Selina solutions for Concise Mathematics [English] Class 9 ICSE chapter 10 - Isosceles Triangles [Latest edition]

In the figure alongside,

AB = AC
∠A = 48°  and
∠ACD = 18° 
Show that BC = CD.

Calculate:

1. ∠ADC
2. ∠ABC
3. ∠BAC

In the following figure, AB = AC; BC = CD and DE are parallel to BC.

Calculate:

1. ∠CDE
2. ∠DCE

Calculate x :

Calculate x :

In the figure, given below, AB = AC.

Prove that: ∠BOC = ∠ACD.

In the figure given below, LM = LN; angle PLN = 110^o.

calculate: (i) ∠LMN
                 (ii) ∠MLN

An isosceles triangle ABC has AC = BC. CD bisects AB at D and ∠CAB = 55°.

Find: 

1. ∠DCB 
2. ∠CBD

Find x :

In the triangle ABC, BD bisects angle B and is perpendicular to AC. If the lengths of the sides of the triangle are expressed in terms of x and y as shown, find the values of x and y.

In the given figure; AE || BD, AC || ED and AB = AC. Find ∠a, ∠b and ∠c.

In the following figure; AC = CD, AD = BD and ∠C = 58°.

Find the angle CAB.

In the figure given below, if AC = AD = CD = BD; find angle ABC.

In triangle ABC; AB = AC and ∠A : ∠B = 8 : 5; find angle A.

In triangle ABC; ∠A = 60^o, ∠C = 40^o, and the bisector of angle ABC meets side AC at point P. Show that BP = CP.

In triangle ABC; angle ABC = 90^o and P is a point on AC such that ∠PBC = ∠PCB.
Show that: PA = PB.

ABC is an equilateral triangle. Its side BC is produced up to point E such that C is mid-point of BE. Calculate the measure of angles ACE and AEC.

In triangle ABC, D is a point in AB such that AC = CD = DB. If ∠B = 28°, find the angle ACD.

In the given figure, AD = AB = AC, BD is parallel to CA and angle ACB = 65°. Find angle DAC.

Prove that a triangle ABC is isosceles, if: altitude AD bisects angles BAC.

Prove that a triangle ABC is isosceles, if: bisector of angle BAC is perpendicular to base BC.

In the given figure; AB = BC and AD = EC.
Prove that: BD = BE.

If the equal sides of an isosceles triangle are produced, prove that the exterior angles so formed are obtuse and equal.

In the given figure, AB = AC.

Prove that:
(i) DP = DQ
(ii) AP = AQ
(iii) AD bisects angle A

In triangle ABC, AB = AC; BE ⊥ AC and CF ⊥ AB.

Prove that:

1. BE = CF
2. AF = AE

In isosceles triangle ABC, AB = AC. The side BA is produced to D such that BA = AD.
Prove that: ∠BCD = 90°

In triangle ABC, AB = AC and ∠A= 36°. If the internal bisector of ∠C meets AB at point D, prove that AD = BC.

If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.

Prove that the bisectors of the base angles of an isosceles triangle are equal.

In the given figure, AB = AC and ∠DBC = ∠ECB = 90°

Prove that: 
(i) BD = CE 
(ii) AD = AE

ABC and DBC are two isosceles triangles on the same side of BC. Prove that:

(i) DA (or AD) produced bisects BC at right angle.

(ii) BDA = CDA.

The bisectors of the equal angles B and C of an isosceles triangle ABC meet at O. Prove that AO bisects angle A.

Prove that the medians corresponding to equal sides of an isosceles triangle are equal.

Use the given figure to prove that, AB = AC.

In the given figure; AE bisects exterior angle CAD and AE is parallel to BC.

Prove that: AB = AC.

In an equilateral triangle ABC; points P, Q and R are taken on the sides AB, BC and CA respectively such that AP = BQ = CR. Prove that triangle PQR is equilateral.

In triangle ABC, altitudes BE and CF are equal. Prove that the triangle is isosceles.

Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. Prove that the triangle so formed is isosceles.

In triangle ABC; AB = AC. P, Q, and R are mid-points of sides AB, AC, and BC respectively.
Prove that: PR = QR

In triangle ABC; AB = AC. P, Q, and R are mid-points of sides AB, AC, and BC respectively.
Prove that: BQ = CP

From the following figure,

prove that: ∠ACD = ∠CBE

From the following figure,

prove that: AD = CE.

Equal sides AB and AC of an isosceles triangle ABC are produced. The bisectors of the exterior angle so formed meet at D. Prove that AD bisects angle A.

ABC is a triangle. The bisector of the angle BCA meets AB in X. A point Y lies on CX such that AX = AY.
Prove that: ∠CAY = ∠ABC.

In the following figure; IA and IB are bisectors of angles CAB and CBA respectively. CP is parallel to IA and CQ is parallel to IB.

Prove that:
PQ = The perimeter of the ΔABC.

Sides AB and AC of a triangle ABC are equal. BC is produced through C up to a point D such that AC = CD. D and A are joined and produced (through vertex A) up to point E. If angle BAE = 108°; find angle ADB.

The given figure shows an equilateral triangle ABC with each side 15 cm. Also, DE || BC, DF || AC, and EG || AB.
If DE + DF + EG = 20 cm, find FG.

If all the three altitudes of a triangle are equal, the triangle is equilateral. Prove it.

In a ΔABC, the internal bisector of angle A meets the opposite side BC at point D. Through vertex C, line CE is drawn parallel to DA which meets BA produced at point E. Show that ΔACE is isosceles.

In triangle ABC, the bisector of angle BAC meets the opposite side BC at point D. If BD = CD, prove that ΔABC is isosceles.

In ΔABC, D is point on BC such that AB = AD = BD = DC.
Show that: ∠ADC : ∠C = 4 : 1.

Using the information given of the following figure, find the values of a and b. [Given: CE = AC] 

Using the information given of the following figure, find the values of a and b.