Properties of Triangles 183

180 x= 15. 12

So, the angles measure (3 x15)°, (4 x15)° and (5x15)°, i.e., 45°, 60° and 75°.

Hence, the angles of the triangle are 45°, 60°, 75.

To angles of a triangle are equal and the third angle measures 70°. Find the

measure of each of the unknown angles.

EXAMPLE 3.

Let the measure of each unknown angle be x° We know that the sum of the angles of a triangle is 180°.

x+*+70 = 180 2x = (180 -70) 2x = 110 x = 55.

Solutton

Hence, each unknown angle is 55°.

EXAMPLE4. In a AABC. f3 LA = 4 ZB 6 2C, calculate LA, LB and 2C.

Solutlon Let 3 2A = 4 2B = 62C =x°. Then,

A 4B-and 20 Now, LA + ZB+ LC =180° I:sum of the 4 of a triangle is 180°]

180

4x + 3x +2x = (180x 12) 180x12

9x = (180 x12)»: = 240. 9 =

A--s0", LB = = 60 and C -20-40 Hence, LA = 80°, LB = 60° and 2C = 40°

EXAMPLE 5. The adjotning figure has been obtained by ustng two triangles. Prove that 2A + LB+2C+ZD+LE + 4F = 360°.

Solution We know that the sum of the angles of a triangle is 180.

From AACE, we have: LA +4C+ ZE = 180°. (i)

From ABDF, we have: LB+ZD +ZF =180°. ...( (ii)

From (i) and (ii), we get D

LA + ZB + 2C + 2D+ LE +F = 360°.

EXERCISE 15A

1. In a AABC, if ZA = 72° and 2B = 63, find 2C.

2. In a ADEF, if ZE =105° and LF = 40°, find 2D.

3. In a AxYz, if ZX =90° and Z = 48°, find Y.

4. Find the angles ofa triangle which are in the ratio 4:3:2.

5. One of the acute angles of a right triangle is 36, find the other.

6. The acute angles of a right triangle are in the ratio2: 1. Find each of these angles 7. One of the angles of a triangle is 100 and the other two angles are equal. Find each of the

equal angles.

184 Mathematics for Class 7

8. Each of the two equal angles of an isosceles triangle is twice the third angle. Find the angles

of the triangle. Hint. Let the third angle be x°, Then, 2 x +2x+x = 180.

9. f one angle of a triangle is equal to the sum of the other two, show that the triangle is

right-angled. Hint. LA = LB+ 2C LA+LA = ZA+ LB+ 2C =180.

10. In a AABC, if 2LA =34B = 62C, calculate 2A, ZB and 2C.

11. What is the measure of each angle of an equilateral triangle? Hint. All the angles of an equilateral triangle are equal.

12. In the given figure, DE || BC. If ZA = 65° and ZB = 55°, find

(1) ZADE (i) LAED Hint. Since DE || BC and ADB is the transversal, so

LADE = LABC = 55° (corresponding angles).

(1il) 2C 55

D

A55 13. Can a triangle have

() two right angles?

(ii) two acute angles?

(v) all angles less than 60°?

3

(ii) two obtuse angles?

(iv) all angles more than 60°?

(vi) all angles equal to 60°?

14. Answer the following in 'Yes' or 'No'.

() Can an isosceles triangle be a right triangle? (i) Can a right triangle be a scalene triangle?

(iii) Can a right triangle be an equilateral triangle?

(iv) Can an obtuse triangle be an isosceles triangle?

15. Fill in the blanks:

(1) A right triangle cannot have an... angle.

(i1) The acute angles of a right triangle are.. (ii) Each acute angle of an isosceles right triangle measures (iv) Each angle of an equilateral triangle measures.. (v) The side opposite the right angle of a right triangle is called.....

(vi) The sum of the lengihs of the sides of a triangle is called its....

= =

EXTERIOR AND INTERIOR OPPOSITE ANGLESs Let the side BC of a AABC be produced to D. Then, ZACD is called an exterior angle. Also, ZBAC and 2ABC are called the interior

opposite angles. D EXTERIOR ANGLE PROPERTY OF A TRIANGLE

Ifa side ofa triangle is produced then the exterior angle so formed is equal to thhe

sum of the two interior opposite angles. THEOREM

Let a side BC of a AABC be produced to D, formtng exterior angle ZACD.

We know that the sum of the angles of a triangle is 180.

4+L2+ 23 = 180°

PROOF

. (i) .

.. (i1) [by linear pair property] But, 3+24 = 180°

From (i) and (iti), we get

4+22 + 23 = L3 + 24.

Hence, A+2 = L4. B D

186 Mathematics for Class 7

On adding the corresponding sides of (1), (ii) and (1ii), we get

A+ 22+ 23 2(LA + ZB+ 2C) 2x180° : sum of the angles of a triangle is 180 = 360

4+22+ 23 = 360°. Hence, the sum of the exterior angles is 360.

EXERCISE 15B

1. In the figure glven alongside, find the measure of ZACD.

45 B

2. In the figure given alongside, find the values of x and y.

68° 130 C D

3. In the figure given alongside, find the values of x and y.

65 C

4. An exterior angle of a triangle measures 1 10° and its interior opposite angles are in the ratio 2:3. Find the angles of the triangle.

5. An exterior angle of a triangle is 100° and its interior opposite angles are equal to each other. Find the measure of each angle of the triangle.

6. In the figure given alongside, find: (i) ACD (i1) LAED

AE

45 A

7. In the figure given alongside, find: (i) LACD (11) ADC (iil) DAE

40 100 B

8. In the figure given alongside, x : y =2:3 and 2ACD = 130°. Find the values of x, y and z. Hint. Let x = 2t and y = 3t. Then, 2t +3t = 130 >t =26.

B D

188

Is it possible to draw a triang whose sides have lengths 10.3 cm, 5.8

Mathematics for Class 7

8 cm and EXAMPLE 2.

4.6 cm?

Solutlon Clearly, we have: (10.3+5.8)> 4.6 (5.8+4.6) > 10.3. (10.3+4.6)> 5.8.

nus, the sum of any two of these numbers is greater than the third.

ence, itis possible to draw a triangle whose sides are 10.3 cm, 5.8 cm and 4.6 cm.

EXAMPLE 3. s t possible to draw a triangle whose sides are 5 cm, 7 cm and 12 cm?

Solution Consider the numbers 5, 7, 12.

Clearly, 5 +7 12. Thus, the sum of two of these numbers is not greater than the tnira.

Hence, it is not possible to draw a triangle whose sides are 5 cm, 7 cm and 12 cm.

Three points A. B, C are collinear, as shown in the figure. Can you draw AABC? f not, why?

EXAMPLE 4.

Since the points A, B, C are collinear, they lie on the same straight line.

AB +BC = AC.

Solution

But, in a triangle, the sum of any two sides must be greater than the third side

Hence, it is not possible to draw AABC

EXAMPLE 5. Two stdes of a triangle are 6 cm and 8 cm long. What can be the length of lts third side?

Let the length of the third side be x cm.

We know that the sum of any two sides of a triangle is greater than the third. (6+8)> x

Also, we know that the dtfference of any two sides of a triangle is less than the

Solution

x <14.

third side.

(8-6) < x > x> 2.

Thus, 2 <x<14. Hence, the length of the third side must be larger than 2 cm and smaller than 14 cm

EXERCISE 15C 1. Is it possible to draw a triangle, the lengths of whose sides are given below?

(1) 1 cm, 1 cm, 1 cm (11) 2 cm, 3 cm, 4 cm

(iv) 3.4 cm, 2.1 cm, 5.3 cm (v) 6 cm, 7 cm, 14 cm

(iil) 7 cm, 8 cm, 15 cm

2. Two sides of a triangle are 5 em and 9 cm long. What can be the length of its third side

3. If P is a point in the interior of AABC then fill in the blanks with

>or < or =.

(i) PA+PB.... AB --

(ii) PB + PC ..... BC

(ii) AC... PA + PC

4. AM is a median of AABC. Prove that (AB + BC+CA) > 2AM.

B

(In A ABM) Hint. (AB+BM)> AM

(In A ACM) (AC +MC)> AM

Now, add the two inequalities.

M

Properties of Triangles 189

. In the glven figure, P is a point on the side BC of AABC. Prove that (AB+ BC + AC) > 2AP.

Hint. In AABP, AB+BP> AP.

In AAPC, PC + AC > AP.

Now, add the corresponding sides.

6. ABCD 1s a quadrilateral.

Prove that (AB + BC + CD +DA) > (AC +BD)

7. If O is a point in the exterior of AABC, show that 2(OA +OB +OC) > (AB +BC +CA). Hint. Join OA, OB, OC.

From AAOB, OA +OB> AB.

From ABOC, OB +OC> BC.

From AAOC, OA +OC> CA.

PYTHAGORAS' THEOREM Pythagoras was an eminent Greek philosopher who was born in 580 BC and died in 500 BC. He gave a wonderful relation between the lengths of the sides of a right triangle, which is known as Pythagoras' theoremn.

Pythagoras' Theorem In a right triangle, the square of the hypotenuse equals the sum f the squares of its remaining two sides.

Thus, in a right AABC in which 2C = 90°, we have:

AB = Bc? + AC2. Thus, if AB = C, BC = a and AC = b, we have:

b

ca+b.

In order to verify the above result, we perform the experiment given below. B

APERIMENT Draw any three right triangles, say T1, T2, and 73. Label each one of them as AABC. with C as the right angle.

b b

B 2 (ii)

Ta (i) (ii)

In each case, measure the sides a, b and the hypotenuse c of the triangle.

Compute a2, b2 and c2, and tabulate the observations as under:

Properties of Triangles 193

Let AB be the tree of height h metres broken at the point C and let CB take tne

position CD as shown in the figure. Then, AC =6 m, AD = 8 m and CD = CB =(h-6) m.

From right A DAC by Pythagoras' theorem, we have CD = AC +AD

(h-6) = 62 +82

Solution

B

(h-6) m

= 36 +64 =100 = (10 6 m

h-6) m

= (h-6) =10 h=(10+6) = 16 m. Hence, the original height of the tree was 16 m.

8 m

EXERCISE 15D 1. Find the length of the hypotenuse of a right triangle, the other two sides of which measure

9 cm and 12 cm.

2. The hypotenuse of a right triangle is 26 cm long. If one of the remaining two sides is 10 cm long. find the length of the other side.

3. The lengih of one side of a right triangle is 4.5 cm and the lengilh of its hypotenuse is 7.5 cm. Find the length of its third side. Hint. Let the third side be x cm. Then,

x = (7.5)2 -(4.5)* = (7.5+4.5)(7.5-4.5) = (12x3) = 36 = (6).

4. The two legs of a right triangle are equal and the square of its hypotenuse is 50. Find the length of each leg.

5. The sides of a triangle measure 15 cm, 36 cm and 39 em. Show that it is a right-angled triangle. Hint. (39)-(36) = (39+36)139 -36) = (75x3) =(5x5x3x3) = (5x3 = (15)?.

6. In right AABC, the lengths of its legs are given asa = 6 cm and b = 4.5 cm. Find the length of its hypotenuse

7. The lengths of the sides of some trlangles are given below. Which of them are right-angled? (1) a =15 cm, b = 20 cm and c =25 cm

11) a = 9 cm, b = 12 cm and c = 16 cm

(1i1) a =10 cm, b = 24 cm and c = 26 cm

8. In a AABC, LB = 35 and C 55°. Write which of the following is true:

(1) AC= AB +BC (11) AB = BC+ AC2

(11) BC AB + AC 38 55

8. A 15-m-long ladder is placed agalnst a wall to reach a window 12 m igh. Find the distance of the foot of the ladder from the wall.

15 m

12 m

X

A5-m-long ladder when set against the wall ot a house reaches a helght of 4.8 m. How far is

the foot of the ladder from the walP

10.

dat -14.8) -15+4.5)5-4.8) 19.8x0.2)-00 =(1.4.

11, Atree i above the ground and its top touches the ground at a distance of 12 m from its foot, fnd ut

e total hetght of the tree before it broke. the

is broken by the wind but does not separate. the point from where it breaks is 9 m

194 Mathematics for Clas8 7

14. wopoles, 18 m and 13 m high. stand upright in a playground. If their leet are 1a n apart, find the distance between their tops. 13 Aman goes 35 m due west and then 12 m due north. How far is he from the starting pont?

n es 3 kn due north and then 4 km due east. How far is he away from his intal position?

16. Plnd the length of diagonal of the rectangde whose sides are 16 cm and 12 cm.

16. Find the perimeter of the rectangle whose length is 40 cm and a diagonal is l Cm.

Hint. (Breadth)* - (41)*-(40) =141+40141-40)=(81x) = 8l = 9. 17. Find the perlineter of a rhombus, the lengths of whose diagonals are

16 cm and 30 cm.

Hiat. The dlagonals ofa rhombus hlsect each other at right angies.

18. Flll In the blanks:

naright triangle, the square of the hypotenuse is equal to the...of the squares of the other two sides.

(11) Tf the square of one slde of a trlangle is cqual to the sum of the squares of the other two sides then the triangle is..

(11) Of all the line segments that can be drawn to a gven Iine from a given point outside 1,

the. 1s the shorlest.

Things to Remember 1. fA, B, Cure three norncollinear pobnls, thejlgure made up by three line seyments AB, BC and CA Ls

called a triangle wtth vertices A, B and C. 2. The three ltne segments Jorming a triargle are called the stdes of the trtangle. 3. The three sides and the three angles o a triangle are together caled the sx parts or elements of

the trlangle. 4. A trlangle ls satd to be

an equilateral triangle f al of ts stdes are equal;

() an tsosceles triangle turo tis sides are equal;

(u) a scalene trlangle all ts sldes are of dferent lengths.

5. A irlangle is said to be (1) an acute triangle each of its angles meusures less than 90;

(1) a rtght triangle one of ts angles measures 90:

(M) an ohtuse triangle fone f its angles measures more than 90 6. The sum of the angles ofa trlangle is 180.

7. fone stde ofa trlangle is extended, the exterior angle sojormed is equal to the sum of the tnterior

oPposite angles. 8. The sum of any luo sides of a triangle is greater than the third side.

9. In a right rlangle, the square o the hupotenuseequais he sun of the squares of the rematntng two sldes. Ths s knour as Pytnagoras theorem

10. In a rlght trlangle, the hypotenuse is the longest side.

11. ofal the line segments that can be draun to a glven Inejron a pount outstde tt the perpendicular ine segmend ts the shortest.

Do the work in copy