1Look at these pictures.

Stacks of coins, trays, tiles.Look at these stacks of books:

Notice the difference?

If you stack up the math textbooks of all kids in your class, willthey all fit nicely, one over another?

What about your math notebooks?

The pages of a book all fit exactly over one another.

What about the pages of different books?

In geometry, figures such as triangles, rectangles, circles and soon, which can be placed one over the other so as to fit exactly aresaid to be congruent figures.

Can you give some examples of congruent and non-congruentfigures?

Even though there is nodifference in shape, their sizesare different, aren’t they? So,they are not congruent.

Congruent Triangles

All equalIn geometry, we use the term“congruent” in the sense that “allmeasures are equal”. In thepicture on the left, the tiles areof different colors; but all arerectangles and all theserectangles have the same lengthand breadth. And preciselybecause of this, they all can bestacked together nicely, eachfitting exactly over another.Thus in general, we can say thatcongruent figures are those withthe same shape and size.For example, two copies of thesame photo have the sameshape and size; and so they arecongruent.

What about a photo and itsenlargement?

8

Class VIII - Mathematics

Let’s measureLook at these figures:

A circle and a square. Can you draw figures congruent tothese in your notebook?

How do you do it?

Copy them using tracing paper; or measure the figures anddraw with the same measurements.

What are the things to be measured?

Look at this rectangle:

To draw a rectangle congruent to this, what all things shouldwe measure?

Appu drew a parallelogram:

Ammu measured its sides and drew a parallelogram withthe same measurements, like this:

Measure the sides of both parallelograms. The lengths areequal, aren’t they? But we can tell at a glance that these arenot congruent.

For parallelograms to be congruent, what more should beequal, apart from the lengths of sides?

Can you draw a parallelogram congruent to the one Appuhad drawn?

Can you cut this into four congruentfigures?

Congruent dissection

Take a square sheet of paper and cut

off a quarter as shown below:

The remaining figure has to be cut intofour congruent pieces. Can you do it?

Another puzzle. Cut out a paperrectangle with one side twice as longas the other.

9

Congruent Triangles

A

B C

P

Q R

Without shifting these, how manydifferent parallelograms can you makeusing the other two with these?

Triangle match

Take a look at these triangles:

Are they congruent?

Copy one of these on a piece of tracing paper and place itover the other at different positions.

They are congruent, aren’t they?

When the triangles were made to fit exactly over each other,which were the sides that matched?

And which angles?

To specify the equal sides and angles, let’s name the triangles.

Complete the table, by writing the last pair of equal sidesand angles.

Equal sides Equal angles

AB = PQ ∠ACB = ∠PRQ

BC = PR ∠BAC = ∠PQR

Do you see any relation between each pair of equal sidesand the pair of equal angles given alongside?

Different angles

Cut out a pair of eerkkil bits, each 3centimeters long and another pair, each2 centimeters long. How many differentkinds of rectangles can you make withthese?

We can only place the same rectanglein different positions; we cannot makedifferent types of rectangles.

Now with the same eerkkil pieces,how many different types ofparallelograms can you make?

As many as we want, right?

Now place one of the longer pair ofeerkkil pieces and one of the shorterpairs, at an angle of 45o, as shownbelow:

10

Class VIII - Mathematics

If two triangles are congruent, then the sides and

angles of one are equal to the sides and angles of

the other; angles opposite to equal sides are equal

and sides opposite to equal angles are equal.

When sides are equal

In Class 7, we have seen how we can draw triangles with

specified measures (see the lesson, Math Drawing).

Can you draw a triangle of sides 4 centimeters, 5 centi-

meters and 6 centimeters?

Given below are pictures of such triangles which some kids

in a class drew.

4 cm5 cm

6 cm

6 cm

5 cm

4 cm

5 cm

4 cm

6 cm

What do you say now?

Fold and turn

Cut out a paper square. If it is foldedalong the diagonal, we get two trianglesand these triangles fit exactly over eachother; that is, these triangles arecongruent.

Now cut out a paper rectangle, whichis not a square. Again fold through thediagonal.

They don’t fit over each other. Can wesay that these triangles are notcongruent?

Cut out these triangles and place oneover the other in various positions.

11

Congruent Triangles

4 cm

5 cm6 cm

If the three sides of a triangle are equal to thethree sides of another triangle, then these trian-gles are congruent.

Two sides and an angle

We can draw a triangle, if two sides and the angle includedbetween them are specified, instead of three sides.

For example, can you draw a triangle with two sides 5centimeters and 3 centimeters long and the angle betweenthem equal to 60o?

We can draw in different ways:

60o

5 cm

3 cm

60o

5 cm

3 cm

Even if angles are equal

We have noted that if all the sides of atriangle are equal to the sides of anothertriangle, then these triangles arecongruent. What if the angles areequal?

We can draw triangles with the samethree angles in different sizes, can’twe?

60o

30o

60o

30o

60o

5 cm

3 cm

Make a copy of one of these on a piece of tracing paperand place it over the others. These triangles are all congruent,aren’t they?

What do we see here?

In other words, if the sides of two tri-angles are equal, their angles are alsoequal; but simply because angles areequal, the sides need not be equal. Lookat this picture:

12

Class VIII - Mathematics

6 cm

4 cm

30o

30o

6 cm

4 cm

We can tell at a glance that these are not congruent. What

do we see here?

Just because two sides and some angle of a triangle are

equal to two sides and some angle of another triangle, the

two triangles need not be congruent.

If two sides of a triangle and their included angleare equal to two sides of another triangle and theirincluded angle, then these triangles are congruent.

As before, check if these are all congruent to one another.You can also check those drawn by other kids in your class.

What do we see here?

If some angle other than the included angle is specified,

then also we can draw a triangle sometimes.

For example, a triangle with two sides 6 centimeters, 4

centimeters and the angle opposite the shorter side equal

to 30o. Do you remember drawing such triangles in

Class 7? (See the section, Another angle in the lesson,

Math Drawing).

Determining a triangle

Make an angle by bending a long pieceof eerkkil.

Now we have to place another pieceof eerkkil over the two sides of theangle, to make a triangle. We can putit in different positions, can’t we?

Let’s mark a spot on the top side ofthe angle. What if we insist that thesecond eerkkil should pass throughthis?

Now let’s mark spots on both the topand bottom sides and want the secondpiece to pass through both these. Howmany triangles can we make?

Once we fix one angle and the lengthsof its two sides, a triangle is determined,isn’t it?

13

Congruent Triangles

4 cm

30o

45o

4 cm30o 45o

4 cm30o45o

If one side and the two angles on it of a triangle areequal to one side of and the two angles on it ofanother triangle, then these triangles are congruent.

6 cm

3 cm

30o 60o 60o30o

So, in these two triangles, a side (3 centimeters) and twoangles (30o, 60o) are equal; but the triangles are notcongruent, are they?

Try with still longer pieces.

3 cm

One side and two anglesWe have seen that we can draw a triangle, if one of thesides and the two angles on it are specified. For example,draw a triangle with one side 4 centimeters long and thetwo angles on it equal to 30o and 45o.

Some such triangles are shown below:

Compare these triangles and also those drawn in your class.

If two triangles have one side and any two angles equal, isit necessary that they should be congruent?

For example, draw a triangle with one side 6 centimetersand the angles on it equal to 30o and 60o. Draw anothertriangle with one side 3 centimeters and the angles on itagain equal to 30o and 60o. Now measure the shortest sideof the larger triangle. It is also 3 centimeters, isn’t it?

How many triangles?

Cut out a 6 centimeter long piece ofeerkkil. Place a long piece of eerkkilat one end of it, making a 30o angle.

Now we are to make a triangle usinganother piece of eerkkil with thisangle. But there are some conditions.One end of this new piece should beat the end of the bottom side of theangle; the other end should just touchthe top side of the angle.

Can we make such a triangle with thethird piece 2 centimeters long?

How about a piece 3 centimeters long?

And a 4 centimeter long piece?

What do you see?

14

Class VIII - Mathematics

A B

D C

A B

D C

Measure their angles. The angles ofthe two triangles are equal, aren’t they?(You can also check this by cutting outthe triangles and placing each angle ofone triangle over the angles of theother.)

Thus in these two triangles five of thesix measures (three angles and twosides) are equal; but they are evidentlynot congruent.

Are the triangles got by drawing a diagonal congruent?

Applications and examples

When we draw a diagonal of a rectangle, we get twotriangles.

And we can see that these two triangles are congruent, bycutting out the triangles and placing one over the other. Nowhow do we prove that this is so for all rectangles?

In the rectangle ABCD shown above, the opposite sidesAB and CD are equal; and the opposite sides AD and BCare equal.

That is, two sides of ∆ACB are equal to two sides of∆ACD. What about the third sides?

In each triangle, the third side is AC. (We can say that ACis a common side of these triangles.)

Thus the three sides of ∆ACB are equal to the three sidesof ∆ACD. And so, these triangles are congruent.

What about a parallelogram?

Incorrect match

A triangle has three sides and threeangles and thus six measures in all. Wehave seen that if in two trianglescertain triples of these measures (threesides, two sides and the included angle,one side and the angles on them) areequal, then these triangles arecongruent; that is, the remainingmeasures are also equal.

Now take a large sheet of paper anddraw a triangle of sides 8, 12 and 18centimeters.

And then a triangle of sides 12, 18 and27 centimeters.

15

Congruent Triangles

A B

D C

A B

D C

Look at the angles on the side AC of each triangle.

∠BAC and ∠DCA are alternate angles formed by the lineAC meeting the pair of parallel lines AB and CD.So,

∠BAC = ∠DCASimilarly,

∠BCA = ∠DAC

being alternate angles formed by the line AC meeting thepair of parallel lines AD and BC. (See the section, Anotherkind of pairing, of the lesson, Lines in Unison in the Class7 textbook.)

Thus, one side of ∆ACB and the two angles on it are equalto one side of ∆ACD and the two angles on it; and sothese triangles are congruent.

We usually use the symbol ≅ for the phrase “is congruentto”. For example, in the figure above,

∆ACB ≅ ∆ACD

The congruency of these two triangles gives another fact:

AB = CD and BC = AD

Thus we have the result:

The opposite sides of a parallelogram are equal.

Here also, AC is a common side of ∆ACB and ∆ACD.But we don’t know whether the opposite sides are equal.

For what position of the dividing lineare the pieces also parallelograms?

Congruent parts

We saw that by joining a pair ofopposite corners of a rectangle, we gettwo triangles, congruent to each other.Suppose instead of joining oppositecorners, we join points equidistant fromopposite corners?

Cut these out and check. Draw severalsuch lines. For what position of the lineare the pieces also rectangles?

Do parallelograms also have thispeculiarity?

16

Class VIII - Mathematics

A M B

Q

P

Prove that AM = MB. Can you suggest a constructionto locate the midpoint of a line, using this idea?

$ Prove that the point of intersection of the two diago-nals of a parallelogram is the midpoint of both the di-agonals.

Right angled triangles

A right angled triangle is a triangle with one angle right (thatis, 90o), right? The longest side of a right angled triangle iscalled its hypotenuse. The other two sides may be calledthe perpendicular sides or the short sides.

In the right angled triangles ABC and PQR shownbelow, PR = BC and QR = AB.

P

Q R

A

B C

A B

D C

Now we have two triangles ABC andADC. The sides AB and BC of ∆ABCare equal to the sides CD and DA of∆ADC; also the third side of eachtriangle is AC. Thus the sides of thesetriangles are equal and so they arecongruent.

So, the angles BAC and DCA, whichare opposite the equal sides BC andDA, are themselves equal.

These are the alternate angles formedby the line AC with the pair of lines ABand CD. Since these angles are equal,the lines AB and CD are parallel.

Similarly, by drawing the diagonal BD,we can show that ∠DAC = ∠BCAand hence AD and BC are parallel.Thus ABCD is a parallelogram.

What have we proved?

If both pairs of opposite sides ofa quadrilateral are equal, thenit is a parallelogram.

Again from the congruency of the triangles ACB and ACDin the above discussion, we get

∠ABC = ∠ADC

Similarly, by drawing the other diagonal BD, we can provethat ∆BDA and ∆BDC are congruent and hence

∠BAD = ∠BCD.

That is,

The opposite angles of a parallelogram are equal.

Now try these problems:

$ In the figure below, AP and BQ are equal and parallel.

Are they congruent?

Parallelogram means...

The general name for a figure of foursides is a quadrilateral. A parallelo-gram is a quadrilateral with both pairsof opposite sides parallel. We provedthat in any parallelogram, both pairs ofopposite sides are equal. So, there is areverse question: is every quadrilateral,with both pairs of opposite side equal,a parallelogram?

Suppose in the quadrilateral ABCD, wehave AB = CD and AD = BC. Drawthe diagonal AC.

17

Congruent Triangles

A

B C

P

Q R

If instead of the perpendicular sides, some other pairs of

sides are equal, would the right angled triangles be

congruent?

In the two right angled triangles shown above, PR = AB

and PQ = BC.

Here we don’t know anything about the included angles

PRQ and BAC.

Let’s take a look at the third sides. Is there any relation

between the sides QR and AC?

A B

D

C

A B

D

C

Two sides of ∆PQR are equal to two sides of ∆ABC.

What more do we need for the triangles to be congruent?

In ∆PQR, the angle PRQ, included between the sides PR

and RQ, is a right angle.

In ∆ABC, the angle ABC, included between the sides BC

and AB is again a right angle. That is,

∠PRQ = 90o = ∠ABC

Thus the sides PR and RQ of ∆PQR and their included

angle PRQ are equal to the sides AB and BC of ∆ABC

and their included angle ABC. So, these triangles are

congruent.

Angles of a quadrilateral

We saw that in a parallelogram, bothpairs of opposite angles are equal. Thereverse question is whether aquadrilateral, in which both pairs ofopposite angles are equal, is aparallelogram.

To prove that this is true, we must firstknow the sum of all the four angles ofa quadrilateral. Look at these figures:

When we draw one diagonal, twoangles of the quadrilateral are split intotwo angles each, giving six angles inall. These six angles are the angles ofthe ∆ABC and ∆ACD. So, the sum ofthese six angles is 180o + 180o = 360o.What do we see here?

The sum of the angles of aquadrilateral is 360o.

18

Class VIII - Mathematics

AB is the hypotenuse of the right angled triangle ABC. So,by Pythagoras Theorem,

AC2 = AB2 − BC2

Similarly, since PR is the hypotenuse of the right angledtriangle PQR, we get

QR2 = PR2 − PQ2

Now by what we have said at the beginning,

PR = AB, and PQ = BC

Putting all these together, we get

QR2 = PR2 − PQ2 = AB2 − BC2 = AC2

From this, we find

QR = AC

Thus the three sides of ∆PQR are equal to the sides of∆ABC. So,

∆ABC ≅ ∆PQR

What general result do we have here?

If the hypotenuse and one other side of a rightangled triangle are equal to the hypotenuse andone other side of another right angled triangle, thenthese two triangles are congruent.

A

B CP

A B

D C

x

xy

y

Isosceles triangles

Any triangle can be split into two right angled triangles, bydrawing a perpendicular from one vertex to the oppositeside.

Angles of a parallelogram

Let’s see how we can prove that if bothpairs of opposite angles are equal in aquadrilateral, then it is a parallelogram.

Consider a quadrilateral with both pairsof opposite angles equal.

In this figure, let’s denote the measureof one pair of opposite angles by xo

and the measure of the other pair ofopposite angles by yo.Since the sum of the angles of aquadrilateral is 360o, we get

x + y + x + y = 360From this we get

x + y = 180

So, we get∠A + ∠D = xo + yo = 180o

∠A and ∠D are co-interior angleswhich the line AD makes with the pairAB, CD of lines. Since their sum is180o, the lines AB and CD are parallel.Since we also have ∠B + ∠D = 180o,we can prove similarly that AD and BCare parallel.Thus both pairs of opposite sides areparallel, and so ABCD is aparallelogram.

If both pairs of opposite anglesin a parallelogram are equal,then it is a parallelogram.

19

Congruent Triangles

We get another thing from this. In the figure, AB = AC in∆ABC. The line AP is drawn perpendicular to the side BC.As seen above, the right angled triangles ABP and ACPare congruent. So, their sides and angles must be equal.For example, since AP is a side of both the triangles, theangles opposite this side in the two triangles must be equal.That is,

∠ABC = ∠ACB

What do we see here?

Congruency in action

Thales was a mathematician andphilosopher who lived in Greece in thesixth century BC.Here’s is a trick heis supposed to haveused to calculate thedistance to a shipanchored at seafrom the shore.

First he stuck a longpole on the shore, directly in front ofthe ship. Then he stuck another poleon the shore, some distance away fromthe first one. A third pole he stuckexactly at the middle of the first twopoles.

He then drew a line from the secondstick, perpendicular to the shore. Hewalked backwards along this line,keeping the ship in sight. Just when themiddle stick came between the shipand himself in the line of sight, hestopped and marked his position.

Thales

Stick 2Stick 3Stick 1

Ship

Now the triangle on sea and thetriangle on shore in the picture arecongruent. (Why?) So, the distancefrom the shore to the ship is equal tothe distance between the spot wherehe stopped and the second stick.

A

B CP

A

B C

A

PB C

In the picture above, ∆ABC is split into two right angledtriangles ABP and ACP, by drawing the perpendicular fromA to BC.

What type of triangle should ABC be, so that these tworight angled triangles are congruent?

In ∆ABP and ∆ACP, the side AP is common. For theseright angled triangles to be congruent, we must have thehypotenuses also equal.

That is, we must have AB = AC.

So, a triangle with two of its sides equal can be split intotwo congruent right angled triangles; we need only drawthe perpendicular from the vertex where the equal sidesmeet, to the opposite side.

20

Class VIII - Mathematics

If two sides of a triangle are equal, then the anglesopposite to these sides are also equal.

A

B CP

Are the triangles ABP and ACP congruent?

For these triangles, AP is a common side. And one angleon this side is 90o in both triangles. What more do we needto claim that the triangles are congruent?

Are ∠BAP and ∠CAP equal?

Since the sum of the angles of any triangle is 180o, we havein ∆ABP

∠ABP + ∠BAP + 90o = 180o

from which we get ∠ABP + ∠BAP = 90o and hence

∠BAP = 90o − ∠ABP

Similarly, from ∆ACP, we get

∠CAP = 90o − ∠ACP

If we use the fact that ∠ABP = ∠ACP also, we get

∠BAP = ∠CAP

Is the reverse true?

That is, if two angles of a triangle are equal, are the sidesopposite them also equal?

In ∆ABC shown below, ∠ABC = ∠ACB. Let’s see whetherAB = AC. As before, we draw the perpendicular fromA to BC.

Angle in a semicircle

In the figure below, AB is a diameterof the circle and C is a point on thecircle.

Can we say anything about ∠ACB?

For this, join C with the center O ofthe circle.

The angles of ∆ABC are xo, yo,(x + y)o. So, x + y + (x + y) = 180 andhence x + y = 90. Thus, ∠ABC = 90o

In this argument, the point C can beanywhere on the circle. So, what dowe see?

In a circle, the angle formed byjoining the endpoints of adiameter with another point onthe circle, is a right angle.

A B

C

A B

C

O

x y

A B

C

O

x yx y

Now ∠ACB is split into two. Let themeasures of these angles be xo and yo.In ∆OAC, we have OA = OC. (Why?)So, ∠OAC = xo. Similarly, in ∆OBC,we have OB = OCand so ∠OBC = yo.

21

Congruent Triangles

115o

A P B

Since QR is a diameter and P is a pointon the circle, QPR is a right angle. Inother words, PR is perpendicular toAB.

17o

33o

70o

67o50o

Thus the side AP of ∆ABP and the angles APB and BAP

on it are equal to the side AP of ∆ACP and the angles

APC and CAP on it. So, the triangles are congruent.

And because of this, the sides AB and AC, opposite the

equal angles APB and APC are equal.

What do we find?

If two angles of a triangle are equal, thenthe sides opposite these angles are also equal.

A triangle in which two sides are equal is called an isosceles

triangle. From what we have seen just now, we can also

say that an isosceles triangle is a triangle with two angles

equal.

Recall that a triangle with all the three sides equal is called

an equilateral triangle. Such triangles are a special class of

isosceles triangles.

Now you can try some problems:

$ Some isosceles triangles are shown below. One angleof each is given. Find the other two angles of each.

Circle and perpendicularWe are given a line AB and a point Pon it.

The problem is to draw a perpendicularto AB, passing through P.

We can draw it using a set square inthe geometry box. We can also drawit using compass.

For this we draw a circle, which passesthrough P and cuts AB at another point.We name this point Q.

Now draw the diameter through Q andjoin its other end with P.

A PQ B

R

A P BQ

22

Class VIII - Mathematics

In an isosceles triangle, the perpendicular from thevertex joining the equal sides to the opposite side,bisects this side and the angle at this vertex.

A

B CP

Rope math

We have mentioned the Elements, theauthorative text on ancient geometry.In this, Euclid considers only figuresthat can be drawn with straight linesand circles; in other words, only figuresthat can be drawn with a straight rodwithout markings and a compass. Whyis this so?

In ancient times, ropes or strings wereused for measuring as well as drawing.The two figures that can be easilydrawn with rope are straight lines andcircles. Lines could be drawn bystretching a rope between two fixedpegs; circles could be drawn by takingout one peg and rotating about theother.

Nowadays, all sorts of tools can bemade to draw different kinds offigures and so, such restriction on toolshas only historic and theoreticalsignificance.

A line which divides a line or an angle into two equal partsis called a bisector. Thus in the figure above, the line AP isa bisector of the line BC; since it is perpendicular to BC, itis called the perpendicular bisector of BC. It is also thebisector of ∠BAC.

$ One angle of an isosceles triangle is 120o. What arethe other two angles?

$ What are the angles of an isosceles right angled trian-gle?

$ What are the angles of an equilateral triangle?

Bisectors

We have seen how an isosceles triangle can be divided intotwo congruent right angled triangles. From this, we can learna bit more.

In ∆ABC shown below, we have AB = AC; and AP isthe perpendicular from A to BC.

Since ∆ABP and ∆ACP are congruent, their sides andangles are equal. We get BP = CP; also, we get ∠BAP =∠CAP, since these are angles opposite to equal sides.

Thus, the line AP divides the side BC into two equal parts;it divides ∠BAC also into two equal parts.

23

Congruent Triangles

Now look at this picture:

A B

A B

A B

A P B

A P BC D

Since CQD is an isosceles triangle, theperpendicular from Q to CD bisectsCD. This means, this perpendicularpasses through the midpoint P of CD.

In other words, the line QP isperpendicular to CD. Since the line CDis a part of the line AB, the line QP isperpendicular to AB also.

A P BC D

Q

A P BC D

Q

Several points equidistant from two points A and B are

marked. Each of these points joined to A and B gives an

isosceles triangle. So all these points lie on the perpendicular

bisector of the line AB. In other words, if we join all these,

we get the perpendicular bisector of AB.

Another way

There is another method to draw theperpendicular to a line from a specifiedpoint on it.

First mark two points C and D on ABitself, at equal distances from P.

Next mark a point Q equidistant fromC and D.

24

Class VIII - Mathematics

We need only two points to draw a line. So, we can drawthe perpendicular bisector of a line AB as shown below:

A B

A B

A

B C C

P

A

B Q

A B

P

Now we need only draw the perpen-dicular bisector of QR. And for that,we need only draw circles of the sameradius, with Q and R as centre.

P

Q RA B

If there is not enough space below AB, we can also drawlike this:

We can use the general result we saw above to draw thebisector of an angle also.

First, we draw an isosceles triangle which contains this angle:

A perpendicular from outside

Through a point on a line, we can drawthe perpedicular, using a compass. Howdo we draw the perpendicular from apoint not on the line?

We have to draw an isosceles trianglewith P as the top vertex and AB as thebottom side. For this, we need onlymark two points on AB, equidistant fromP. For this draw a circle with P ascenter to cut AB at Q and R.

P

Q R

S

25

Congruent Triangles

C

P

A

B Q

A

B C

P

A

B C

P

Q

S

xx

Since BP is the bisector of ∠ABC,we have ∠ABP = ∠CBP.

If we take this as xo, then

∠BQS = ∠BQR = 90o − xo (Why?)

Now we can prove that ∆BQS and∆BQR are congruent. (How?)

So QS = QR.

The perpendiculars from a point on thebisectors of an angle, to the sides ofthe angle, are equal.

D

A B

C

P

Q

R

Now we need only draw the perpendicular bisector of theside PQ of ∆PBQ.

Here there’s a convenience. The perpendicular bisectorwe want to draw passes through B (why?) So, we needonly mark one more point on this bisector.

Now try these problems:

$ Prove that in a parallelogram with all four sides equal,

the diagonals are perpendicular bisectors of each other.

$ Can you draw a line of length 2.25 centimeters using a

ruler? How about using ruler and compass?

$ How can we draw a circle with a given line as diam-

eter, without actually measuring the line?

$ How do we draw an angle of 22o1

2?

$

In the figure, ABCD is a parallelogram and AP = CQ.Prove that PD = BQ. Prove also that the quadrilateralPBQD is a parallelogram.

$ Prove that if one pair of opposite sides of a quadrilat-eral are equal and parallel, then it is a parallelogram.

Equidistant bisectorIn the figure, PQ is the bisector of∠ABC.

Mark a point Q on BP and draw per-pendiculars from Q to AB and BC.

26

Class VIII - Mathematics

$ Are the triangles below congruent? Give the reason.

60o

70o

50o

70o

B CR

A

Q

P

Congruent dissections

$ How many different (non-congruent) isosceles trian-gles can be drawn with one angle 80o and one side8 centimeters?

$ In the figure below, PQ = PR. Prove that the point Pis on the bisector of ∠ABC.

3.5 cm

3.5 cm