NCERT Exemplar Solutions
Class 7 - Mathematics

Unit 6: Triangles

The ______ triangle always has altitude outside itself.

The sum of an exterior angle of a triangle and its adjacent angle is always ______.

The longest side of a right angled triangle is called its ______.

Median is also called ______ in an equilateral triangle.

Measures of each of the angles of an equilateral triangle is ______.

In an isosceles triangle, two angles are always ______.

In an isosceles triangle, angles opposite to equal sides are ______.

If one angle of a triangle is equal to the sum of other two, then the measure of that angle is ______.

Every triangle has at least ______ acute angle(s).

Two line segments are congruent, if they are of ______ lengths.

Two angles are said to be ______, if they have equal measures.

Two rectangles are congruent, if they have same ______ and ______.

Two squares are congruent, if they have same ______.

If ΔPQR and ΔXYZ are congruent under the correspondence QPR ↔ XYZ, then

In Fig. 6.22, ΔPQR ≅ Δ ______

In Fig. 6.23, ΔPQR ≅ Δ ______

In Fig. 6.24, Δ ______ ≅ Δ PQR

In Fig. 6.25, ΔARO ≅ Δ ______

In Fig. 6.26, AB = AD and ∠BAC = ∠DAC. Then

In Fig. 6.27,

In a triangle, sum of squares of two sides is equal to the square of the third side.

Sum of two sides of a triangle is greater than or equal to the third side.

The difference between the lengths of any two sides of a triangle is smaller than the length of third side.

In ΔABC, AB = 3.5 cm, AC = 5 cm, BC = 6 cm and in ΔPQR, PR = 3.5 cm, PQ = 5 cm, RQ = 6 cm. Then ΔABC ≅ ΔPQR.

Sum of any two angles of a triangle is always greater than the third angle.

The sum of the measures of three angles of a triangle is greater than 180°.

It is possible to have a right-angled equilateral triangle.

If M is the mid-point of a line segment AB, then we can say that AM and MB are congruent.

It is possible to have a triangle in which two of the angles are right angles.

It is possible to have a triangle in which two of the angles are obtuse.

It is possible to have a triangle in which two angles are acute.

It is possible to have a triangle in which each angle is less than 60°.

It is possible to have a triangle in which each angle is greater than 60°.

It is possible to have a triangle in which each angle is equal to 60°.

A right-angled triangle may have all sides equal.

If two angles of a triangle are equal, the third angle is also equal to each of the other two angles.

In Fig. 6.28, two triangles are congruent by RHS.

The congruent figures super impose each other completely.

A one rupee coin is congruent to a five rupee coin.

The top and bottom faces of a kaleidoscope are congruent.

Two acute angles are congruent.

Two right angles are congruent.

Two figures are congruent, if they have the same shape.

If the areas of two squares is same, they are congruent.

If the areas of two rectangles are same, they are congruent.

If the areas of two circles are the same, they are congruent.

Two squares having same perimeter are congruent.

Two circles having same circumference are congruent.

If three angles of two triangles are equal, triangles are congruent.

If two legs of a right triangle are equal to two legs of another right triangle, then the right triangles are congruent.

If two sides and one angle of a triangle are equal to the two sides and angle of another triangle, then the two triangles are congruent.

If two triangles are congruent, then the corresponding angles are equal.

If two angles and a side of a triangle are equal to two angles and a side of another triangle, then the triangles are congruent.

If the hypotenuse of one right triangle is equal to the hypotenuse of another right triangle, then the triangles are congruent.

If hypotenuse and an acute angle of one right triangle are equal to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent.

AAS congruence criterion is same as ASA congruence criterion.

In Fig. 6.29, AD ⟂ BC and AD is the bisector of angle BAC. Then, ΔABD ≅ ΔACD by RHS.

The measure of three angles of a triangle are in the ratio 5 : 3 : 1. Find the measures of these angles.

In Fig. 6.30, find the value of x.

In Fig. 6.31(i) and (ii), find the values of a, b and c.

In triangle XYZ, the measure of angle X is 30° greater than the measure of angle Y and angle Z is a right angle. Find the measure of ∠Y.

In a triangle ABC, the measure of angle A is 40° less than the measure of angle B and 50° less than that of angle C. Find the measure of ∠A.

I have three sides. One of my angle measures 15°. Another has a measure of 60°. What kind of a polygon am I? If I am a triangle, then what kind of triangle am I?

Jiya walks 6 km due east and then 8 km due north. How far is she from her starting place?

Jayanti takes shortest route to her home by walking diagonally across a rectangular park. The park measures 60 metres × 80 metres. How much shorter is the route across the park than the route around its edges?

In ΔPQR of Fig. 6.32, PQ = PR. Find the measures of ∠Q and ∠R.

In Fig. 6.33, find the measures of ∠x and ∠y.

In Fig. 6.34, find the measures of ∠PON and ∠NPO.

In Fig. 6.35, QP ∥ RT. Find the values of x and y.

Find the measure of ∠A in Fig. 6.36.

In a right-angled triangle if an angle measures 35°, then find the measure of the third angle.

Each of the two equal angles of an isosceles triangle is four times the third angle. Find the angles of the triangle.

The angles of a triangle are in the ratio 2 : 3 : 5. Find the angles.

If the sides of a triangle are produced in an order, show that the sum of the exterior angles so formed is 360°.

In ΔABC, if ∠A = ∠C, and exterior angle ABX = 140°, then find the angles of the triangle.

Find the values of x and y in Fig. 6.37.

Find the value of x in Fig. 6.38.

The angles of a triangle are arranged in descending order of their magnitudes. If the difference between two consecutive angles is 10°, find the three angles.

In ΔABC, DE ∥ BC (Fig. 6.39). Find the values of x, y and z.

In Fig. 6.40, find the values of x, y and z.

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

In ΔPQR, if 3∠P = 4∠Q = 6∠R, calculate the angles of the triangle.

In ΔDEF, ∠D = 60°, ∠E = 70° and the bisectors of ∠E and ∠F meet at O. Find (i) ∠F (ii) ∠EOF.

In Fig. 6.41, ΔPQR is right-angled at P. U and T are the points on line QRF. If QP ∥ ST and US ∥ RP, find ∠S.

In each of the given pairs of triangles of Fig. 6.42, applying only ASA congruence criterion, determine which triangles are congruent. Also, write the congruent triangles in symbolic form.

In each of the given pairs of triangles of Fig. 6.43, using only RHS congruence criterion, determine which pairs of triangles are congruent. In case of congruence, write the result in symbolic form:

In Fig. 6.44, if RP = RQ, find the value of x.

In Fig. 6.45, if ST = SU, then find the values of x and y.

Check whether the following measures (in cm) can be the sides of a right-angled triangle or not: 1.5, 3.6, 3.9

Height of a pole is 8 m. Find the length of rope tied with its top from a point on the ground at a distance of 6 m from its bottom.

In Fig. 6.46, if y is five times x, find the value of z.

The lengths of two sides of an isosceles triangle are 9 cm and 20 cm. What is the perimeter of the triangle? Give reason.

Without drawing the triangles write all six pairs of equal measures in each of the following pairs of congruent triangles.

In the following pairs of triangles of Fig. 6.47, the lengths of the sides are indicated along the sides. By applying SSS congruence criterion, determine which triangles are congruent. If congruent, write the results in symbolic form.

ABC is an isosceles triangle with AB = AC and D is the mid-point of base BC (Fig. 6.48).

(a) State three pairs of equal parts in the triangles ABD and ACD.

(b) Is ΔABD ≅ ΔACD? If so why?

In Fig. 6.49, it is given that LM = ON and NL = MO. (a) State the three pairs of equal parts in the triangles NOM and MLN. (b) Is ΔNOM ≅ ΔMLN? Give reason.

Triangles DEF and LMN are both isosceles with DE = DF and LM = LN, respectively. If DE = LM and EF = MN, then, are the two triangles congruent? Which condition do you use? If ∠E = 40°, what is the measure of ∠N?

If ΔPQR and ΔSQR are both isosceles triangles on a common base QR such that P and S lie on the same side of QR, are triangles PSQ and PSR congruent? Which condition do you use?

In Fig. 6.50, which pairs of triangles are congruent by SAS congruence criterion? If congruent, write the congruence of the two triangles in symbolic form.

State which of the following pairs of triangles are congruent. If yes, write them in symbolic form.

In Fig. 6.51, PQ = PS and ∠1 = ∠2.

(i) Is ΔPQR ≅ ΔPSR? Give reasons.

(ii) Is QR = SR? Give reasons.

In Fig. 6.52, DE = IH, EG = FI and ∠E = ∠I. Is ΔDEF ≅ ΔHIG? If yes, by which congruence criterion?

In Fig. 6.53, ∠1 = ∠2 and ∠3 = ∠4. (i) Is ΔADC ≅ ΔABC? Why? (ii) Show that AD = AB and CD = CB.

Observe Fig. 6.54 and state the three pairs of equal parts in triangles ABC and DBC.

(i) Is ΔABC ≅ ΔDCB? Why?

(ii) Is AB = DC? Why?

(iii) Is AC = DB? Why?

In Fig. 6.55, QS ⟂ PR, RT ⟂ PQ and QS = RT.

(i) Is ΔQSR ≅ ΔRTQ? Give reasons.

(ii) Is ∠PQR = ∠PRQ? Give reasons.

Points A and B are on the opposite edges of a pond as shown in Fig. 6.56. To find the distance between the two points, the surveyor makes a right-angled triangle as shown. Find the distance AB.

Two poles of 10 m and 15 m stand upright on a plane ground. If the distance between the tops is 13 m, find the distance between their feet.

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

In Fig. 6.57, state the three pairs of equal parts in ΔABC and ΔEOD. Is ΔABC ≅ ΔEOD? Why?