Properties of An Isosceles Triangle - Triangle, Class 9, Mathematics | Extra Documents & Tests for Class 9 PDF Download

1. In ΔBAD and ΔCAD
    (i) AB = AC
    (ii) ∠BAD = ∠CAD
    (iii) AD = AD

Given
Construction
Common

STATEMENT

REASON

1. AB = AC = BC

ΔABC is an equilateral triangle.

2. ∠B = ∠C

AB = AC

3. ∠A = ∠C

AB = BC

4. ∠A = ∠B = ∠C

From (2) and (3)

5. ∠A + ∠B + ∠C = 180°
⇒ ∠A=∠B=∠C= 1/3 × 180°=60°

Angle sum property.

1. In ΔABD and ΔACD
(i) ∠ABD = ∠ACD
(ii) ∠ADB = ∠ADC
(iii) AD = AD

Given
Each = 90°
Common

STATEMENT

REASON

1. AB = A C 

  ⇒ ∠ABC = ∠ACB

Given

∠s opposite to equal sides of a D are equal

2. A B = AD

Given

3. AC = A D
 ⇒ ∠CD A = ∠AC D
or ∠CDB = ∠ACD

From (1) & (2)

∠s opposite to equal sides of a D are equal

4. ∠A B C + ∠C D B = ∠A C B + ∠A C D
 ⇒ ∠AB C + ∠CD B = ∠BC D

Adding (1) & (3)

5. In ΔBCD ∠BCD + ∠DBC + ∠CDB = 180°
 ⇒ ∠BCD + ∠ABC + ∠CDB = 180°
 ⇒ ∠BCD + ∠BCD = 180°
 ⇒ 2∠BCD = 180°
 ⇒ ∠BCD = 90°
 ⇒ ∠BCD is a right angle.

Angle sum property

Using (4)

STATEMENT

REASON

1. In ΔABD and ΔABC
(i) BD = BC
(ii) ∠ABD = ∠ABC
(iii) AB = AB

By construction.

Each = 90°

Common

2. ΔABD ≌  ΔA B C

By SAS criteria

3. AD = AC, ∠DAB = ∠CAB = x°
 ⇒ ∠DA C = ∠AC B

C.P.C.T

4. A D = D C

Sides opposite to equal ∠s are equal.

5. A C = D C

AD = AC, from (3)

6. AC = 2 BC

DC = BD + BC = 2BC, as BD = BC.

STATEMENT

REASON

1. A B = A D

By construction

2. ∠A B D = ∠B D A

∠s opposite to equal sides of a D are equal

3. ∠B D A > ∠D C B

(Ext. ∠ of ΔBCD) > (Each of its int. Opp. ∠s)

4. ∠A B D > ∠D C B

Using (2)

5. ∠AB C > ∠AB D

∠ABD is a part of ∠ABC.

6. ∠AB C > ∠DC B

Using (5)

7. ∠ABC > ∠AC B

∠DCB = ∠ACB.

STATEMENT

REASON

We may have three possibilities only : (i) AC = AB
(ii) AC < AB
(iii) AC > AB
Out of these exactly one must be true.

AC = AB.

Case-I. 

 ⇒ ∠ABC = ∠ACB.

This is contrary to what is given.∴ AC = AB is not true.

∠s opposite to equal sides of a D are equal

Case-II. AC < AB.  ⇒ AB > AC
 ⇒ ∠ACB > ∠ABC

This is contrary to what is given. ∴ AC < AB is not true.
Thus, we are left with the only possibility AC > AB, which must be true.

 Greater side has greater angle opp. to it.

STATEMENT

REASON

1. A D = A C
 ⇒ ∠AC D = ∠AD C

By construction

2. ∠B C D > ∠A C D

 ∠s opposite to equal sides of a D are equal

3. ∠B C D > ∠A D C
 ⇒ BD > B C
 ⇒ BA + AD > BC
 ⇒ BA + AC > BC
or AB + AC > BC.

Using (1) & (2)

Greater angle has greater side opp. to it.

BAD is a straight line, BD = BA + AD.

4. Similarly, AB + BC > AC and BC + AC > AB.

AD = AC, by construction

STATEMENT

REASON

1. ∠A + ∠B = 100°
 ⇒ ∠A + 40° = 100°
 ⇒ ∠A = 60°

Ext. ∠ = sum of int. opt. ∠s
Linear pair of angles.

2. ∠C + 100° = 180°
 ⇒ ∠C = 80°

∠C = 80° and ∠B = 40°
Greater angle has greater side opp. to it.

3. ∠C > ∠B
 ⇒ AB > AC

∠C = 80° and ∠A = 60°
Greater angle has greater side opp. to it.

4. ∠C > ∠A
 ⇒ AB > BC

∠A = 60° and ∠B = 40°

5. ∠A > ∠B
 ⇒ BC > AC

STATEMENT

REASON

1. ∠PMN = 90°

P M ⊥ A B

2. ∠PNM < 90°

In a right D, one angle measures 90° and each one of the remaining two is acute.

3. ∠P N M < ∠P M N

From (1) and (2)

4. P M < P N

Side opp. the smaller angle is smaller.

THINGS TO REMEMBER

1. Two figures are congruent, if they are of the same shape and of the same size.

2. Two circles of the same radii are congruent.

3. Two squares of the same sides are congruent.

4. If two triangles ABC and PQR are congruent under the correspondence A ⇔  P, B ⇔  Q and C ⇔  R, then symbolically, it is expressed as Δ ABC ≌ Δ PQR.

5. If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent (SAS congruence rule).

6. If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent (ASA congruence rule).

7. If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent (AAS Congruence rule).

8. Angle opposite to equal sides of a triangle are equal.

9. Sides opposite to equal angles of a triangle are equal.

10. Each angle of an equilateral triangle is of 60°.

11. If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent (SSS congruence rule).

12. If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent (RHS congruence rule).

13. In a triangle, angle opposite to the longer side is larger (greater).

14. In a triangle, side opposite to the larger (greater) angle is longer.

15. Sum of any two sides of a triangle is greater than the third side.