Solution of Chapter 7. Triangles (NCERT - Mathematics Book)

In quadrilateral ACBD,

AC = AD and AB bisects ∠ A (see Fig. 7.16). Show that Δ ABC ≅Δ ABD.

What can you say about BC and BD?

ABCD is a quadrilateral in which AD = BC and∠ DAB = ∠ CBA (see Fig. 7.17). Prove that

(i) Δ ABD ≅Δ BAC

(ii) BD = AC

(iii) ∠ ABD = ∠ BAC.

AD and BC are equal perpendiculars to a line segment AB (see Fig. 7.18). Show that CD bisects AB.

l and m are two parallel lines intersected by another pair of parallel lines p and q(see Fig. 7.19). Show that Δ ABC ≅Δ CDA.

Line l is the bisector of an angle ∠ A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠ A (see Fig. 7.20). Show that:

(i) Δ APB ≅Δ AQB

(ii) BP = BQ or B is equidistant from the arms of ∠ A.

In Fig. 7.21, AC = AE, AB = AD and∠ BAD = ∠ EAC. Show that BC = DE.

AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that∠ BAD = ∠ ABE and ∠ EPA = ∠ DPB (see Fig. 7.22). Show that

(i) Δ DAP ≅Δ EBP

(ii) AD = BE

In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B(see Fig. 7.23). Show that:

(i) Δ AMC ≅Δ BMD

(ii) ∠DBC is a right angle.

(iii) Δ DBC ≅Δ ACB

(iv) CM =AB

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠ B and ∠ C intersect each other at O. Join A to O. Show that:

(i) OB = OC

(ii) AO bisects ∠ A

In Δ ABC, AD is the perpendicular bisector of BC (see Fig. 7.30). Show that Δ ABC is an isosceles triangle in which AB = AC.

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see Fig. 7.31). Show that these altitudes are equal.

ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see Fig. 7.32) Show that:

(i) Δ ABE ≅Δ ACF

(ii) AB = AC, i.e., ABC is an isosceles triangle

ABC and DBC are two isosceles triangles on the same base BC (see Fig. 7.33). Show that∠ ABD = ∠ ACD

ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see Fig. 7.34). Show that ∠ BCD is a right angle.

ABC is a right angled triangle in which ∠ A = 90°and AB = AC. Find ∠ B and ∠ C.

Show that the angles of an equilateral triangle are 60° each.

Δ ABC and Δ DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig. 7.39). If AD is extended to intersect BC at P, show that

(i) Δ ABD ≅Δ ACD

(ii) Δ ABP ≅Δ ACP

(iii) AP bisects ∠ A as well as ∠ D.

(iv) AP is the perpendicular bisector of BC.

AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that

(i) AD bisects BC

(ii) AD bisects ∠ A

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of Δ PQR (see Fig. 7.40). Show that:

(i) Δ ABM ≅Δ PQN

(ii) Δ ABC ≅Δ PQR

BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.

ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that∠ B = ∠ C.

Show that in a right angled triangle, the hypotenuse is the longest side.

In Fig. 7.48, sides AB and AC of Δ ABC are extended to points P and Q respectively. Also, ∠ PBC <∠ QCB. Show that AC > AB.

In Fig. 7.49, ∠ B <∠ A and ∠ C <∠ D. Show that AD < BC

AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (see Fig. 7.50). Show that ∠ A >∠ C and∠ B >∠ D

In Fig 7.51, PR > PQ and PS bisects ∠ QPR. Prove that ∠ PSR >∠ PSQ

Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

ABC is a triangle. Locate a point in the interior of Δ ABC which is equidistant from all the vertices of Δ ABC.

In a triangle locate a point in its interior which is equidistant from all the sides of the triangle.

In a huge park, people are concentrated at three points (see Fig. 7.52):

A: Where there are different slides and swings for children,

B: Near which a man-made lake is situated,

C: Which is near to a large parking and exit?

Where should an ice cream parlour be set up so that maximum number of persons can approach it?

(Hint: The parlour should be equidistant from A, B and C)

Complete the hexagonal and star shaped Rangolies [see Fig. 7.53 (i) and (ii)] by filling them with as many equilateral triangles of side 1 cm as you can. Count the number of triangles in each case. Which has more triangles?