In Fig. 10.22, the sides BA and CA have been produced such that BA = AD and CA = AE. Prove that segment DE||BC.

In a Δ PQR, if PQ=QR and L, M and N are the mid points of the sides PQ, QR and RP respectively, Prove that LN=MN.

In Fig. 10.23, PQRS is a square and SRT is an equilateral triangle. Prove that

(i) PT = QT (ii) ∠TQR =15°

Prove that the medians of an equilateral triangle are equal.

In a Δ ABC, if ∠A =120° and AB=AC. Find ∠B and ∠C.

In a Δ ABC, if AB =BC 120° and ∠B = 70°, Find ∠A.

The vertical angle of an isosceles triangle is 100°. Find its base angles.

In Fig. 10.24, AB = AC and ∠ACD = 105°, find ∠BAC.

Find the measure of each exterior angle of an equilateral triangle.

If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.

In Fig. 10.25, AB = AC and DB = DC, find the ratio ∠ABD:∠ACD.

Determine the measure of each of the equal angles of a right angled isosceles triangle.

OR

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

AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig. 10.26). Show that the line PQ is perpendicular bisector of AB.

In Fig. 10.40, it is given that RT=TS, ∠1 =2∠2 and ∠4 = 2∠3. Prove that Δ RBT ≅ Δ SAT.

Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.

BD and CE are bisectors of ∠B and ∠C of an isosceles Δ ABC with AB = BC. Prove that BD = CE.

In two triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are congruent.

If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.

In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.

PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.

In a Δ ABC, it is given that AB = AC and the bisectors of ∠B and ∠C intersect at O. If M is a point on BO produced, prove that ∠MOC = ∠ABC.

P is a point on the bisector of an angle ∠ABC. If the line through P parallel to AB meets BC at Q, prove that the triangle BPQ is isosceles.

Prove that each angle of an equilateral triangle is 60°.

Angles A, B, C of a triangle ABC are equal to each other. Prove that Δ ABC is equilateral.

ABC is a triangle in which ∠B=2∠C. D is a point on BC such that AD bisects ∠BAC and AB = CD. Prove that ∠BAC = 72°.

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

In Fig. 10.92, it is given that AB = CD and AD = BC. Prove that Δ ADC ≅ Δ CBA.

In Δ PQR, if PQ=QR and L, M and N are the mid-point of the sides PQ, QR and RP respectively. Prove that LN=MN.

ABC is a triangle and D is the mid-point of BC. The perpendicular from D to AB and AC are equal. Prove that the triangle is isosceles.

ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE=CF, prove that ΔABC is isosceles.

If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.

In Fig. 10.99, AD ⊥ CD and CB ⊥ CD. If AQ=BP and DP =CQ, prove that ∠DAQ = ∠CBP.

ABCD is a square, X and Y are points on sides AD and BC respectively such that AY = BX. Prove that BY = AX and ∠BAY = ∠ABX.

Which of the following statements are true (T) and which are false (F):

(i) Sides opposite to equal angles of a triangle may be unequal.

(ii) Angles opposite to equal sides of a triangle are equal.

(iii) The measure of each angle of an equilateral triangle is 60°.

(iv) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.

(v) The bisectors of two equal angles of a triangle are equal.

(vi) If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.

(vii) The two altitudes corresponding to two equal sides of a triangle need not be equal.

(viii) If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent.

(ix) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.

Fill in the blanks in the following so that each of the following statements is true.

(i) Sides opposite to equal angles of a triangle are ………

(i) Sides opposite to equal angles of a triangle are …………..

(iii) In an equilateral triangle all angles are ……….

(iv) In a Δ ABC, if ∠A=∠C, then AB = ………

(v) If altitudes CE and BF of a triangle ABC are equal, then, AB = ……..

(vi) In an isosceles triangle ABC with AB=AC, if BD and CE are its altitudes, then BD is ….CE.

(vii) In right triangles ABC and DEF, if hypotenuse AB=EF and side AC=DE, then Δ ABC ≅ Δ …….

In Δ ABC, if ∠A=40° and ∠B=60°. Determine the longest and shortest sides of the triangle.

In a Δ ABC, if ∠B=∠C =45°, which is the longest side?

In Δ ABC, side AB is produced to D so that BD=BC. If ∠B=60° and ∠A=70°, prove that:

(i) AD > CD (ii) AD > AC

Is it possible to draw a triangle with sides of length 2 cm, 3 cm, and 7 cm?

In Δ ABC, ∠B=35°, ∠C=65° and the bisector of ∠ABC meets BC in P. Arrange AP, BP and CP in descending order.

O is any point in the interior of Δ ABC. Prove that

(i) AB + AC > OB + OC

(ii) AB + BC + CA > OA + OB + OC

(iii) OA + OB + OC > (AB + BC + CA)

Prove that the perimeter of a triangle is greater than the sum of its altitudes.

Prove that in a quadrilateral the sum of all the sides is greater than the sum of its diagonals.

In Fig. 10.131, prove that:

(i) CD + DA + AB + BC > 2AC

(ii) CD+DA +AB > BC

Which of the following statements are true (T) and which are false (F)?

(i) Sum of the three sides of a triangle is less than the sum of its three altitudes.

(ii) Sum of any two sides of a triangle is greater than twice the median drawn to the third side.

(iii) Sum of any two sides of a triangle is greater than the third side.

(iv) Difference of any two sides of a triangle is equal to the third side.

(v) If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it.

(vi) Of all the line segments that can be drawn from a point to a line not containing it, the perpendicular line segment is the shortest one.

Fill in the blanks to make the following statements true:

(i) In a right triangle the hypotenuse is the …….side.

(ii) The sum of three altitudes of a triangle is …… than its perimeter.

(iii) The sum of any two sides of a triangle is ….. than the third side.

(iv) If two angles of a triangle are unequal, then the smaller angle has the …. side opposite to it.

(v) Difference of any two sides of a triangle is ……. than the third side.

(vi) If two sides of a triangle are unequal, then the larger side has ……. angle opposite to it.

In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.

In two triangles ABC and DEF, it is given that ∠A = ∠D, ∠B = ∠E and ∠C = ∠F. Are the two triangles necessarily congruent?

If ABC and DEF are two triangles such that AC = 2.5 cm, BC = 5 cm, ∠C = 75°, DE = 2.5 cm, DF = 5 cm and ∠D = 75°. Are two triangles congruent?

In two triangles ABC and ADC, if AB = AD and BC = CD. Are they congruent?

In triangles ABC and CDE, if AC = CE, BC = CD, ∠A= 60°, ∠C= 30° and ∠D = 90°. Are two triangles congruent?

ABC is an isosceles triangle in which AB = AC. BE and CF are its two medians. Show that BE=CF.

Find the measure of each angle of an equilateral triangle.

CDE is an equilateral triangle formed on a side CD of a square ABCD. Show that Δ ADE ≅ Δ BCE.

Prove that the sum of three altitudes of a triangle is less than the sum of its sides.

In Fig. 10.134, if AB = AC and ∠B = ∠C. Prove that BQ = CP.

If Δ ABC ≅ Δ LKM, then side of Δ LKM equal to side AC of Δ ABC is

If Δ ABC ≅ Δ ACB, then Δ ABC is isosceles with

If Δ ABC ≅ Δ PQR and Δ ABC is not congruent to Δ PQR, then which of the following not true:

In triangle ABC and PQR three equality relations between some parts are as follows:

AB = QP, ∠B = ∠P and BC = PR

State which of the congruence conditions applies:

In triangles ABC and PQR, if ∠A=∠R, ∠B=∠P and AB=RP, then which one of the following congruence conditions applies:

In Δ PQR ≅ Δ EFD then ED =

In Δ PQR ≅ Δ EFD then ∠E =

In a Δ ABC, if AB = AC and BC is produced to D such that ∠ACD =100°, then ∠A =

In an isosceles triangle, if the vertex angle is twice the sum of the base angles, then the measure of vertex angle of the triangle is

D, E, F are the mid-point of the sides BC, CA and AB respectively of Δ ABC. Then Δ DEF is congruent to triangle

Which of the following is not criterion for congruence of triangles?

In Fig. 10.135, the measure of ∠B’AC’ is

If ABC and DEF are two triangles such that Δ ABC≅ Δ FDE and AB =5 cm, ∠B = 40° and ∠A =80°, Then, which of the following is true?

In Fig. 10.136, AB ⊥ BE and FE ⊥ BE. If BC=DE and AB=EF, then Δ ABD is congruent to

ABC, is an isosceles triangle such that AB=AC and AD is the median to base BC. Then, ∠BAD=

In Fig. 10.138, if AE||DC and AB=AC, the value of ∠ABD is

In Fig. 10.139, ABC is an isosceles triangle whose side AC is produced to E. Through C, CD is drawn parallel to BA. The value of x is

In Fig. 10.140, X is a point in the interior of square ABCD.AXYZ is also a square. If DY = 3 cm and AZ = 2 cm, then BY =

In Fig. 10.141, ABC is a triangle in which ∠B =2∠C. D is a point on side such that AD bisects ∠BAC and AB=CD. BE is the bisector of ∠B. The measure of ∠BAC is

[Hint: Δ ABE ≅ Δ DCE]

In Fig. 10.142, if AC is bisector of ∠BAD such that AB=3 cm and AC=5 cm, then CD=