R d sharma solutions mathematics class 7 Solutions for Chapter 15 properties of triangle

Take three non-collinear points A. B and C on a page of your notebook. Join AB, BC and CA. What figure do you get? Name the triangle. Also, name

(i) The side opposite to ∠B

(ii) The angle opposite to side AB

(iii) The vertex opposite to side BC

(iv) The side opposite to vertex B.

Take three collinear points A, B and C on a page of your note book. Join AB. BC and CA. Is the figure a triangle? If not, why?

Distinguish between a triangle and its triangular region

D is a point on side BC of a △CAD is joined. Name all the triangles that you can observe in the figure. How many are they?

A, B. C and D are four points, and no three points are collinear. AC and BD intersect at O. There are eight triangles that you can observe. Name all the triangles

What is the difference between a triangle and triangular region?

Explain the following terms:

(i) Triangle

(ii) Parts or elements of a triangle

(iii) Scalene triangle

(iv) Isosceles triangle

(v) Equilateral triangle

(vi) Acute triangle

(vii) Right triangle

(viii) Obtuse triangle

(ix) Interior of a triangle

(x) Exterior of a triangle

In Fig. 11, the length (in cm) of each side has been indicated along the side. State for each triangle angle whether it is scalene, isosceles or equilateral:

In Fig. 12, there are five triangles. The measures of some of their angles have been indicated. State for each triangle whether it is acute, right or obtuse.

Fill in the blanks with the correct word/symbol to make it a true statement:

(i) A triangle has ……. sides.

(ii) A triangle has ……. vertices.

(iii) A triangle has ……. angles.

(iv) A triangle has …….. parts.

(v) A triangle whose no two sides are equal is known as ………

(vi) A triangle whose two sides are equal is known as ……….

(vii) A triangle whose all the sides are equal is known as ………

(viii) A triangle whose one angle is a right angle is known as ……..

(ix) A triangle whose all the angles are of measure less than 90′ is known as ………

(x) A triangle whose one angle is more than 90′ is known as ……….

In each of the following, state if the statement is true (T) or false (F):

(i) A triangle has three sides.

(ii) A triangle may have four vertices.

(iii) Any three line-segments make up a triangle.

(iv) The interior of a triangle includes its vertices.

(v) The triangular region includes the vertices of the corresponding triangle.

(vi) The vertices of a triangle are three collinear points.

(vii) An equilateral triangle is isosceles also.

(viii) Every right triangle is scalene.

(ix) Each acute triangle is equilateral.

(x) No isosceles triangle is obtuse.

Two angles of a triangle are of measures 150 degree and 30 degree. Find the measure of the third angle.

One of the angles of a triangle is 130, and the other two angles are equal. What is the measure of each of these equal angles?

The three angles of a triangle are equal to one another. What is the measure of each of the angles?

If the angles of a triangle are in the ratio 1: 2: 3, determine three angles

The angles of a triangle are (x − 40) ⁰ , (x − 20) ⁰ and (1/2 − 10) ⁰ . Find the value of x.

The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angles is 10 degree. Find the three angles.

Two angles of a triangle are equal and the third angle is greater than each ofthose angles by 30 degree. Determine all the angles of the triangle

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

If each angle of a triangle is less than the sum of the other two, show that the triangle is acute angled.

In each of the following, the measures of three angles are given. State in which cases the angles can possibly be those of a triangle:

(i) 63 ⁰ , 37 ⁰ , 80 ⁰

(ii) 45 ⁰ , 61 ⁰ , 73 ⁰

(iii) 59 ⁰ , 72 ⁰ , 61 ⁰

(iv) 45 ⁰ , 45 ⁰ , 90 ⁰

(v) 30 ⁰ , 20 ⁰ , 125 ⁰

The angles of a triangle are in the ratio 3: 4: 5. Find the smallest angle

Two acute angles of a right triangle are equal. Find the two angles.

One angle of a triangle is greater than the sum of the other two. What can you say about the measure of this angle? What type of a triangle is this?

In the six cornered figure, (fig. 20), AC, AD and AE are joined. Find ∠FAB + ∠ABC +

∠BCD + ∠CDE + ∠DEF + ∠EFA.

Find x, y, z (whichever is required) from the figures (Fig. 21) given below:

If one angle of a triangle is 60 degree and the other two angles are in the ratio 1: 2, find the angles.

It one angle of a triangle is 100 degree and the other two angles are in the ratio 2: 3. Find the angles.

In △ABC, if 3∠A = 4∠B = 6∠C, calculate the angles.

Is it possible to have a triangle, in which

(i) Two of the angles are right?

(ii) Two of the angles are obtuse?

(iii) Two of the angles are acute?

(iv) Each angle is less than 60?

(v) Each angle is greater than 60?

(vi) Each angle is equal to 60?

In△ ABC, ∠A = 100o, AD bisects ∠A and AD ⊥ BC. Find ∠B

In △ABC, ∠A = 50, ∠B = 100 and bisector of ∠C meets AB in D. Find the angles of the triangles ADC and BDC

In ΔABC, ∠A = 60, ∠B = 80, and the bisectors of ∠B and ∠C, meet at O. Find

(i) ∠C

(ii) ∠BOC

The bisectors of the acute angles of a right triangle meet at O. Find the angle at O between the two bisectors

In △ABC, ∠A = 50 and BC is produced to a point D. The bisectors of ∠ABC and

∠ACD meet at E. Find ∠E.

In ΔABC, ∠B = 60°, ∠C = 40°, AL ⊥ BC and AD bisects ∠A such that L and D lie on side BC. Find ∠LAD

Line segments AB and CD intersect at O such that AC ∥ DB. It ∠CAB = 35 and ∠CDB

= 55. Find ∠BOD.

In Fig. 22, ΔABC is right angled at A, Q and R are points on line BC and P is a point such that QP ∥ AC and RP ∥ AB. Find ∠P

1. In Fig. 35, ∠CBX is an exterior angle of ΔABC at B. Name

(i) The interior adjacent angle

(ii) The interior opposite angles to exterior ∠CBX

Also, name the interior opposite angles to an exterior angle at A.

In the fig. 36, two of the angles are indicated. What are the measures of ∠ACX and

∠ACB?

In a triangle, an exterior angle at a vertex is 95 and its one of the interior opposite angles is 55. Find all the angles of the triangle.

One of the exterior angles of a triangle is 80, and the interior opposite angles are equal to each other. What is the measure of each of these two angles?

The exterior angles, obtained on producing the base of a triangle both waysare 104 and 136. Find all the angles of the triangle.

In Fig. 37, the sides BC, CA and BA of a △ABC have been produced to D, E and F respectively. If ∠ACD = 105 and ∠EAF = 45; find all the angles of the △ABC.

In Fig. 38, AC perpendicular to CE and C ∠A: ∠B: ∠C= 3: 2: 1. Find the value of ∠ECD.

A student when asked to measure two exterior angles of △ABC observed that the exterior angles at A and B are of 103 and 74 respectively. Is this possible? Why or why not?

In Fig.39, AD and CF are respectively perpendiculars to sides BC and AB of △ABC. If

∠FCD = 50, find ∠BAD

In Fig.40, measures of some angles are indicated. Find the value of x.

In Fig. 41, ABC is a right triangle right angled at A. D lies on BA produced and DE perpendicular to BC intersecting AC at F. If ∠AFE = 130, find

(i) ∠BDE

(ii) ∠BCA

(iii) ∠ABC

ABC is a triangle in which ∠B = ∠C and ray AX bisects the exterior angle DAC. If

∠DAX = 70. Find ∠ACB.

The side BC of △ABC is produced to a point D. The bisector of ∠A meets side BC in

L. If ∠ABC= 30o and ∠ACD = 115o, find ∠ALC

D is a point on the side BC of △ABC. A line PDQ through D, meets side AC in P and AB produced at Q. If ∠A = 80, ∠ABC = 60 and ∠PDC = 15, find

(i) ∠AQD

(ii) ∠APD

Explain the concept of interior and exterior angles and in each of the figuresgiven below. Find x and y (Fig. 42)

Compute the value of x in each of the following figures:

In each of the following, there are three positive numbers. State if these numbers could possibly be the lengths of the sides of a triangle:

(i) 5, 7, 9

(ii) 2, 10, 15

(iii) 3, 4, 5

(iv) 2, 5, 7

(v) 5, 8, 20

In Fig. 46, P is the point on the side BC. Complete each of the following statements using symbol ‘=’,’ > ‘or ‘< ‘so as to make it true:

(i) AP… AB+ BP

(ii) AP… AC + PC

(iii) AP…. ½ (AB + AC + BC)

P is a point in the interior of △ABC as shown in Fig. 47. State which of the following statements are true (T) or false (F):

(i) AP + PB < AB

(ii) AP + PC > AC

(iii) BP + PC = BC

O is a point in the exterior of △ABC. What symbol ‘>’,’<’ or ‘=’ will you see to complete the statement OA+OB….AB? Write two other similar statements andshow that OA + OB + OC > ½ (AB + BC +CA)

In △ABC, ∠A = 100, ∠B = 30, ∠C = 50. Name the smallest and the largest sides of the triangle.

State Pythagoras theorem and its converse.

In right △ABC, the lengths of the legs are given. Find the length of the hypotenuse

(i) a = 6 cm, b = 8 cm

(ii) a = 8 cm, b = 15 cm

(iii) a = 3 cm, b = 4 cm

(iv) a = 2 cm, b =1.5 cm

The hypotenuse of a triangle is 2.5 cm. If one of the sides is 1.5 cm. find the length of the other side.

A ladder 3.7 m long is placed against a wall in such a way that the foot of the ladder is 1.2 m away from the wall. Find the height of the wall to which the ladder reaches.

If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the triangle is right-angled triangle.

The sides of certain triangles are given below. Determine which of them areright triangles.

(i) a = 7 cm, b = 24 cm and c= 25 cm

(ii) a = 9 cm, b = 16 cm and c = 18 cm

Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12 m. Find the distance between their tops.

A man goes 15 m due west and then 8 m due north. How far is he from the starting point?

The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its top reach?

A ladder 50 dm long when set against the wall of a house just reaches a windowat a height of 48 dm. How far is the lower end of the ladder from the base of the wall?

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

Verity that the following numbers represent Pythagorean triplet:

(i) 12, 35, 37

(ii) 7, 24, 25

(iii) 27, 36, 45 (iv) 15, 36, 39

In △ABC, ∠ABC = 100°, ∠BAC = 35° and BD ⊥ AC meets side AC in D. If BD = 2 cm, find ∠C, and length DC.

In a △ABC, AD is the altitude from A such that AD = 12 cm. BD = 9 cm and DC = 16 cm. Examine if △ABC is right angled at A.

Draw a triangle ABC, with AC = 4 cm, BC = 3 cm and ∠C = 105. Measure AB.

Is (AB)^2 = (AC)^2 + (BC)^2? If not which one of the following is true:

(AB)^2 > (AC)^2 + (BC)^2 or (AB)^2 < (AC)^2 + (BC)^2?

Draw a triangle ABC, with AC = 4 cm, BC = 3 cm and ∠C = 80°. Measure AB.

Is (AB)^2 = (AC)^2 + (BC)^2? If not which one of the following is true:

(AB)^2 > (AC)^2 + (BC)^2 or (AB)^2 < (AC)^2 + (BC)^2?