or ∠CAB.

It is at times convenient to refer the angle BAC, simply as ∠A. However this cannot be done if there are more than one angle, with the same vertex A.

Also, it is convenient to denote angles by symbols such as ∠1, ∠2, ∠3, etc.

A B

C

D E

3 2

1

Types of angles

Vertical angles

When two lines intersect, four angles are formed. The angles opposite to each other are called vertical angles and are equal to each other.

a b c

d

a and c are vertical angles, ∠ a = ∠ c.

b and d are vertical angles, ∠ b = ∠ d.

Straight Angle

A straight angle has its sides lying along a straight line. It is always equal to 180°.

A B C

∠ ABC = ∠ B = 180°.

i.e., ∠ B is a straight angle.

Adjacent angles

Two angles are adjacent if they share the same vertex and a common side but no angle i s inside the other angle. ∠ABC and ∠CBD are adjacent angles. Even though they share a common vertex B and a common side AB, ∠ ABD and ∠ ABC are not adjacent angles because one angle is inside the other.

B D

C

A

Supplementary angles

If the sum of two angles is a straight angle (180°), the angles are supplementary and each angle is the supplement of the other.

G b a

∠ G is a straight angle = 180° ∴ ∠ a + ∠ b = 180°. ⇒ ∠ a and ∠ b are supplementary angles.

E1. What is the supplement of an angle whose measure is 57°?

Sol. The supplement = 180° – 57° = 123°.

E2. The supplement of an angle is one third of the angle. Find the supplement of the angle.

Sol. Let the angle be x. So, the supplement is x/3. Now, x + x/3 = 180° or 4x/3 = 180° or x = 135°.

Hence, the supplement = 135/3 = 45°.

Right angles

If two supplementary angles are equal, they are both right angles. A right angle is onehalf of 180°. Its measure is 90°. A

right angle is symbolised by .

a b G

K ∠G is a straight angle.

∠b + ∠a = ∠G and ∠a = ∠b.

∴ ∠a and ∠b are right angles.

Complementary angles

If the sum of two angles is 90°, the angles are complementary and each angle is the complement of the other.

b a

Y

For example, if ∠ Y is a right angle and

∠ a + ∠ b = ∠ Y = 90°, then

∠ a and ∠ b are complementary angles.

E3. What is the complement of an angle whose measure is 47°?

Sol. The complement = 90° – 47° = 43°.


E4. The complement of an angle is two third of the angle. Find the complement of the angle.

Sol. Let the angle be x. So, the complement is 2x/3. Now, x + 2x/3 = 90° or 5x/3 = 90° or x = 54°.

Hence, the complement = 54 2 3

36 × = ° .

Acute angles

Acute angles are the angles whose measure is less than 90°. No two acute angles can be supplementary. Two acute angles can be complementary angles.

C or C

∠ C is an acute angle.

Obtuse angles

Obtuse angles are the angles whose measure is greater than 90° and less than 180°.

D D

or

∠ D is an obtuse angle.

Reflex angle

An angle with measure more than 180° and less than 360° is called a reflex angle.

a

180°< a < 360°

E5. Categorise the following angles as acute, right, obtuse, reflex and straight angle : 37°, 49°, 138°, 90°, 180°, 89°, 91°, 112°, 45°, 235°.

Sol. Acute angles : 37°, 49°, 45°, 89°. Right angle : 90°. Obtuse angles : 138°, 91°, 112°. Reflex angle : 235°.

Straight angle : 180°.

Some properties of angles

Linear pair of angles

From the figure ∠ AOC and ∠ BOC are adjacent angles. Look at the noncommon arms, OA and OB. These are two opposite rays or these are collinear. We have a name for them also.

A B

C

O

Definition: Two adjacent angles are said to form a linear pair of angles if their noncommon arms are two opposite rays. Now, look at the figure below

A

C

O

B

A B

C

O

(i) (ii)

In each of the figures above, ∠ AOC and ∠ COB form a pair of adjacent angles. In figure (ii) we have a linear pair of angles. If adjacent angles ∠ AOC and ∠ COB form a linear pair, then

∠ AOC + ∠ COB = 180°.

Imp.

Two adjacent angles are a linear pair of angles if and only if they are supplementary.

Parallel lines

Two lines meet or intersect if there is one point that is on both lines. Two different lines may either intersect at a single point or never intersect, but they can never meet in more than one point.

a

P b

Two lines in the same plane that never meet no matter how far they are extended are said to be parallel, for which the symbol is ||. In the following diagram a || b.

a

b


Imp.

Ø If two lines in the same plane are parallel to a third line, then they are parallel to each other. Since a || b and b || c, we know that a || c.

a

b

c

Ø Two lines that meet each other at right angle are said to be perpendicular, for which the symbol i s ⊥ . Line a is perpendicular to line b.

a

b 90° 90°

Ø Two lines in the same plane that are perpendicular to the same line are parallel to each other.

a

c 90° 90°

b

Ø Line ‘a’ ⊥ line ‘c’ and line ‘b’ ⊥ line ‘c’. ∴ a || b. A line intersecting two other lines is called a transversal. Line ‘c’ is a transversal intersecting lines ‘a’ and ‘b’.

c a

b

The four angles between the given lines are called interior angles and the four angles outside the given lines are called exterior angles. If two angles are on opposite sides of the transversal, they are called alternate angles.

y xw z

q p r s

∠z, ∠w, ∠q and ∠p are interior angles.

∠y, ∠x, ∠r and ∠s are exterior angles.

∠z and ∠p are alternate interior angles, so are ∠w and ∠q.

∠y and ∠s are alternate exterior angles, so are ∠x and ∠r.

Pairs of corresponding angles are ∠y and ∠q; ∠z and ∠r; ∠x and ∠p and ∠w and ∠s.

Two lines intersected by a transversal are parallel, if

1 . the corresponding angles are equal, or

2. the alternate interior angles are equal, or

3. the alternate exterior angles are equal, or

4. interior angles on the same side of the transversal are supplementary.

y xw z

q p r s

Line a

Line b

If a || b, then 1. ∠y = ∠q. ∠z = ∠r, ∠x = ∠p and ∠w = ∠s.

2. ∠z = ∠p and ∠w = ∠q. 3. ∠y = ∠s and ∠x = ∠r.

4 . ∠z + ∠q = 180° and ∠p + ∠w = 180°.

Since vertical angles are equal, ∴ ∠p = ∠r, ∠q = ∠s, ∠y = ∠w and ∠x = ∠z. If any one of the four conditions for equality of angles holds true, then the lines are parallel.

E6. In the given figure, two parallel lines are intersected by a transversal. Find the measure of angle y.

3x+50°

2x

y

Sol. The two labelled angles are supplementary. ∴ 2x + (3x + 50°) = 180°. 5x = 130° ⇒ x = 26°.

Since ∠y is vert ical to the angle whose measurement is 3x + 50°, it has the same measurement.

∴ y = 3x + 50° = 3(26°) + 50° = 128°.


Mini Revision Test # 01

DIRECTIONS: Fill in the blanks.

1. A part of a line with one end point is called a..... 2 . Two adjacent angles are a linear pair of angles if

and only if they are........ 3 . A line intersecting two other lines is called a..... 4 . Two angles are said to be supplementary if their

sum is ..... 5 . Two angles are said to be complementary if their

sum is ..... 6 . The angle formed by the two rays AB and AC, is

denoted by the symbol ....... or ....... 7. When two lines intersect, four angles are formed.

The angles opposite each other are called .......

8 . Two angles are ....... if they share the same vertex and a common side but no angle is inside another angle.

9. If a ray stands on a line, the sum of the two adjacent angles so formed is .......

10. Two lines in the same plane that are perpendicular to the same line are ...... to each other.

Circle

The following figure shows the parts of a circle.

Diameter

Radius

Centre

Circle

Centre of a circle is the point within that circle from which all straight lines drawn to the circumference are equal in length to one another.

Circumference of a circle is the outer periphery of the circle. It is also known as the perimeter of the circle.

Radius of a circle is the distance from the centre of the circle to the circumference.

Diameter of a circle is a line that runs from the edge of the circle, through the centre, to the edge on the other side. The diameter is twice the length of the radius.

Chord of a circle is a straight line that divides the circle into two parts.

In case of the diameter, it is the largest possible chord that can be drawn in a circle.

Imp.

The problem of finding lengths and areas related to figures with curved boundaries is of practical importance in our daily life.

I f A and C denote the area and per imeter (a lso cal led circumference) respectively, of a circle of radius r, then

A = πr 2 , C = 2πr

Here π (a letter of Greek alphabet) is the ratio of circumference to the diameter of a circle. It stands for a particular irrational number whose value is given, approximately to two decimal

places, by 3.14 or by the fraction 22 7 .

E7. Find the circumference of a circle with area 25π sq ft.

Sol. Area of the circle = πr 2 = 25π.

∴ r = 5 ft.

∴ circumference = 2πr = 10π ft.

E8. Find the area of a circle with circumference 30π cm.

Sol. Circumference = 2πr = 30π

r = 15 cm.

∴ area = πr 2 = π × 15 2 = 225π sq. cm.

Sector and segment of a circle: See the figures given below.

B

O

C

A

θ

Q

O

S

P

R

(i) (ii)

In figure (i) note that the area enclosed by a circle is divided into two regions, viz., OBA (shaded) and OBCA (nonshaded). These regions are called sectors.

A minor sector has an angle θ, subtended at the centre of the circle. The boundary of a sector consists of arc of the circle and the two lines OA and OB.

In fig (ii), the area enclosed by a circle is divided into two segments. Here segment PRQ (shaded) is the minor one and segment PQS (nonshaded) is the major one. The boundary of a segment consists of an arc of the circle and the chord dividing the circle into segments.


Area of a sector and length of arc

The arc length l and the area A of a sector of angle θ in a circle of radius r, are given by

l r

A r

= = π θ π θ 180 360

2

, . We also note that A lr = 1 2

.

Theorems on Circles

Ø If ON is ⊥ from the centre O of a circle to a chord AB, then AN = NB.

N B A

O

(⊥ from centre bisects chord)

Ø If N is the midpoint of a chord AB of a circle with centre O, then ∠ONA = 90°. (Converse, ⊥ from centre bisects chord)

Ø Equal chords in a circle are equidistant from the centre. If the chords AB and CD of a circle are equal and if OX ⊥ AB, OY ⊥ CD, then OX = OY.

A C

O X Y

B D

(Eq. chords are equidistant from centre)

Ø If OX ⊥ chords AB, OY ⊥ chord CD and OX = OY, then chords AB = CD (Refer the figure given above). (Chords equidistant from centre are eq.)

Ø A circle with centre O, with ∠AOB at the centre, ∠ACB at the circumference, standing on the same arc AB, then ∠AOB = 2∠ACB.

O

D

B

A

C

(∠ at centre = 2∠ at Circumference on the same arc AB)

Ø In a circle with centre O, ∠ACB and ∠ADB are the angles at the circumference, by the same arc AB, then ∠ACB = ∠ADB.

(∠s in same arc or ∠s in same seg.) Ø In a circle with centre O and diameter AB, if C is any point at

the circumference, then ∠ACB = 90°.

A B

C

O

(∠ in semicircle)

Ø If ∠APB = ∠AQB and if P, Q are on the same side of AB, then A, B, Q, P are concyclic.

P

A

B

Q

O

(AB subtends equal ∠s at P and Q on the same side)

Ø In equal circles (or in the same circle) if two arcs subtend equal angles at the centre (or at the circumference), then the arcs are equal. P

Q

Y X A

B

O

If ∠BOA = ∠XOY, then AB = XY or if f ∠BPA = ∠XQY, then

AB = XY .

(Eq. ∠s at centre or at circumference stand on equal arcs)

In the same or equal circles if chords AB = CD, then arc AB = arc CD.

||

||

A B

C D

O

(Eq. chords cut off equal arcs)


Few important results

Ø If two chords AB & CD intersect externally at P then

P

B

D

A

C

+ PA × PB = PC × PD

+ ∠P = 1 2 [m (arc AC) – m (arc BD)]

Ø If PAB is a secant and PT is a tangent then,

P

B

Y

A

T

X

+ PA × PB = PT 2

+ ∠P = 1/2[m(arc BXT) – m(arc AYT)] E9. Complete the following table.

Radius r r 2

Area A

Circumference C

1 1

2 4

3 9

4 16

Sol.

Radius r r 2 Area = πr 2 C = 2πr

1 1 π 2π

2 4 4π 4π

3 9 9π 6π

4 16 16π 8π

E10.The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand in 10 minutes. (Take π = 22/7).

Sol. The minute hand describes a circle of radius 7 cm.

It describes an angle of 360 60

6 °

= ° per minute.

Thus, we get a sector of angle 6° and radius 7 cm per minute.

Its area π θ r 2

360 22 7

7 7 6 360

7730 °

= × × ×

= = 2.567 sq. cm.

∴ Area of the sector described in one minute = 2.567 sq. cm.

⇒ Area described in 10 minutes = 2.567 × 10 sq. cm = 25.67 sq. cm.

E11.A sector is cut from a circle of radius 21 cm. The angle of the sector is 150°. Find the length of its arc and area.

Sol. Length of the arc of the sector,

l r = × 150 180

π cm.

= × × 150 180

22 7

21 cm = 55 cm.

Area of the sector = × 150 360

2 πr sq. cm.

= × × × 150 360

22 7

21 21 2 cm = 577.5 sq. cm.


DIRECTIONS: Determine whether the given statements are TRUE or FALSE.

1. The sum of the interior angles of a pentagon is 540°.

2. The perimeter of a regular hexagon with side a is 6a.

3. The circumference of a circle whose radius is 14 cm is 176 cm.

4. The perimeter of a regular hexagon with side 8 cm is 48 cm.

5. The area of a quadrant of a circle with radius 14 cm is 148 sq. cm.


6. All circles measure .... degrees.

7. The area of a regular hexagon with side 8 cm is .....

8 . If the radius of a circle is decreased by 50%, its area will decrease by ....

9 . If the diameter of a circle is increased by 100%, its area is increased by ...

10. The length of minute hand on a wall clock is 14 cm. The area swept by it in 30 minutes is ...


Polygon

Definition: A polygon is a closed figure bounded by straight lines. Each of the linesegments forming the polygon is called its side. The angle determined by two sides meeting at a vertex is called an angle of the polygon.

A

B C C D

B

A

E

A F

E

D C

B

Remark: For n = 3, the polygon has the special name triangle and similarly for n = 4, the polygon has the special name quadrilateral.

Some other special names are

Pentagon (n = 5).

Hexagon (n = 6).

Definition: A polygon P 1 P 2 .... P n is called convex if for each side of the polygon, the line containing that side has all the other vertices on the same side of it.

All the three figures given (triangle, pentagon and hexagon) are convex polygons.

Regular polygon: A polygon is called a regular polygon if all its sides and angles are equal.

In the given figure, the hexagon is a regular polygon.

Congruent and similar polygons

If two polygons have equal corresponding angles and equal corresponding sides, they are said to be congruent. Congruent polygons have the same size and shape. The symbol for congruence is ≅.

C

D

B

A

≅

G

H

F

E

When two sides of the polygons are equal, we indicate the fact by drawing the same number of short lines through the equal sides.

B

C D

A F

G H

E

This indicates that AB = EF and CD = GH.

Two polygons with equal corresponding angles and corresponding sides in proportion are said to be similar. The symbol for similar is ~.

B

A D

C

~ F

E H

G

Similar figures have the same shape but not necessarily the same size.

Triangles and their types

A triangle is a polygon with three sides. Triangles are classified by measuring their sides and angles.

The sum of the angles of a plane triangle is always 180°. The symbol for a triangle is ∆. The sum of any two sides of a triangle is always greater than the third side.

Equilateral

Equilateral triangles have equal sides and equal angles. Each angle measures 60°.

In the ∆ ABC,

C B

A

AB = AC = BC.

∴ ∠A = ∠B = ∠C = 60°.

Isosceles

Isosceles triangles have two sides equal. The angles opposite to the equal sides are equal. The two equal angles are sometimes called the base angles and the third angle is called the vertex angle. Note that an equilateral triangle is isosceles.

In the ∆FGH, FG = FH.

H G

F

FG ≠ GH.

∠G = ∠H. ∠F is vertex angle. ∠G and ∠H are base angles.


Scalene

Scalene triangles have all three sides of different length and all angles of different measure. In a scalene triangle, the shortest side is opposite the angle of smallest measure and the longest side is opposite the angle of greatest measure.

In the ∆ABC,

AB > BC > CA.

∴ ∠C > ∠A > ∠B.

C B

A

Right angled triangle

Right angled triangle contains one right angle. Since the right angle is 90°, the other two angles are complementary. They may or may not be equal to each other. The side of a right triangle opposite the right angle is called the hypotenuse. The other two sides are called legs. The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

In the ∆ABC,

AC is the hypotenuse.

C B a

A

c b AB and BC are legs.

∠B = 90°.

∴ ∠A + ∠C = 90°.

⇒ a 2 + c 2 = b 2 .

E12.ABC is a right triangle with ∠B = 90°. If AB = 12 and BC = 5, then what is the length of AC?

Sol. AB 2 + BC 2 = AC 2 . ⇒ AC 2 = 12 2 + 5 2 = 144 + 25 = 169. ∴ AC = 13.

E13.Show that a triangle with sides 15, 8 and 17 is a right triangle.

Sol. The triangle will be a right triangle if a 2 + b 2 = c 2 .(as shown above)

∴ 15 2 + 8 2 = 17 2 . ⇒ 225 + 64 = 289.

∴ the triangle is a right triangle and 17 is the length of the hypotenuse.

Properties of triangles

Ø Sum of the three interior angles of a triangle is 180°.

Ø When one side is extended in any direction, an angle is formed with another side. This is called the exterior angle.

Ø Interior angle + corresponding exterior angle = 180°.

Ø An exterior angle = Sum of the other two interior angles not adjacent to it.

Ø Sum of any two sides is always greater than the third side.

Ø A triangle must have at least two acute angles.

Properties of similar triangles

If two triangles are similar, then ratio of sides = ratio of heights = ratio of perimeters = ratio of medians = ratio of angle bisectors = ratio of inradii = ratio of circumradii.

In the above result, rat io of s ides refers to the rat io of corresponding sides etc.

Also, ratio of areas = ratio of squares of corresponding sides.

Right triangle

90°

90° B

A

D C

ABC is a right triangle with ∠A = 90°.

If AD is perpendicular to BC, then

(a) Triangle ABD ~ Triangle CBA and BA 2 = BC × BD.

(b) Triangle ACD ~ Triangle BCA and CA 2 = CB × CD.

(c) Triangle ABD ~ Triangle CAD and DA 2 = DB × DC.

Two triangles are similar

Two triangles are similar if the angles of one of the triangles are respect ively equal to the angles of the other and the corresponding sides are proportional.


Theorems

(Equiangular triangles) A.A.A.

If two triangles are equiangular, their corresponding sides are proportional. In triangles ABC and XYZ,

if ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z, then AB/XY = AC/XZ = BC/YZ.

A

B C

X

Y Z

(3 sides proportional)

If 2 triangles have their corresponding sides proportional, then they are equiangular. In triangles ABC and XYZ,

if AB/XY = AC/XZ = BC/YZ, then ∠A =∠X, ∠B =∠Y, ∠C =∠Z.

(Ratio of two sides and included angle)

If one angle of the triangle is equal to one of the angles of the other tr iangle and the sides containing these angles are proportional, then the triangles are similar.

If ∠A = ∠X and AB/XY = AC/XZ,

then ∠B = ∠Y, ∠C = ∠Z.

If tr iangles ABC and XYZ are similar, then it is denoted by ∆ABC ~ ∆XYZ.

Congruency of triangles

Two triangles are said to be ‘CONGRUENT’ if they are equal in all respects (sides and angles).

A

B C

X

Y Z

The three sides of one of the triangle must be equal to the three sides of the other, respectively.

The three angles of the first triangle must be equal to the three angles of the other respectively. Thus, if ∆ ABC and ∆ XYZ are congruent, (represented as ∆ ABC ≅ ∆ XYZ), then

AB = XY, AC = XZ, BC = YZ and

∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z.

Theorems

(Side . angle . side) S.A.S. If AB = XY, ∠A = ∠X, AC = XZ, then triangle ABC ≅ triangle XYZ.

(Angle . angle . side) A.A.S. If ∠B = ∠Y, ∠C = ∠Z, AC = XZ, then triangle ABC ≅ triangle XYZ.

(Angle . side . angle) A.S.A. If ∠B = ∠Y, BC = YZ, ∠C = ∠Z, then triangle ABC ≅ triangle XYZ.

(Side . side . side) S.S.S. If AB = XY, AC = XZ, BC = YZ, triangle ABC ≅ triangle XYZ.

(Right angle hypotenuse side) (R.H.S)

A

B C

X

Y Z

If ∠B = ∠Y = 90°, Hypotenuses AC = XZ, AB = XY, then

triangle ABC ≅ triangle XYZ.

Note: Congruent triangles are definitely similar but similar triangles may not be congruent.


DIRECTIONS: Answer the questions.

1. What is the measure of each acute angle of an isosceles right angled triangle?

2. One of the acute angles of a right triangle is 42°. Find the measure of the other acute angle.

3. Two angles of a tr iangle measure 92° and 58° respectively. Find the measure of the third angle.

4. One of the acute angles of a right triangle is double the other. Find the measure of its smallest angle.

5. Each of the equal angles of an isosceles triangle is double the third angle. What is the measure of the third angle?

6. The measures of the angles of a triangle are in the ratio 2 : 3 : 4. Find the measure of its greatest angle.

7. One of the angles of a triangle is equal to the sum of the other two. Find the measure of the greatest angle.

8. What is the measure of each angle of an equilateral triangle?

9. The measures of the acute angles of a right triangle are in the ratio 1 : 5. Find the measure of its angles.

10. Match the following

i. Right triangle i. 120°, 28°, 32°. ii. Isosceles right triangle ii. 60°, 60°, 60°.

iii. Obtuse scalene triangle iii. 90°, 42°, 48°. iv. Equilateral triangle iv. 108°, 36°, 36°. v. Acute angled isosceles v. 45°, 45°, 90°.

triangle vi. Obtuse isosceles triangle vi. 65°, 50°, 65°.


Quadrilaterals

A quadrilateral is a polygon of four sides. The sum of the interior angles of a quadrilateral is 360°.

Parallelogram

P

A

B C

D

AD || BC ∠A + ∠B = 180°

AD = BC ∆ ABD ≅ ∆ CDB AB || DC ∆ ABC ≅ ∆ CDA AB = DC AP = PC

∠D = ∠B BP = PD

∠A = ∠C

Rhombus

If in a parallelogram, all the sides are equal, then the parallelogram so formed is a rhombus.

A a

D

a

B C

P

Some important properties

Ø ∠DAB = ∠DCB & ∠ADC = ∠ABC.

Ø AP = PC, DP = PB & ∠APD = 90°.

Ø ∠ADP = ∠PDC, ∠ABD = ∠PBC.

Ø ∠ADC + ∠DCB = 180° & ∠DAB + ∠ABC = 180°.

Rectangle

A

B

D

C

b P

l


Ø AD = BC, DC = AB, ∠A = ∠B = ∠C = ∠D = 90°.

Ø DP = PB = AP = PC.

Square

A D

B C

a

a


Ø ∠A = ∠B = ∠C = ∠D = 90°.

Ø AB = BC = CD = DA.

Trapezium (Trapezoid)

C

D A

B b 2

b 1

h

AD || BC.

AD and BC are bases. AB and DC are legs. h = altitude.

Imp.

If a quadrilateral is not a parallelogram or a trapezoid but it is irregularly shaped, its area can be found by dividing it into triangles, attempting to find the area of each and adding the results.

The chart below shows at a glance the properties of each type of quadrilateral.

Quadrilateral

Rectangle Opposite sides are equal. Each angle is 90°.

Diagonals bisect each other.

Diagonals are of equal length.

Square All sides are

equal. Each angle is 90°. Diagonals bisect each other at right angles. Diagonals are

equal.

Rhombus All sides are

equal. Opposite angles are equal. Diagonals bisect each other at right angles.

Trapezium (One pair of opposite sides are parallel)

Parallelogram (Opposite sides and opposite angles are equal. Diagonals

bisect each other)




1. A quadrilateral has ..... vertices, no three of which are .....

2 . A quadri lateral is . . . . ., if for any side of the quadrilateral, the line containing it has the remaining .. . . .

3 . The diagonals of a rectangle are ..... 4 . The diagonals of a parallelogram ..... each other.

5. The line segment joining any two points lying on two ..... sides of a quadrilateral, except the two ..... lies in the interior of the quadrilateral.

6 . A ..... has one pair of opposite sides parallel to each other.

7. In a parallelogram, a pair of opposite sides are ..... and .....

8 . In a rhombus, the diagonals ..... each other ..... 9 . Each diagonal of a rhombus ..... the angles through

which it passes. 10. A diagonal of a parallelogram divides it into two .....

triangles.

DIRECTIONS: For each statement given below, indicate whether it is true (T) or false (F).

11. A diagonal of a quadrilateral is a line segment joining the two opposite vertices.

12. A square is a rhombus. 13. A square is a parallelogram.

14. A rectangle is a square. 15. The diagonals of a rectangle are perpendicular.

+ Important results and formulae

Ø Only one circle can pass through three given points.

Ø There is one and only one tangent to the circle passing through any point on the circle.

Ø From any exterior point of the circle, two tangents can be drawn on to the circle.

Ø The lengths of two tangents segment from the exterior point to the circle, are equal.

Ø The tangent at any point of a circle and the radius through the point are perpendicular to each other.

Ø When two circles touch each other, their centres & the point of contact are collinear.

Ø If two circles touch externally, distance between centres = sum of radii.

Ø If two circles touch internally, distance between centres = difference of radii

Ø Circles with same centre and different radii are concentric circles.

Ø Points lying on the same circle are called concyclic points.

Ø Measure of an arc means measure of central angle.

m(minor arc) + m(major arc) = 360°.

Ø Angle in a semicircle is a right angle.

Ø The sum of all the angles of a triangle is 180°.

Ø The sum of any two sides of a triangle is greater than the third side.

Ø Exterior angle of a triangle is equal to the sum of opposite interior angles.

Ø Conditions of congruence of triangles are SAS, SSS, ASA and RHS.

Ø Angles opposite to equal sides in a triangle are equal and vice versa.

Ø If two sides of a triangle are unequal, the greater side has greater angle opposite to it and vice versa.

Ø If the square of the longest side of a triangle is equal to the sum of squares of the other two sides, the triangle is right angled. Angle opposite to the longest side is a right angle.

Ø If the square of the longest side of a triangle is greater than the sum of squares of the other two sides, the triangle is obtuseangled. Angle opposite to the longest side is obtuse.

Ø If the square of the longest side of a triangle is less than the sum of squares of the other two sides, the triangle is acuteangled.

Ø Perimeter of a tr iangle is greater than the sum of its medians.

Ø Triangles on the same or congruent bases and of the same altitude are equal in area.

Ø Triangles equal in area and on the same or congruent bases are of the same altitude.

Ø Parallelograms have opposite sides and opposite angles equal.

Ø Diagonal of a parallelogram divides it into two congruent triangles.

Ø Diagonals of parallelogram bisect each other. If a pair of opposite sides of a quadrilateral are parallel and equal, the quadrilateral is a parallelogram.

Ø The line–segment joining midpoints of two sides of a triangle is parallel to the third side and half of it. Conversely, the line drawn through the midpoint of one side of a triangle parallel to the other, passes through the midpoint of the third.

Ø Conditions of similarity of two triangles are SAS, SSS, ASA and AA.

Ø Areas of similar triangles are in the ratio of squares of corresponding sides, altitudes or medians.

Ø The diagonals of a rhombus bisect each other at right angles. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle.

Ø The diagonals of a rectangle are equal and bisect each other.

Ø The diagonals of a square are equal and bisect each other at right angles.


DIRECTIONS: For each of the following questions, please give the complete solution.

1. Given that AB || CD, EF || CD and CE || FG, then ∠EFG will be

A B

E G

F

D C

65°

35°

2. In the pentagon given below, sum of the angles marked is

A B

C

D

E O

3. Calculate the value of x.

8x° 3x°

x°


37°

68°

x

x x


133° x 79°

6. In the given figure, find x and y.

40° 40°

y x

7. In the given figure, AB is parallel to EF, ∠ABC = 65°, ∠CDE = 50°, ∠DEF = 30°. Find ∠BCD.

65° 50°

30°

A

B D

F

E

C

8. In the given figure, APD, BPE and CPF are straight lines, find the measure of (a) the angle x, if x = c + d.

(b) the angle e, if b = c = d = e = f = x. (c) the values of a + b + c + d + e + f.

P

SOLVED EXAMPLES – LECTURE # 09 The following questions will help you get a thorough hold on the basic fundamentals of the chapter and will also help you develop your concepts. The following questions shall be covered in the class by the teacher. I t is also advised that a sincere student should solve each of the given questions at leas t twice to develop the required expert ise.


9. In the figure, AD, BE and CF are the altitudes of ∆ ABC intersecting at G. Find the sum of ∠BAC and ∠BGC.

B

b r

D C

E

A

G F

s

10. Find the number of sides of a polygon, if the sum of its angles is three times that of an octagon.

11. Each interior angle of a regular polygon exceeds the exterior angle by 150°. Find the number of sides of the polygon.

12. In the figure, AB, BC, CD are three sides of a regular polygon and ∠BAC is 20°. Calculate (a) the exterior angle of the polygon,

(b) the number of sides of the polygon, (c) ∠ADC

B

E

C

20° b

c

A D

x

13. In the figure ABCD and ABEF are parallelograms. Prove that ∆ADF and ∆BCE are congruent.

A

F E

B

C D

14. In the figure, if AC : CB = 1 : 3, AP = 4.5 and BQ = 7.5, PA || RC || QB, then find CR.

4.5

P R

A C

B

7.5

Q

15. In the following figure, equal sides BA and CA of a triangle ABC are produced to Q and P respectively, so that AP = AQ. Prove that PB = QC.

Q

C B

P

A

16. ABC is an isosceles triangle such that AB = AC. Side BA is produced to a point D such that BA = AD. What is the measure of ∠BCD?

B

A

D

C b

a a

b

17. In the figure, ABC is an equilateral triangle of side 3a. P, Q and R are points on BC, CA and AB respectively such that AR = BP = CQ = a.

A

R

a Q a C

P B a

Prove that PQR is an equilateral triangle.

18. In a trapezium ABCD, AB || DC, AB = 2CD. If the diagonals intersect at O, show that area of ∆AOB = 4 × area of ∆COD.

19. In the given figure, PQ || BC and ∠BPC = 90°. If PQ = 15 cm, BC = 25 cm and AP = 10.5 cm, then calculate the lengths of (a) AB and

(b) PC.

A

15 10.5

P

B 25

C

Q

20. P is a point on the base BC of the equilateral ∆ABC such

that BP = 1 3 BC. Prove that 9AP 2 = 7AB 2 .

A

D C P

B


1 . ∠EFG = ∠FEC, ∠FEC + ∠ECD = 180°, ∠ECD = 30°, hence ∠EFG = 150°.

2 . Required sum = 5 × 360° – Sum of angles of a pentagon = 1800° – 540° = 1260°.

3 . We have 8x° + 3x° + x° = 180°. ∴ 12x° = 180° ⇒ x = 15°.

4 . We have 37° + x° + x° + 68° + x° = 360°. ⇒ 105° + 3x° = 360°. ⇒ 3x° = 255° ⇒ x = 85°.

5 . We have 133° + 90° + 79° + x° = 360°. ⇒ 302° + x° = 360° ⇒ x = 58°.

6 . x = 40° (vertical ly opposite ∠s ) y = 180° – 40° – x (adjacent ∠s on straight l ine) = 180° – 40° – 40° = 100°.

7 . Draw CG | | AB. ∴ CG | | EF Draw DH | | EF, ∴ CG | | DH.

B

E

30°

F

50°

D

C

65°

A

y x

r s

H

G

s = 30° (al ternate ∠s, DH | | EF). ∴ r = 50° – 30° = 20°. x = r = 20°. (al ternate ∠s, DH | | CG) y = 65° (al ternate ∠s, CG | | AB). ∴ ∠BCD = x + y = 20° + 65° = 85°.

8 . ( a ) x = ∠CPD (vertically opposite ∠s). c + d + ∠CPD = 180° (∠sum of ∆). ∴ c + d + x = 180°. ∴ x + x = 180° (given x = c + d). ∴ 2x = 180° ⇒ x = 90°.

Solutions Solved Examples – Lecture # 09

Answers to Mini Revision Test # 01

1. ray 2. supplementary. 3. transversal 4. 180° 5. 90°

6. ∠BAC or ∠CAB 7. vertical angles 8. adjacent 9. 180° 10. parallel


1. 45° 2. 48° 3. 30° 4. 30° 5. 36°

6. 80° 7. 90° 8. 60° 9. 15°, 75°, 90° 10. iiii; iiv; ii ii, ivii; vvi; viiv


1. four, collinear 2. convex, vertices 3. equal 4. bisect

5. dif ferent, endpoints 6. trapezium 7. equal, parallel

8. intersect, at 90° 9. bisects 10. congruent 11.True

12. True 13. True 14. False 15. False


1. True 2. True 3. False 4. True 5. False

6. 360 7. 96 3 sq.cm 8. 75% 9. 300% 10. 308 sq.cm

Answers Mini Revision Tests

Success is where preparation and

opportunity meet...


( b ) x = ∠CPD (vertically opposite ∠s). x + c + d = 180° (∠sum of ∆). ⇒ 3x = 180° (Q c = d = x). ⇒ x = 60°. ∴ e = f = 60° (given e = f = x).

(c ) a + b + c + d + e + f

= 3 × 180° – 360 2

o F H G

I K J = 540° – 180° = 360°.

9 . Consider ∆BGC ∠BGC + r + s = 180°. ∴ ∠BGC = 180° – r – s ....(1) Consider ∆BAE. ∠BAE + b + 90° = 180°. ⇒ ∠BAE = 90° – b ....(2) Adding equation (1) and (2), ∠BGC + ∠BAC = 270° – (b + r + s). = 270° – 90° = 180°.

10. Let n be the number of sides of the polygon. (n – 2) 180° = [(8 – 2)× 180°] × 3. (n – 2) 180° =6 × 180° × 3. n – 2 = 18 ⇒ n = 20.

11. Let n be the number of sides of the polygon.

Each exterior angle = 360 o

n .

Each interior angle = − × n

n 2 180 c h o

.

As per the given condition

360 150 2 180 °

+ = − × °

n n

n c h

.

360 + 150n = 180n – 360. 30n = 720 ⇒ n = 24.

12. ( a ) c = 20° (base ∠s, isosceles ∆). b = 180° – 20° – 20° (∠sum of ∆) = 140°. ∴ the exterior angle = 180° – 140° (adjacent angles on l ine). i .e. x = 40°.

( b ) Let n be the number of sides.

n n

− × ° = °

2 180 140

c h or 180n – 360 = 140n.

40n = 360 ⇒ n = 9. (c ) Produce AB and DC to meet at E.

∠EBC = ∠ECB = 180° – 140° = 40°. ∴ EB = EC (sides opposite equal angles). ∠BEC = 180° – 40° – 40° = 100°. EB + BA = EC + CD ∴ EA = ED. ∴ ∠EDA = ∠EAD (base angles, isosceles triangle).

∴ ∠EAD = − 180 100 2

o o = 40° i .e. ∠ADC = 40°.

13. AB | | DC || EF (opposite sides of a | |gram). AB = DC = EF (opposite sides of a | |gram). ∴ CDFE is a | |gram (opposite sides are | |). ∴ DF = CE (opposite sides of a | |gram). In ∆ADF and BCE, ( a ) AD = BC (opposite sides of a | |gram). ( b ) AF = BE (opposite sides of a | |gram). (c ) DF = CE (proved). ∴ ∆ADF ≅ ∆BCE (S.S.S.).

14. Join AQ cutting CR at K.

Consider ∆s ACK and ABQ,

CK CK 7.5

1 4

7.5 4

= ⇒ = .

Consider ∆ KRQ and APQ.

⇒ KR KR 4.5

3 4

13.54

= ⇒ = P R

4.5 K

A 1 C

3 B

7.5

Q

∴ CR = CK + KR = 7.5 4

13.54

21 4

5 1 4

+ = = .

15. In ∆PAB and ∆QAC since, PA = QA, BA = CA and ∠QAC = ∠PAB (vertically opposite angles) ∴ ∆s PAB and QAC are congruent. Hence PB = QC (corresponding sides of congruent triangles).

16.

B

A

D

C b

a a

b

If AB = AC, then ∠ABC = ∠ACB = a°(say). Also if AC = AD, then ∠ACD = ∠ADC = b°(say). Now in ∆BCD, a + a + b + b = 180° ⇒ a + b = ∠BCD = 180/2 = 90°.

17. In ∆s ARQ, BPR and CQP, AR = BP = CQ = a (given). Also ∠RAQ = ∠PBR = ∠QCP = 60° (given) AQ = BR = CP = 3a – a = 2a. ∴ ∆ARQ ≅ ∆BPR ≅ ∆CQP (S.A.S.). i .e. ∆PQR is equi lateral.

18. A

D C

B

x

O

2x

∆s AOB and ∆COD are similar. The ratio of the proportional sides is 1 : 2. Hence the ratio of the areas of the triangles COD and AOB is 1 : 4.

19. ( a ) In triangles APQ and ∆ABC ( i ) ∠A = ∠A (common) ( i i ) ∠APQ = ∠ABC (corresponding ∠s, PQ | | BC) (i i i ) ∠AQP = ∠ACB (corresponding ∠s, PQ | | BC)

∴ ∆APQ ~ ∆ABC (A.A.A.)

∴ APAB

PQ BC

= ⇒ 10.5 15

25 AB = ⇒ AB = 17.5 cm.

( b ) BP = AB – AP = 17.5 – 10.5 = 7 cm. In ∆BPC, PC 2 = BC 2 – BP 2 (Pythagoras’ Theorem) ∴ PC 2 = 25 2 – 7 2 = 576 ⇒ PC = 24 cm.

20. Draw AD ⊥ BC.

AB = BC = CA (given). BP = 1 3 BC (given).

BD = DC = 1 2

BC (⊥ bisector theorem).

A

D C

P B

Also, PD = BD – BP.

= − = = 1 2

1 3

1 6

1 6

BC BC BC AB .

Also, AP 2 = AD 2 + PD 2 (Pythagoras’ Theorem).

= − + = AB AB AB 2 2 2

4 36 7 9 AB 2

∴ 9AP 2 = 7AB 2 .

Plane Geometry - PT education · PDF fileIC : PTtkmml09 (1) of (20) Point, line and plane...

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