Triangles 9th Class

Two distinct intersecting lines cannot be parallel to the same line.

Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.

ABC is an isosceles triangle with AB = AC and BD and CE are its two medians. Show that BD = CE.

Q is a point on the side SR of a ∆PSR such that PQ = PR. Prove that PS > PQ.

S is any point on side QR of a ∆PQR. Show that: PQ + QR + RP > 2 PS.

Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Prove that ∠MOC = ∠ABC.

Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC.

Which of the following is not a criterion for congruence of triangles? (A) SAS (B) ASA (C) SSA (D) SSS

 If AB = QR, BC = PR and CA = PQ, then (A) ∆ABC ≅ ∆ PQR (B) ∆CBA ≅ ∆ PRQ (C) ∆BAC ≅ ∆ RPQ (D) ∆PQR ≅ ∆ BCA

In ∆ ABC, AB = AC and ∠B = 50°. Then ∠C is equal to (A) 40° (B) 50° (C) 80° (D) 130°

In ∆ ABC, BC = AB and ∠B = 80°. Then ∠A is equal to (A) 80° (B) 40° (C) 50° (D) 100°

In ∆PQR, ∠R = ∠P and QR = 4 cm and PR = 5 cm. Then the length of PQ is (A) 4 cm (B) 5 cm (C) 2 cm (D) 2.5 cm

D is a point on the side BC of a ∆ABC such that AD bisects ∠BAC. Then (A) BD = CD (B) BA > BD (C) BD > BA (D) CD > CA

It is given that ∆ABC ≅ ∆FDE and AB = 5 cm, ∠B = 40° and ∠A = 80°. Then which of the following is true? (A) DF = 5 cm, ∠F = 60° (B) DF = 5 cm, ∠E = 60° (C) DE = 5 cm, ∠E = 60° (D) DE = 5 cm, ∠D = 40°

Two sides of a triangle are of lengths 5 cm and 1.5 cm. The length of the third side of the triangle cannot be (A) 3.6 cm (B) 4.1 cm (C) 3.8 cm (D) 3.4 cm

In ∆PQR, if ∠R > ∠Q, then (A) QR > PR (B) PQ > PR (C) PQ < PR (D) QR < PR

In triangles ABC and PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are (A) isosceles but not congruent (B) isosceles and congruent (C) congruent but not isosceles (D) neither congruent nor isosceles

In triangles ABC and DEF, AB = FD and ∠A = ∠D. The two triangles will be congruent by SAS axiom if (A) BC = EF (B) AC = DE (C) AC = EF (D) BC = DE

Complete the hexagonal and star shaped rangolies (see the given figures) 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?

In a huge park people are concentrated at three points (see the given figure)

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 parlor should be equidistant from A, B and C)

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

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

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

In the given figure, PR > PQ and PS bisects ∠ QPR. Prove that ∠ PSR >∠ PSQ.

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

In the given figure, ∠ B < ∠ A and ∠ C < ∠ D. Show that AD < BC.

In the given figure sides AB and AC of ΔABC are extended to points P and Q respectively. Also, ∠ PBC < ∠ QCB. Show that AC > AB.

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

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

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

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see the given figure). Show that:

(i) ΔABM ≅ ΔPQN

(ii) ΔABC ≅ ΔPQR

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

(i) AD bisects BC (ii) AD bisects ∠ A.

ΔABC and ΔDBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see the given figure). If AD is extended to intersect BC at P, show that

(i) ΔABD ≅ ΔACD

(ii) ΔABP ≅ ΔACP

(iii) AP bisects ∠ A as well as ∠ D.

(iv) AP is the perpendicular bisector of BC.

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

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

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

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

ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that

(i) ABE ≅ ACF

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

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

In ΔABC, AD is the perpendicular bisector of BC (see the given figure). Show that ΔABC is an isosceles triangle in which AB = AC.

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 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 the given figure). Show that:

(i) ΔAMC ≅ ΔBMD

(ii) ∠DBC is a right angle.

(iii) ΔDBC ≅ ΔACB

(iv) CM = AB

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 the given figure). Show that

(i) ΔDAP ≅ ΔEBP

(ii) AD = BE

In the given figure, AC = AE, AB = AD and ∠ BAD = ∠ EAC. Show that BC = DE.

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 the given figure). Show that:

(i) ΔAPB ≅ ΔAQB

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

l and m are two parallel lines intersected by another pair of parallel lines p and q (see the given figure). Show that ΔABC ≅ ΔCDA.

AD and BC are equal perpendiculars to a line segment AB (See the given figure). Show that CD bisects AB.

ABCD is a quadrilateral in which AD = BC and ∠ DAB = ∠ CBA (See the given figure). Prove that

(i) ΔABD ≅ ΔBAC

(ii) BD = AC

(iii) ∠ ABD = ∠ BAC.

In quadrilateral ACBD, AC = AD and AB bisects ∠ A (See the given figure). Show that ΔABC ≅ ΔABD. What can you say about BC and BD?