Triangle

A closed figure with three sides is called a Triangle. It has three vertex, sides and Angles.

Points to remember:

• Triangle is a two-dimensional and three-sided polygon that has three edges and three vertices.
• The sum of all the angles of a triangle is 180°.
• Scalene Triangle, Isosceles Triangle, and Equilateral Triangle are the three types of triangles based on their sides.
• On the basis of angles, the types of triangles are Right-angled Triangle, Acute-angled Triangle, and Obtuse-angled Triangle.
• Area of a triangle is given as A = ½ x Base x Height.

Ques. State any three important properties of triangles. (3 Marks)

Ans. The three important properties of triangles are as follows:

• The sum of the angles of the triangle is equal to 180 degrees.
• The exterior angles of a triangle give a sum of 360 degrees.
• The shortest side of a triangle is always opposite the smallest angle of a triangle.

Pythagoras Theorem

The square of the hypotenuse in a right-angled triangle is equal to the sum of squares of the other two sides.

In ABC, if  C is a right angle triangle ,  then C² = a² + b²

1. There are three types of triangles on the basis of the length of the sides.

2. There are three types of triangles on the basis of angles.

Congruence

If the shape and size of two figures are same then these are called Congruent.

1. Two circles are congruent if their radii are same.

2. Two squares are congruent if their sides are equal.

Congruence of Triangles

A triangle will be congruent if its corresponding sides and angles are equal. The symbol of congruent is “≅”.

Two triangles are said to be congruent if their all-corresponding sides and angles are equal.

Sides: AB=BC=AC=PQ=QR=PR

Angles: ∠A=∠B=∠C=∠P=∠Q=∠R

So, we can say both triangles ABC and PQR are congruent.

AB = DE, BC = EF, AC = DF

m∠A = m∠D, m∠B = m∠E, m∠C = m∠F

Here ∆ABC ≅ ∆DEF

Criteria for Congruence of Triangles:

1. SSS (Side-Side-Side) – Two triangles are said to be congruent by SSS condition if all three sides are equal.
2. SAS(Side-Angle-Side)- Two triangles are said to be congruent by SAS condition if their two sides and 1 angle are equal.
3. ASA(Angle-Side-Angle)- Two triangles are said to be congruent by ASA condition if their two angles and 1 side are equal.
4. AAS(Angle-Angle-Side)- Two triangles are said to be congruent by AAS condition if their two angles and 1 side are equal.
5. RHS (Right angle- Hypotenuse-Side)- If the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent

Remark

1. SSA and ASS do not show the congruency of triangles.

2. AAA is also not the right condition to prove that the triangles are congruent.

Triangle Similarity Theorems

Triangle Similarity Theorems aim at proving the similarity between any two given triangles. Under these theorems, various conditions are stated according to which two or more triangles can be proved similar. Similar triangles are depicted using the congruence sign as ΔABC~ ΔQPR.

The three main triangle similarity theorems are:

SSS Similarity Theorem

• It is the Side-Side-Side similarity theorem.
• When all three sides of two triangles have the same length or are in the same proportion, they are said to be congruent.
• For the sides to be congruent, all three sides need to be proportionate.

AA Similarity Theorem

• It is the Angle-Angle similarity theorem.
• When two or three angles of the triangle are of the same measure, they are congruent.
• Even when only two of the angles are the same, by the Triangle Sum Property, the third angle is itself the same.

SAS Similarity Theorem

• It is the Side-Angle-Side similarity theorem.
• When two sides are proportional in a triangle, and the angle between the two sides is equal, then the triangles are congruent.

Example

Find the ∠P, ∠R, ∠N and ∠M if ∆LMN ≅ ∆PQR.

Solution

If ∆ LMN ≅ ∆PQR, then

∠L=∠P

∠M =∠Q

∠N =∠R

So,

∠L=∠P = 105°

∠M =∠Q = 45°

∠M + ∠N + ∠L = 180° (Sum of three angles of a triangle is 180°)

45° + 105° + ∠N = 180°

∠N = 180°- 45° + 105°

∠N = 30°

∠N = ∠R = 30°

Ques1: In triangle ABC, if AB=AC and ∠B= 70°, then what is the value of ∠A? (4 Marks)

Ans. In a Δ ABC

AB=AC (Given)

∠B=∠C= 70°, (Angle opposite to equal sides are equal)

As per angle sum property some of 3 angles of a triangle is 180 degrees.

∠A + ∠B +∠C= 180°

∠A+70°+70°= 180°

∠A = 180° – 140°

∠A = 40°

Ques2: What is CPCT? (2 Marks)

Ans. CPCT stands for Corresponding Parts of Congruent Triangles. It is a theorem which states that if two or more triangles which are congruent to each other then the corresponding angles and sides are also congruent to each other.

Ques3: Is it possible two equilateral triangles are always congruent? (2 Marks)

Ans. No it is not always possible. Equilateral triangle angles are of 60 degrees each but it is noy necessary that their sides are always equal.

Ques4: The length of two sides of an isosceles triangle are 5cm and 8 cm, find perimeter of triangle. (3 Marks)

Ans. If we consider two sides of isosceles triangles are 5 cm and third one 8 cm then perimeter will be 5 cm+5 cm+8 cm= 18 cm

If we consider two sides of isosceles triangles are 8 cm and third one 5 cm then perimeter will be 8 +8 +5 = 21 cm.

Ques 5: In the given figure, find these two triangles LMN and ONM are congruent and by which rule? (4 Marks)

Ans. In ΔLMN and ΔONM

LM=ON

LN=OM

MN=NM

By SSS rule, ΔLMN and ΔONM are congruent.

Ques 5: The triangle ABC is an isosceles triangle. In the given figure, AB = AC and AP = AQ. Show that CP = BQ. (4 Marks)

Ans.

Given: In ΔABQ and ΔACP,

AB = AC (Stated in question)

∠BAQ = ∠CAP (Common angle)

AQ = AP (Stated in question)

Hence using SAS congruence,

ΔABQ $\cong$ΔACP

So, CP = BQ (Corresponding parts of congruent triangles)

Ques 6: ΔABC and ΔDBC are 2 isosceles triangles on the same base, as shown in the figure. The line segment BC and vertices A and D are on the same side of BC and AD has been extended to intersect BC at P. Prove that : (i) ΔABD $\cong$ ΔACD (ii) ΔABP $\cong$ ΔACP (4 Marks)

Ans: (i) In the triangles ABD & ACD,

It is given that AB = AC and BD = CD.

AD is common to both triangles.

So, using SSS congruence,

ΔABD $\cong$ ΔACD

(ii) In the triangles ABP & ACP,

It is given that AB = AC

∠BAP = ∠CAP [corresponding parts of congruent triangles, ΔABD $\cong$ ΔACD]

AP is common to both triangles.

So SAS congruence rule,

ΔABP $\cong$ ΔACP

Ques 7: In the figure given below, AE = AD & BD = CE. Show that ΔAEB $\cong$ ΔADC. (4 Marks)

Ans: It is given that AE = AD … (a)

& CE = BD … (b)

Adding (a) and (b),

We get AE + CE = AD + BD

So, AC = AB … (c)

In ΔAEB & ΔADC,

It is given that AE = AD

AB = AC (from (c))

∠A is common in both the triangles.

So, using SAS congruence criteria,

ΔAEB is congruent to ΔADC

Some Properties of a Triangle

If a triangle has two equal sides then it is called an Isosceles Triangle.

1. Two angles opposite to the two equal sides of an isosceles triangle are also equal.

2. Two sides opposite to the equal angles of the isosceles triangle are also equal. This is the converse of the above theorem.

Triangle Inequality Theorem

According to Triangle Inequality Theorem, one of the sides of the triangle is shorter than the other two sides. This theorem is an essential criterion to form a triangle. A triangle cannot be created without satisfying this theorem.

Let us assume a Triangle ABC with sides AB, BC, and CA.

According to Triangle Inequality Theorem,

Triangle Sum Theorem

According to the Triangle Sum Theorem, the sum of the internal angles of a triangle is equal to 180^o.

Assume a triangle ABC, then, according to the theorem,

Basic Proportionality Theorem

Basic Proportionality Theorem states that when a line is drawn parallel to one side of a triangle to intersect the midpoints of the other two sides, then the two sides are divided in the same ratio.

Assume that ABC is a triangle and DE is a line parallel to BC such that it intersects AB at D and AC at E.

Thus, according to Basic Proportionality Theorem,

Inequalities in a Triangle

Theorem 1: In a given triangle if two sides are unequal then the angle opposite to the longer side will be larger.

a > b, if and only if ∠A > ∠B

Longer sides correspond to larger angles.

Theorem 2: In the given triangle, the side opposite to the larger angle will always be longer. This is the converse of above theorem.

Theorem 3: The sum of any two sides of a triangle will always be greater than the third side.

Example

Show whether the inequality theorem is applicable to this triangle or not?

Solution

The three sides are given as 7, 8 and 9.

According to inequality theorem, the sum of any two sides of a triangle will always be greater than the third side.

Let’s check it

7 + 8 > 9

8 + 9 > 7

9 + 7 > 8

This shows that this theorem is applicable to all the triangles irrespective of the type of triangle.

Q1. In the given figure, PS is the median then ∠QPS?

a) 40^o

b) 50^o

c) 80^o

d) 90^o

Q2. In the given figure, AB = AC and ∠B = 50° then; ∠A is:

a) 50^o

b) 80^o

c) 100^o

d) 130^o

Q3. Find the value of the missing angle x.

1. 70°
2. 110°
3. 55°
4. 130°

Explanation: 

In triangle ABC

AB=BC (Triangle ABC is an Isosceles Triangle)

So, ∠A = ∠C (Opposite angles and sides are equal in an Isosceles Triangle)

∠A= ∠C and ∠B = 70°

Using angle sum property of triangle, we know:

∠A+∠B+∠C = 180°

2∠A+∠B=180°

2∠A = 180-∠B = 180-70 = 110°

∠A = 55°

Q7.E and F are midpoints of equal sides AB and AC of triangle ABC. Then:

1. BF=AC
2. BF=AF
3. CE=AB
4. BF=CE

Explanation: In triangle ABC

AB=AC (Triangle ABC is an Isosceles Triangle)

∠A =∠A (Angle is common)

AE=AF (Half of the equal sides that is AB and AC)

Using Side Angle Side Property,

ABF ≅ ACE

So, BF=CE (corresponding sides of corresponding triangles)

Q8.In an Isosceles Triangle ABC altitude BE and CF is drawn to the equal sides of the isosceles triangle – AB and AC respectively, then:

1. BE>CF
2. BE<CF
3. BE= CF
4. None of these 

Explanation: In triangle ABC

AB=AC (Triangle ABC is an Isosceles Triangle)

∠A =∠A (Angle is common)

∠AEB =∠AFC = 90° (BE and CF are altitudes)

Using Angle Angle Side (AAS) Property,

ΔAEB ≅ ΔAFC

So, BE = CF (corresponding sides of corresponding triangles)

Q9.If triangle ABC and DBC are two isosceles triangles on the same base BC. Then:

1. ∠ABD = ∠ACD
2. ∠ABD > ∠ACD
3. ∠ABD < ∠ACD
4. None of these

Explanation: In triangle ABC

AB=AC (Triangle ABC is an Isosceles Triangle)

AD=AD (Common arm)

BD=CD (Triangle DBC is an Isosceles Triangle)

Using Side Angle Side Property,

So,

ΔABD ≅ ΔACD.

∴∠ABD = ∠ACD (corresponding sides of corresponding triangles)

Q.10. If triangle ABC is an equilateral triangle, then each angle equals to:

1. 90°
2. 180° 
3. 120°
4. 60°

Explanation: Triangle ABC is an equilateral triangle,

In an equilateral triangle, all sides are equal to each other

So, AB=BC=AC

Thus, all angles are also equal to each other

∠A=∠B=∠C

By angle sum property of triangle:

∠A+∠B+∠C = 180°

3∠A = 180° (∠A=∠B=∠C)

∠A = 60°

Thus, each angle measures 60°.

Q.11. AD is an altitude of an isosceles triangle, then:

1. BD=CD
2. BD>CD
3. BD<CD
4. None of these

Explanation: In triangle ABC

AB=AC (Triangle ABC is an Isosceles Triangle)

Now in triangle ABD and triangle ACD

∠ADB = ∠ADC = 90° (AD is an altitude)

AD=AD (common side in triangle ABD and triangle ACD)

AB=AC (Triangle ABC is an Isosceles Triangle)

Using RHS Congruence property

ΔABD ≅ ΔACD

BD=CD (corresponding sides of corresponding triangles)

Q.12. In a right-angle triangle ABC, the longest side is:

1. Perpendicular
2. Hypotenuse
3. Base
4. None of these

Explanation: In a right triangle ABC

∠B=90° (the right angle at B)

Using angle sum property of triangle, we know:

∠A+∠B+∠C = 180°

∠A+90°+∠C = 180°

∠A+∠C = 90°

Clearly, ∠B is the largest angle in the triangle

Since the side opposite to the largest angle is the longest

The hypotenuse is the longest side of the right-angle triangle ABC.

Q.13. In triangles ABC and PQR, AB=BC, ∠C=∠P, and ∠B=∠Q then:

1. Isosceles and congruent
2. Only Isosceles
3. Only Congruent
4. None of these

Explanation: In triangles ABC and PQR

AB=BC, ∠C=∠P, and ∠B=∠Q,

The two triangles are isosceles because an isosceles triangle has opposite sides and angles equal to each other.

But triangles ABC and PQR are not congruent.

Q.14.In triangle PQR,∠R = ∠P, QR=4 and PR = 5. Find PQ.

1. 2
2. 2.5
3. 4
4. 5

Explanation: In triangle PQR

∠R = ∠P, QR=4 and PR = 5

As ∠R = ∠P

PQ=QR (Equal angles have equal sides)

PQ=QR= 4

Q.15.If AB=BC, BC=PR, CA=PQ, then,

1. PQR ≅ BCA
2. BAC ≅ RPQ
3. CBA ≅ PRQ
4. ABC ≅ PQR

Explanation: In triangle ABC and PQR,

AB=BC (given)

BC=PR (given)

CA=PQ (given)

Using Side-Side-Side Congruence Property,

CBA ≅ PRQ.

Q.16.In triangle ABC, BC=AB and ∠B = 80°. Find ∠A

1. 40°
2. 50°
3. 80°
4. 100°

Explanation:

In triangle ABC

AB=BC (Triangle ABC is an Isosceles Triangle)

So, ∠A = ∠C (Opposite angles and sides are equal in an Isosceles Triangle)

∠A= ∠C and ∠B = 80°

Using angle sum property of triangle, we know:

∠A+∠B+∠C = 180°

2∠A = 100°

∠A = 50°

Q.17.The sides of a triangle are of lengths 5cm and 1.5 cm. The length of the third side cannot be:

1. 3.4
2. 3.6
3. 3.8
4. 4.1

Explanation: The sides of the triangle are of lengths 5cm and 1.5 cm. The sum of two sides of a triangle is more than the third side. The difference between the two sides should be less than the third side. Hence Answer: a

Q.18.In triangle ABC, AB=AC and ∠B = 40°. Find ∠A

1. 40°
2. 70°
3. 80°
4. 130°

Explanation:

In triangle ABC

AB=AC (Triangle ABC is an Isosceles Triangle)

So, ∠A = ∠C (Opposite angles and sides are equal in an Isosceles Triangle)

∠A= ∠C and ∠B = 50

Using angle sum property of triangle, we know:

∠A+∠B+∠C = 180°

2∠C = 140°

∠C = 70°

Q.19.If ABC ≅ FDE, ∠B = 40° and ∠A = 80°. Which of the following is true?

1. DF= 5cm and ∠F = 60°
2. DF= 5cm and ∠E = 60°
3. DE= 5cm and ∠E = 60°
4. DE= 5cm and ∠D= 40°

Explanation: ABC ≅ FDE

∠B = 40° and ∠A = 80°

Using the CPCT rule

DF =AB=5cm

∠E =∠C

Using angle sum property of triangle, we know:

∠E+∠F+∠D = 180°

∠E = 180° - 120°

∠E = 60°

Multiple Choice Questions