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Ganit Learning Guides

Advanced GeometryLines, Angles, Shapes, Circles, Polygons, Solids

Author: Raghu M.D.

Advanced-Geometry 1 of 41 2014, www.learningforknowledge.com/glg

Contents

GEOMETRY ....................................................................................................... 2

Lines and Angles ............................................................................................................................... 2

Shapes .............................................................................................................................................. 10

Loci ................................................................................................................................................... 16

Cyclic Quadrilaterals ...................................................................................................................... 23

Area of Polygons ............................................................................................................................. 28

Quadrilateral ................................................................................................................................... 29

Curved figures ................................................................................................................................. 30

Solids ................................................................................................................................................ 35

Mensuration Table .......................................................................................................................... 36

Advanced-Geometry 2 of 41 2014, www.learningforknowledge.com/glg

A

B

GEOMETRY

Lines and Angles

A line segment is the shortest distance of a path between two points. A line segment is a part

of line. Usually a line segment is denoted by the points, for example, A and B.

All the points on the line are collinear.

Pairs of lines can be parallel or intersecting.

Parallel Intersecting

Angles

Angles are a measure of separation of two lines. Consider AB and BC intersecting and

terminating at point B.

Rules of Angles

Angles formed by two or more intersecting lines and polygons follow a set of rules. Polygons

are plane figures formed by a set of intersecting lines.

1. Vertically opposite angles are equal.

2. Angles around a point add up to 360.

A

B C

Advanced-Geometry 3 of 41 2014, www.learningforknowledge.com/glg

3. Angles at a point on a straight line add up to 180.

4. Angles in a triangle add up to 180.

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

6. Base angles in an isosceles triangle are equal.

7. Angles in an equilateral triangle are each equal to 60.

8. Angles in a quadrilateral add up to 360.

9. Interior angles in a rectangle or square are each equal to 90.

10. In a parallelogram opposite angles are equal.

Advanced-Geometry 4 of 41 2014, www.learningforknowledge.com/glg

11. In any polygon exterior angles add up to 360.

12. When a pair of parallel lines are intercepted by a transversal, pairs of alternate and corresponding angles are equal.

Angle Calculations

In a given situation, knowing the value of some of the angles can be used to find the value of

other angles.

Example 1: From the figure given below, write an equation and solve for x.

Working: (3x+30) and (2x) are values of angles on a straight line.

(3x+30) + (2x) = 180

5x + 30 = 180

=

=

= 30

Answer: x = 30

A bearing is an angle used to indicate a direction. Bearings are measured from North

direction. They are useful in reading maps.

Example 2: The main road in a town runs north to south. A sketch below shows the path of a

car traveling from A to C through B. Find the bearing from C to A.

Line DE lies in North to South.

Working: Consider ABC.

AB = BC = 2 km

ABC is an isosceles triangle.

Hence BAC = BCA

But exterior angle = 60 (given)

3x+30 2x

N

S

W E 60

D

B

C

A

2km

Advanced-Geometry 5 of 41 2014, www.learningforknowledge.com/glg

BAC + BCA = 60 (sum of opposite interior angles)

BAC + BAC = 60 ( BAC = BCA, by above)

or 2 BAC = 60

BAC = 30

Since DE is North to South, BAC is the bearing of point C from point A.

Answer: Bearing of C is equal to 30.

Example 3: In the figure of a pentagon below, pairs of exterior and interior angles are

marked as, (a, b), (c, d), (e, f), (g, h) and (i, j). Show that the sum of interior angles

(b, c, d, f, h, j) is 540.

Working:

(a+b) = 180 (Angles on a straight line)

Similarly,

(c+d) = (e+f) = (g+h) = (i+j) = 180

(a+b) + (c+d) + (e+f) + (g+h) + (i+j) = 5180

or, (a+c+ e+ g+ i) + (b+d+f+h+j) = 900

But, (a+c+ e+ g+ i) = 360 (Exterior angles of a polygon add up to 360)

360 + (b+d+f+h+j) = 900

b+d+f+h+j = 900 360 = 540

where, b, d, f, h and j are interior angles of the pentagon.

Answer: The sum of the interior angles of a pentagon = 540

Example 4: PQR is an equilateral triangle. PS is a line to QR. Show that PS divides the

angle QPR into two equal parts.

Construction: Draw an equilateral PQR and a perpendicular to QR

from P intersecting at S.

a

b c

d

e f

g

h

i j

P

Q R S

Advanced-Geometry 6 of 41 2014, www.learningforknowledge.com/glg

Working: In PSQ, PSQ = 90 (Given PS QR)

PQS = PQR = 60 (PQR is equilateral)

PSQ + PQS + QPS = 180 (Angles in a triangle)

90 + 60 + QPS = 180

150 + QPS = 180

QPS = 180 150 = 30

Similarly, In PSQ, PSR = 90 (Given PS QR)

PRS = PRQ = 60 (PQR is equilateral)

PSR + PRS + RPS = 180 (Angles in a triangle)

90 + 60 + RPS = 180

150 + RPS = 180

RPS = 180 150 = 30

Hence, QPS = RPS = 30

But, QPR = 60 (PQR is equilateral)

QPR = 60 = QPS + RPS

= 30 + 30 = 60

Answer: The line PS bisects angle APR and divides it into two equal parts.

Example 5: The figure below shows a parallel lines cut by a pair of intersecting lines. Find

the value of the angle marked x.

Construction: Mark the points of intersection as A, B

and C, and the parallel lines as AD and CG. Draw a line

EF through point B, parallel to AD and CG.

Working: Since EF || CG,

EBC = BCG (Alternate angles)

EBC = BCG = 30 (given BCG = 30)

Since EF || AD, DAB = ABE (Alternate angles)

x

30

60

A D

C

BE

F

G

Advanced-Geometry 7 of 41 2014, www.learningforknowledge.com/glg

But, DAB = ABE = x

ABC = ABE + EBC

or, ABE + EBC = 60 (given ABC = 60)

Hence, DAB + 30 = 60

or, DAB = 60 30 = 30

DAB = x = 30

Answer: The angle x = 30

EXERCISE AdvGeoAngles

Angles

1. Find the value of a and b in the figure below.

Ans: a = 60, b = 30

2. Find the value of x and y in the triangle shown below.

Ans: x = 90

3. PQRS is a quadrilateral. Diagonal PS bisects angles P and S. Given QPS = 65 and RSP = 35, find the value of P and S.

Ans: P = S = 80

4. Find the value of the interior angle of a regular octagon.

Ans: 135

90

2a

a

b

x 140

130

Advanced-Geometry 8 of 41 2014, www.learningforknowledge.com/glg

5. In the figure below, show that line AB is parallel to line CD, given A + C = 125.

(Hint: A + B + C = 180)

6. Timothy travels in a car 30 kms North and then turns East and travels a further distance of

30 kms. Find the bearing of his destination with respect to his starting point.

Ans: 45

7. Find the value of x in the figure below.

Ans: 60

8. Find the value of a in the figure below.

Ans:

9. Calculate the value of angles marked a and b.

Ans:

x 80

45

A

B

C

D

E

x

2x

a

2a

2a30

90

90

a b

40 45

Advanced-Geometry 9 of 41 2014, www.learningforknowledge.com/glg

10. In the figure below write the value of x in terms of a and b.

Ans: x = ()

a

x

b

x

Advanced-Geometry 10 of 41 2014, www.learningforknowledge.com/glg

Shapes

Plane figures are broadly classified according to their shapes. A plane figure is defined as an

area bound by intersecting lines and curves. Simple figures are formed by intersecting lines,

for example, polygon or circle. Composite shapes are formed by a combination of polygons

and curves.

Similar Shapes

Although the size of a polygon can vary, its shape is determined by the lengths, the number

of sides and the angles formed at its vertices. Any two polygons which have exactly the same

shape, but not nece