Class 8 Maths Chapter 2 Equal Triangles Questions and Answers Kerala Syllabus

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SCERT Class 8 Maths Chapter 2 Solutions Equal Triangles

Class 8 Kerala Syllabus Maths Solutions Chapter 2 Equal Triangles Questions and Answers

Equal Triangles Class 8 Questions and Answers Kerala Syllabus

Sides and Angles (Page No. 18, 19)

Question 1.
In each pair of triangles below, find the angles of the second triangle equal to the angles of the first triangle, and write these pairs.

Answer:
(i) ∠M = ∠Z, ∠N = ∠X, ∠L = ∠Y
(ii) ∠A = ∠R, ∠C = ∠Q, ∠B = ∠P

Question 2.
In the triangles shown below,
AB = QR, BC = RP, CA = PQ

Calculate angle ∠C of triangle ABC and all angles of triangle PQR.
Answer:
In triangle ABC,
∠C =180 – (40 + 60)
= 180 – 100
= 80°
In triangle PQR,
∠P = ∠C = 80°
∠Q = ∠A = 40°
∠R = ∠B = 60°

Question 3.
In the triangle below,
AB = QR, BC = PQ, CA = RP

Calculate the other two angles of each triangle.
Answer:
AB = QR ⇒ ∠C = ∠P = 50°
CA = RP ⇒ ∠B = ∠Q = 70°
BC = PQ ⇒ ∠A = ∠R = 60°

Question 4.
The diagonal of a quadrilateral splits it into two triangles as shown below:

Do these triangles have the same angles? Why?
Answer:
The diagonal is common to both triangles.
Sides of upper triangles are equal to the sides of lower triangles.
Angles opposite to equal sides are equal.

Question 5.
In the quadrilateral ABCD shown below,
AB = AD, CB = CD

Calculate all the angles of the quadrilateral.
Answer:
AD = AB, CD = CB,
AC is a common Sides of triangle ABC are equal to the sides of triangle ADC
Angles opposite to equal sides are equal.
∠ACD = 50°, ∠BAC = 30°, ∠D = 100°, ∠B = 100°
Angles are ∠A = 60°, ∠B = 100°, ∠C = 100°, ∠D = 100°

Two Angles (Page No. 24)

Question 1.
In each pair of triangles below, find the sides of the triangle on the right equal to the sides of the triangle on the left:

Answer:
(i) BC = PQ, AC = PR, ∠C = ∠P
(ii) MN = XZ, ∠Z = ∠M, LM = YZ

Question 2.
In the picture, the top two sides and the bottom side of a pentagon, with equal angles and equal sides, are extended to form a triangle:

(i) Are the sides of the small triangle on the left equal to the sides of the small triangle on the right? Why?
(ii) Are the left and right sides of the large triangle equal? Why?
Answer:
(i) Look at the picture

Sides and angles of ABCDE are equal.
Since ∠A = ∠B, then ∠PAE = ∠QBC
∠PEA = ∠QCB
In triangle PAE, side AE and angles at the ends, side BC and angles at the ends in triangle QBC are equal.
So triangles are equal.

(ii) In triangle DPQ, ∠P = ∠Q.
Sides opposite to equal angles are equal.
PD = QD

Question 3.
The sides of a triangle are equal to the sides of another triangle.
(i) Is the height from each side of one triangle to the opposite vertex equal to the height from the equal side of the other triangle to its opposite vertex? Why?
(ii) Are the areas of the two triangles equal? Why?
Answer:
(i) In the figure

AB = PQ, AC = PR, BC = QR
AM is the altitude to the side BA.
∠AMB = 90°,
PN is the altitude to the side QR.
∠PNQ = 90°
In triangle AMB and triangle PNQ,
∠B = ∠Q, ∠M = ∠N
⇒ ∠A = ∠P
Side AB of triangle AMB and angles at the ends are equal to side PQ of triangle PNQ and angles at the ends.
Triangles are equal.
Sides opposite to equal angles are equal.
So AM = PN

(ii) In triangle ABC, side BC and altitude to BC are equal to side QR and altitude to QR are equal.
Areas are equal.

Two Sides (Page No. 28, 29)

Question 1.
The green lines in the picture below are parallel and of the same length; one is drawn from the end of the horizontal blue, and the other is drawn from the midpoint of the blue line:

(i) Are the lengths of the red lines in the picture equal? Why?
(ii) Are the red lines parallel?
Answer:

(i) AD = BE, AB = BC, ∠DAB = ∠EBC
In the triangles ABD and BCE, two sides and the angle between them are equal.
Sides opposite to equal angles are equal: BD = CE

(ii) Angle ABD, angle BCE are equal.
Side BD and CE are parallel.

Question 2.
Is the quadrilateral below a parallelogram? Why?

Answer:
∠BAC = ∠DCA, AB = CD
Line AB and CD are equal and parallel.
In triangle ABC and in triangle ADC line AC is common.
Two sides and the angle between them are equal.
AD = BC, ∠DAC = ∠BCA
Opposite sides of ABCD are equal and parallel.
ABCD is a parallelogram.

Question 3.
In the figure below, the lines AB and CD are parallel. M is the midpoint of AB.

(i) Are the quadrilaterals AMCD and MBCD parallelograms? Why?
(ii) Calculate all the angles of the triangles AMD, MBC, and DCM.
Answer:
(i) Opposite sides are equal and parallel
AM = CD = 3, AM is parallel to CD
AMCD is a parallelogram. MBCD is also a parallelogram.

(ii) ∠MCD = 40°, ∠MDC = 60°,
∠ADM = 140 – 60 = 80°
∠AMD = 180 – (40 + 80) = 60°
∠MCB = 120 – 40 = 80°
∠AMC = 140°
∠DMC = 180 – (60 + 40) = 80°

Question 4.
In the picture, O is the centre of the circle and A, B, C, D are points on the circle:

Are the lines AB and CD equal? Why?
Answer:
In triangle OAB and triangle OCD
OA = OC, OB = OD.
The angle between them is 80°.
AB = CD

Question 5.
In the picture, O is the centre of the circle and A, B, and C are points on the circle. ∠AOB and ∠AOC are equal:

Are the lines AB and AC equal? Why?
Answer:
In triangle AOB and triangle AOC, OA is the common side.
OB = OC.
Including angles are equal.
Triangles are equal.
Sides opposite to equal angles are equal.
AB = AC

Isosceles Triangles (Page No. 32)

Question 1.
Find the other angles in each of the isosceles triangles below?

Answer:
(a) AB = AC
⇒ ∠B = ∠C = 30°
∠A = 180 – 60 = 120°

(b) PQ = PR
⇒ ∠Q = ∠R = \(\frac {140}{2}\) = 70°

(c) LM = MN
⇒ ∠L = 45°, ∠N = 45°

(d) XY = YZ
⇒ ∠X = ∠Z = 40°

Textbook Page No. 35, 36

Question 1.
One angle of an isosceles triangle is 120°. What are the other two angles?
Answer:
One angle is 120°.
This will be the angle between equal sides.
So the other two angles are opposite equal sides.
Each is equal to 30°.

Question 2.
The picture shows a triangle drawn by joining the centre of a circle and two points on the circle:

Calculate the other two angles of the triangle.
Answer:
OA and OB are radii, equal.
So angles opposite to radii are equal to 60°.
All angles are 60°.

Question 3.
The picture shows the triangle drawn by joining three points on a circle. Two of the angles formed by joining these points to the centre of the circle are also given:

(i) Calculate the third angle at the centre.
(ii) Calculate all the angles of the large (green) triangle in the circle.
Answer:
(i) In triangle OAB,
OA = OB.
The angles opposite to these sides are 35°.
∠OBA = ∠OAB = 35°
In triangle OAC,
OA = OC.
Sides opposite to these sides are equal.
Each angle is 30°.
In triangle OBC,
∠BOC = 360 – 230 = 130°
∠OBC = ∠OCB = 25°
(ii) ∠A = 65°, ∠B = 60°, ∠C = 55°

Question 4.
Calculate the areas of each of the triangles below:

Answer:
(i) Draw AM perpendicular to BC.
Triangle AMB is a right triangle.
MB = CM = 4 cm
AM² = 5² – 4² = 9
⇒ AM = 3
Area of triangle = \(\frac {1}{2}\) × 8 × 3 = 12 cm

(ii) Draw PN perpendicular to QR.
Triangle PNQ is a right triangle.
NQ = RN = 3 cm
PN² = 5² – 3² = 9
⇒ PN = 4
Area of triangle = \(\frac {1}{2}\) × 6 × 4 = 12 cm

Question 5.
Given that one angle of an isosceles triangle is 70°. What can we say about the other angles?
Answer:
Two possibilities arise
(a) If two angles are 70°, then third angle will be 40°.
Angles are 70°, 70°, 40°.

(b) If one angle is 70° and the other two angles are x, then
70 + 2x = 180
⇒ 2x = 110
⇒ x = 55°
Possible triangles
70°, 70°, 40°
70°, 55°, 55°

Question 6.
How many non-equal isosceles triangles can be drawn with one angle 70° and one side 8 centimetres?
Answer:
(a) One side is 8 cm, and the angles at the ends are 40°. So the third angle will be 70°.
(b) One side is 8 cm, and the angles at the ends are 70°.
(c) Two sides are 8 cm, and the angle between them is 40°. The third angle will be 70°.
(d) Two sides are 8 cm, and the angle between them is 70°.

Class 8 Maths Chapter 2 Kerala Syllabus Equal Triangles Questions and Answers

Class 8 Maths Equal Triangles Questions and Answers

Question 1.
Four statements are given below. Identify the true statement.
(a) Equal triangles have equal area.
(b) Equiangular triangles (Triangles having equal angles) are equal triangles.
(c) Equality: one angle and any two sides of two triangles make them equal.
(d) One side and an angle at the ends can fix a triangle.
(a) (a) and (b) are true
(b) (a) and (c) are true
(c) (b) and (c) are true
(d) (a) and (d) are true
Answer:
(d) (a) and (d) are true
Equality means being identical or having the same shape and size.

Question 2.
Read the following statements and identify the correct option.
p1: A diagonal divides a parallelogram into two equal triangles.
p2: The line joining a corner to the midpoint of the opposite side of any triangle makes the triangle into two equal triangles.
(a) p1 and p2 are true
(b) p1 and p2 are false
(c) p1 is true p2 is false
(d) p1 is false p2 is true
Answer:
(c) p1 is true p2 is false

Question 3.
The base and altitude to the base of an isosceles triangle are 10 cm and 12 cm. The length of the equal sides
(a) 13 cm
(b) 15 cm
(c) 10 cm
(d) 8 cm
Answer:
(a) 13 cm
Half of the base, altitude, and one of the equal sides make a right triangle.

Question 4.
The angle around the centre of a circle is divided into three equal parts, each 120°, by drawing radii. The ends of the radii are joined to make a triangle. The triangle so formed is
(a) Right triangle
(b) Isosceles triangle
(c) Equilateral triangle
(d) None of these
Answer:
(c) Equilateral triangle
Triangles formed around the centre of the circle are equal, one angle is 120°, the Other two angles are 30°. So the angles of the triangle will be 60°.

Question 5.
Two angles of a triangle are equal, and the third is half of one of the equal angles. The small angle of the triangle is
(a) 30°
(b) 40°
(c) 36°
(d) 70°
Answer:
Divide 180° into five equal parts.
Each part is 36°.
Angles of the triangle are 72°, 72°, 36°.

Question 6.
In the diagram, AB = AD, BC = DC.

(a) If ∠BAC = 20°, then what is ∠A?
(b) Name pairs of equal triangles in the figure.
Answer:
(a) Triangle BAC and triangle DAC are equal.
∠BAC = ∠DAC = 20°

(b) Triangle AOB and triangle AOD are equal.
Triangle BOC and triangle DOC are equal.
Triangle ABC and triangle ADC are equal.
Triangle ABD and triangle CBD are equal.

Question 7.
In the figure, AB = AC, BD = CD.
(a) Name two equal triangles in the figure.
(b) If ∠BAC = 40°, ∠ABD = 30°, then what are the angles of triangle ADB?

Answer:
(a) AB = AC, BD = CD, AD is common.
Triangle ABD and triangle ADC are equal.

(b) In triangle ADB,
∠A = 20°, ∠B = 30°, ∠D = 130°

Question 8.
In the figure, AB = CF, EF = BD, ∠and DBC = ∠AFE.

(a) Are the triangles AFE and triangle BDC equal? How do you know this?
(b) What are the equal sides and equal angles of these triangles?
Answer:
(a) AB = CF
∴ AB + BF = CF + BF
AF = CB,
BD = EF,
∠DBC = ∠AFE
Triangles are equal.

(b) CD = AE, EF = BD, ∠DBC = ∠AFE

Question 9.
In the figure, AB = AC, E is the midpoint of AB, and F is the midpoint of AC.

(a) What are the equal triangles in the figure?
(b) If BF = 12 cm, then what is CE?
Answer:
(a) AB = AC
⇒ ∠B = ∠C
BE = CF, BC is common.
Triangle BEC and triangle CFB are equal.

(b) Sides opposite to equal angles are equal.
CE = 12 cm

Question 10.
In the figure, PQ = PR = 10 cm, ∠QPS = ∠RPS.

(a) What are the equal triangles in the figure?
(b) If QS = 18 cm, then what is SR?
(c) Find the perimeter of PQRS.
Answer:
(a) PQ = PR
PS is common.
∠QPS = ∠RPS
Triangles PQR and triangle PRS have equal angles and sides.
(b) 18 cm
(c) 18 + 10 + 18 + 10 = 56 cm

Question 11.
In the figure, AB = AC. The lines BO and CO divide angle B and angle C equally, intersect at O.

(a) Prove that OB = OC.
(b) Does AO divide the angle A into two equal parts?
Answer:
(a) BO divides ∠B equally, and CO divides ∠C equally.
∠OBC = ∠OCB
⇒ BO = CO

(b) Sides of triangle OAB are equal to sides of triangle AOC.
Triangles are equal.
∠BAO = ∠CAO
OA is the bisector of A.

Question 12.
In the figure, D is the midpoint of BC, DL is perpendicular to AB, DM is perpendicular to AC, and DL = DM.

(a) Prove that triangle BLD and triangle CMD have the same side and shape.
(b) Is ∠B = ∠C?
(c) If AB = 12 cm, then what is AC?
Answer:
(a) BD = CD = x, DL = DM = y
Triangle BLD and triangle CMD are right triangles
BD² = BL² + DL², CD² = DM² + CM²
Since D is the midpoint, BD = CD
BL² + DL² = DM² + CM², BL² = CM²
⇒ BL = CM
Three sides of triangle BLD are equal to three sides of triangle CMD.
So these triangles are equal.

(b) Sides opposite to equal angles are equal,
∠B = ∠C

(c) Since ∠B = ∠C we can write AB = AC.
⇒ AB = AC = 12 cm

Class 8 Maths Chapter 2 Notes Kerala Syllabus Equal Triangles

→ If all sides of a triangle are equal to the sides of another triangle, then the triangles are equal.

→ Angles opposite to equal sides are also equal.

→ If one side and angles at the ends of that side are equal to one side and angles at the ends of another triangle, then the triangles are equal. Sides opposite to equal angles are equal. Angles opposite to equal sides are also equal.

→ A quadrilateral having opposite sides parallel and equal is called a parallelogram.

→ If two lines have equal length and are parallel, then joining ends on the same side makes a parallelogram.

→ If two sides and the angle between them in a triangle are equal to two sides and including angle of another triangle, then the triangles are equal.

→ Sides opposite to equal angles are equal.

→ Angles opposite to equal sides are equal.

→ Triangles having two sides equal are called isosceles triangles. Angles opposite to equal sides are equal.

→ Triangles having all sides equal are called equilateral triangles. All angles of an equilateral triangle are 60° each.

→ In an isosceles triangle perpendicular from the apex forming equal angles to the opposite side divides the triangle into two equal angles.

Triangles having the same shape and size are called equal triangles. The geometric conditions for becoming two triangles equal and related concepts are discussed in this unit. To fix a triangle, three measurements are essential. Three sides or one side and angles at the ends, or two sides and including angles are these measurements.

Triangles can be classified into isosceles triangles, equilateral triangles, and scalene triangles. If all sides are equal, then the triangle is called an equilateral triangle. Triangles having two sides equal are called isosceles triangles. If all sides are different, it will be a scalene triangle.

Sides and Angles
If the lengths of the sides of two triangles are the same, then their angles are also the same.
Look at these triangles:

Here, the length of the sides of the two triangles is the same, and their angles are also the same.
That is, each angle in triangle ABC is equal to one of the angles in triangle PQR. Here,

• The longest side is 6 centimetres
• Medium side is 5 centimetres
• Shortest side is 3 centimetres

Thus, in each triangle,

• The largest angle is opposite the 6-centimetre side
• Medium angle is opposite the 5-centimetre side
• The smallest angle is opposite the 3-centimetre side

Therefor the,

• Largest angle is, ∠A = ∠R
• Medium angle is, ∠B = ∠P
• Smallest angle is, ∠C = ∠Q

If the lengths of the sides of two triangles are the same, then the angles opposite to sides of equal length are also equal.

Worksheet – 1

Question 1.
In each pair of triangles below, find all pairs of matching angles and write them down.

Answer:
(i) ∠A = ∠R, ∠B = ∠P, ∠C = ∠Q
(ii) ∠L = ∠Y, ∠N = ∠X, ∠M = ∠Z

Worksheet – 2

Question 1.
Two triangles are given below.

(a) Are the triangles equal? Why?
(b) If the triangles are equal, which are the matching angles?
Answer:
(a) Yes, these two triangles are equal.
Because three sides of ∆ABC are equal to three sides of ∆PQR.
ie. AB = PQ = 8 cm
BC = PR = 6 cm
AC = RQ = 4 cm
(b) ∠C = ∠R, ∠A = ∠Q, ∠B = ∠P

Two Angles
If the angles of a triangle are the same as the angles of another triangle, the lengths of their sides may not be the same.

If in two triangles, the length of one side and the two angles at its ends are the same, then their third angles and the lengths of the other two sides are also the same.

If two angles of a triangle are the same as two angles of another triangle, then the third angles are also the same.

In two triangles, even if all the angles and one side are the same, the other two sides may or may not be the same.
A parallelogram is a quadrilateral with each pair of opposite sides parallel.

The opposite sides of any parallelogram are equal.
The opposite angles of any parallelogram are equal.

Worksheet – 3

Question 1.
In each pair of triangles below, And the pairs of sides and write their names.


Answer:
(i) AB = RQ
BC = QP
AC = RP
Because the two triangles are equal, the sides opposite to the equal angles are also equal.
(ii) LN = YX, MN = ZX, LM = YZ

Two Sides
If in two triangles, the lengths of two sides and the angle between them are the same, then the length of the third side and the remaining two angles are also the same.
If two sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

Equal Triangles: If two sides and the angle between them in a triangle are equal to two sides and the included angle of another triangle, then the triangles are equal. Sides opposite to equal angles are equal. Angles opposite to equal sides are equal.

Worksheet – 4

Question 1.
In the figure below, AC and BE are parallel lines.

(i) Are the lengths of BC and DE equal? Why?
(ii) Are the lengths of BC and DE equal? Why?
Answer:
AC = BE
AB = BD
Since AC is parallel to BE
∠BAC = ∠DBE [corresponding angles]
So ∆ABC and ∆BDE are equal.
(i) Since the two triangles are equal, BC = DE
(ii) Since ∠ABC = ∠BDE
BC is parallel to DE.

Question 2.
Is ACBD in the figure a parallelogram? Why?

Answer:
AC = BD = 6 cm
AB = BA
∠BAC = ∠ABD = 35°
Also, AC is parallel to BD
So ∆BAC and ∆ABD are equal.
So AD = BC and parallel.
∴ So ACBD is a parallelogram.

Worksheet – 5

Question 1.
Are the angles of ∆ABC and ∆ABD equal in the figure below? Why?

Answer:
Yes, because ∆ABC and ∆ABD are equal triangles.
AC = AD, BC = BD, also AB = AB itself.
So ∠C = ∠D, ∠CAB = ∠DAB,
∠CBA = ∠DBA (∵ In equal triangles, corresponding angles are equal)

Isosceles Triangles
A triangle with two of its sides equal is called an isosceles triangle.
See this isosceles triangle, the left and right sides are the same length.

Look at the sides of the small triangles ABM and ACM on the left and right.

• AB = AC as mentioned above.
• BM = CM since M is the midpoint of BC.
• AM is a side of both these triangles.

Thus, the lengths of the sides of the triangle ABM and ACM are the same.
So, the angles opposite equal sides must also be the same.

If two sides of a triangle are equal, then their opposite angles are also equal.

A triangle with all three sides equal is called an equilateral triangle.
In any equilateral triangle, each angle is 60°.

If two angles of a triangle are equal, then their opposite sides are also equal.

In any isosceles triangle, the perpendicular from the vertex joining the equal sides to its opposite side splits this side and the angle at this vertex into equal parts.