Selina solutions concise maths class 9 icse Solutions for Chapter 10 chapter 10 isosceles triangle

Exercises in Chapter 10 chapter 10 isosceles triangle Grade 9

Exercise 10(A)

Exercise 10(B)

Questions in Exercise 10(A)

$\begin{aligned} &\text { In the figure alongside, }\\ &A B=A C\\ &\angle \mathrm{A}=48^{\circ} \text { and }\\ &\angle \mathrm{ACP}=18^{\circ} \end{aligned}$

Show that: BC=CD.

Calculate:

In the given figure, AB=AC. Prove that:

(i) DP=DQ

(ii) AP=AQ

(iii) AD bisects angle A

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

Calculate x:

In the figure, given below, AB=AC.

In the figure given below, LM=LN; angle PLN=1100. Calculate:

An isosceles triangle ABC has AC=BC. CD bisects AB at D and CAB=550.

Find:

$(i)<\mathrm{DCB}$

$\text { (ii) } \angle \mathrm{CBD}$

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.

$\text { In the given figure; AE\|BD, AC IIED and AB=AC. Find } \angle \mathrm{a}, \angle \mathrm{b} \text { and } \angle \mathrm{c} \text { . }$

$\text { In the figure of Q.no.11, given above, if } \mathrm{AC}=\mathrm{AD}=\mathrm{CD}=\mathrm{BD} ; \text { find } \angle \mathrm{ABC} \text { . }$

In the following figure, BL=CM.

$\begin{aligned} &\text { In triangle } \mathrm{ABC} ; \angle \mathrm{ABC}=90^{\circ}, \text { and } \mathrm{P} \text { is a point on } \mathrm{AC} \text { such that } \angle \mathrm{PBC}=\angle \mathrm{PCB} \text { . }\\ &\text { Show that } P A=P B \text { . } \end{aligned}$

ABC is an equilateral triangle. Its side BC is produced up to point E such that C is

the midpoint of BE. Calculate the measure of angles ACE and AEC.

$\text { In triangle } A B C, D \text { is a point in } A B \text { such that } A C=C D=D B \text { . If } \angle B=28^{0}, \text { find the angle }$ ACD.

Prove that a triangle ABC is isosceles, if:

$\text { (i) Altitude AD bisects } \angle \mathrm{BAC} \text { , or }$

$\text { (ii) Bisector of } \angle \mathrm{BAC} \text { is perpendicular to base BC. }$

In the given figure;

AB=BC and AD=EC.

Prove that: BD=BE.

Questions in Exercise 10(B)

If the equal sides of an isosceles triangle are produced, prove that the exterior

angles so formed are obtuse and equal.

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

(i) BE=CF

(ii) AF=AE

An isosceles triangle ABC, AB=AC. The side BA is produced to D such that

BA=AD. Prove that: $\angle B C D=90^{\circ}$

(i) In a triangle ABC, AB=AC, and ∠A=360. If the internal bisector of ∠C meets

AB at point D, proves that AD=BC.

(ii) 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.

The bisectors of the equal angles B and C of an isosceles triangle ABC meet at O.

Prove that AO bisects angle A.

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

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

(ii) ∠BDA=∠CDA

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 is 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:

(i) PR= QR

(ii) BQ = CP

Equal sides AB and AC of an isosceles triangle ABC are produced. The bisectors

of the exterior angles 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.

$\text { Prove that: } \angle \mathrm{CAY}=\angle \mathrm{ABC}$

In the following figures; 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 triangle ABC.

Sides AB and AC of a triangle ABC are equal. BC is produced through C up to

point D such that AC=CD. D and A are joined and produced (through vertex A)

up to point E. If angle BAE=1080; 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 all the three altitudes of a triangle are equal, the triangles are equilateral. Prove

it.

In a triangle, ABC, 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 triangle ACE is isosceles.

In triangle ABC, the bisector of angle BAC meets the opposite side BC at point D. If BD =

CD, prove that triangle ABC is isosceles.

In triangle ABC, D is a point on BC such that AB = AD = BD= DC. Show that:

$\angle \mathrm{ADC}: \angle \mathrm{C}=4: 1$

Using the information given in each of the following figures, find the value of a

and b.

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Chapters in this book

Chapter 1- Rational and Irrational Numbers

Chapter 2- Compound Interest [Without Using Formula

Chapter 3- Compound Interest [Using Formula

Chapter 4- Expansions

Chapter 5- Factorisation

Chapter 6- Simultaneous Equations

Chapter 7- Indices [exponents]

Chapter 8- Logarithms

Chapter 9- Triangles [Congruency in Triangles]

Chapter 10- Isosceles Triangle

Chapter 11- Inequalities

Chapter 12- Mid-Point and Its Converse

Chapter 13- Pythagoras Theorem

Chapter 14- Rectilinear Figures

Chapter 15- Construction of Polygons

Chapter 16- Area Theorems

Chapter 17- Circles

Chapter 18- Statistics

Chapter 19- Mean and Median

Chapter 20- Area and Perimeter of Plane Figures

Chapter 21- Solids

Chapter 22- Trigonometrical Ratios

Chapter 23- Trigonometrical Ratios of Standard Angles

Chapter 24- Solution Of Right Triangles

Chapter 25- Complementary angles

Chapter 26- Co-ordinate Geometry

Chapter 27- Graphical Solution

Chapter 28- Distance Formula