The document discusses different types and properties of triangles. It begins by defining a triangle as a three-sided polygon with three angles and three vertices. It then describes various triangle classifications based on side lengths (scalene, isosceles, equilateral) and angle measures (acute, right, obtuse). Various properties of triangles are outlined, such as angle sum, exterior angles, and relationships between sides and angles. Formulas for calculating perimeter and area of triangles are also provided. The document concludes by presenting theoretical proofs and experimental verifications of several triangle theorems.

1. Plane Figures
Triangle
Class 8
By: Roji Maharjan

2. • Triangle:
A triangle is a three-sided polygon that consists of three
edges and three vertices. The most important property of a
triangle is that the sum of the internal angles of a triangle is
equal to 180 degrees. This property is called angle sum
property of triangle.
• A triangle is a type of polygon, which has three sides, and the
two sides are joined end to end is called the vertex of the
triangle. An angle is formed between two sides.

3. Properties of triangle:
• A triangle has three sides and three angles.
• The sum of the angles of a triangle is always 180 degrees.
• The exterior angles of a triangle always add up to 360 degrees.
• The sum of consecutive interior and exterior angle is
supplementary (180).
• The sum of the lengths of any two sides of a triangle is greater
than the length of the third side. Similarly, the difference
between the lengths of any two sides of a triangle is less than
the length of the third side.
• The shortest side is always opposite the smallest interior angle.
Similarly, the longest side is always opposite the largest
interior angle.

4. • Types
On the basis of length of the sides, triangles are classified into
three categories:
• Scalene Triangle
• Isosceles Triangle
• Equilateral Triangle
On the basis of measurement of the angles, triangles are
classified into three categories:
• Acute Angle Triangle
• Right Angle Triangle
• Obtuse Angle Triangle

5. • Scalene Triangle
A scalene triangle is a type of triangle, in which all the three
sides have different side measures. Due to this, the three angles
are also different from each other.

6. • Isosceles Triangle
In an isosceles triangle, two sides have equal length. The
two angles opposite to the two equal sides are also equal
to each other.

7. • Equilateral Triangle
An equilateral triangle has all three sides equal to each
other. Due to this all the internal angles are of equal
degrees, i.e. each of the angles is 60°.

8. • Acute Angled Triangle
An acute triangle has all of its angles less than 90°.

9. • Right Angled Triangle
In a right triangle, one of the angles is equal to 90° or
right angle.

10. • Obtuse Angled Triangle
An obtuse triangle has any of its one angles more than
90°.

11. Perimeter of Triangle
• A perimeter of a triangle is defined as the total length of
the outer boundary of the triangle. Or we can say, the
perimeter of the triangle is equal to the sum of all its
three sides. The unit of the perimeter is same as the unit
of sides of the triangle.
• Perimeter = sum of all sides
• If ABC is a triangle, where AB, BC and AC are the
lengths of its sides, then the perimeter of ABC is given
by:
• Perimeter = AB+BC+AC

12. Area of a Triangle
• The area of a triangle is the region occupied by the triangle in
2d space. The area for different triangles varies from each
other depending on their dimensions. We can calculate the
area if we know the base length and the height of a triangle. It
is measured in square units.
• Suppose a triangle with base ‘B’ and height ‘H’ is given to
us, then, the area of a triangle is given by-
Area of triangle = Half of Product of
Base and Height
Area = 1/2 × Base × Height

13. “The sum of two sides of a triangle is greater than the third
side.”
Experimental Verification
Three triangles ABC of different shapes and sizes are drawn with the
help of scale and pencil.
Conclusion: It is experimentally verified that the sum of two
sides of a triangle is greater than the third side.

14. “In a triangle, the side opposite to the greater angle is longer than the
side opposite to the smaller angle.”
Experimental Verification
Three triangles PQR of different shapes and sizes with ∠P greater than
other angles ∠Q and ∠R are drawn with the help of scale and pencil.
Conclusion: Hence it is experimentally verified that the side opposite
to greater angle is longer than the side opposite to smaller angle.

15. “The sum of three angles of a triangle is equal to two right angles.”
Experimental Verification
Three triangles ABC of different shapes and sizes are drawn with the
help of scale and pencil.
Conclusion: Hence it is experimentally verified that the sum of the
angles of a triangle is two right angles.

16. Theoretical Proof:
Given: ABC is a triangle.
To prove: ∠ABC+∠BAC+∠ACB = 180°
Construction: Through A, PQ parallel to BC is drawn.
Proof:
Statements Reasons
1. ∠PAB = ∠ABC ------> Alternate angles
2. ∠QAC = ∠ACB -------> Alternate angles
3. ∠PAB+∠BAC+∠QAC = 180° -----> Straight angle
4. ∠ABC+∠BAC+∠ACB = 180° -----> From statements 1, 2 and 3.
Proved.

17. “The exterior angle of a triangle is equal to the sum of the two interior
opposite angles.”
Experimental Verification
Three triangles PQR with exterior angle ∠PRS are drawn with the help of scale
and pencil.
Table,
Conclusion: Hence it is experimentally verified that the exterior angle of a
triangle is equal to the sum of the two interior opposite angles.

18. Theoretical Proof:
Given: In ∆ABC, ∠ACD is an exterior angle.
To prove: ∠ACD = ∠A + ∠B
Proof:
Statements Reasons
1. ∠ACB + ∠ACD = 180° -----> Straight angle.
2. ∠A+∠B+∠ACB = 180° -----> Sum of angles of a ∆.
3. ∠ACB+∠ACD = ∠A+∠B+∠ACB -----> From statements 1 and 2.
4. ∠ACD = ∠A+∠B ------> Removing ∠ACB from both sides.
Proved.

19. “Base angles of an isosceles triangle are equal.” OR “If two sides of a
triangle are equal, then the angles opposite to them are also equal.”
Experimental Verification
Three isosceles triangles PQR of different shapes and sizes with PQ = PR are
drawn with the help of scale, pencil and compass.
Table,
Conclusion: Hence it is experimentally verified that if two sides of a triangle
are equal then the angles opposite to them are also equal.

20. Theoretical Proof:
Given: ∆XYZ is an isosceles triangle in which XY = XZ.
To prove: ∠XYZ = ∠YZX
Construction: XA⊥YZ drawn.
Proof:
Statements Reasons
1. In ∆XYA and ∆XZA
i. ∠XAY = ∠XAZ (R) ------> Both right angles.
ii. XY = XZ (H) ------> Given.
iii. XA = XA (S) ------> Common side.
2. ∆XYA ≅ ∆XZA ------> By RHS axiom.
3. ∠XYA = ∠XZA -----> Corresponding angles of congruent triangles
4. ∠XYZ = ∠YZX -----> From statement 3.
Proved.

21. “In a triangle, the sides opposite to the equal angles are also equal.”
Experimental Verification
Three triangles PQR of different shapes and sizes with ∠Q = ∠R are drawn
with the help of scale, pencil and protractor.
Table,
Conclusion: Hence it is experimentally verified that the sides opposite to equal
angles in a triangle are equal.

22. Theoretical Proof:
Given: In ∆XYZ, ∠Y = ∠Z.
To prove: XY = XZ
Construction: XA⊥YZ drawn.
Proof:
Statements Reasons
1. In ∆XYA and ∆XZA
i. XA = XA (S) -------> Common side.
ii. ∠XAY = ∠XAZ (A) -------> Both right angles.
iii. ∠Y = ∠Z (A) -------> Given.
2. ∆XYA ≅ ∆XZA -------> By SAA axiom.
3. XY = XZ -------> Corresponding sides of congruent triangles.
Proved.

23. “The bisector of the vertical angle of an isosceles triangle is perpendicular
to the base and bisects the base.”
Experimental Verification
Three triangles PQR with bisector PX of different shapes and sizes are drawn
with the help of scale, pencil and compass.
Table,
Conclusion: Hence it is experimentally verified that the bisector of vertical
angle of an isosceles triangle is perpendicular to the base and bisects the base.

24. Theoretical Proof:
Given: ∆XYZ is an isosceles triangle in which XY = XZ. XA is bisector of ∠YXZ
i.e. ∠AXY = ∠AXZ.
To prove: XA⊥YZ and YA = ZA
Proof:
Statements Reasons
1. In ∆XYA and ∆XZA
i. XY = XZ (S) -------> Given.
ii. ∠AXY = ∠AXZ (A) -------> Given.
iii. XA = XA (S) -------> Common side.
2. ∆XYA ≅ ∆XZA -------> By SAS axiom.
3. ∠XAY = ∠XAZ -------> Corresponding angles of congruent triangles.
4. XA⊥YZ -------> Being adjacent angles equal (statement 3).
5. YA = ZA ------> Corresponding sides of congruent triangles.
Proved.

25. “The line joining the mid-points of the base of an isosceles triangle to the
opposite vertex is perpendicular to the base and bisects the vertical angle.”
Experimental Verification
Three triangles PQR with A as mid-point of QR of different shapes and sizes are
drawn with the help of scale, pencil and compass.
Table,
Conclusion: Hence it is experimentally verified that the line joining mid-point of the
base of an isosceles triangle to the opposite vertex is perpendicular to the base and
bisects the vertical angle.

26. Theoretical Proof:
Given: ∆XYZ is an isosceles triangle in which XY = XZ. A is the mid-point of YZ
i.e. AY = AZ.
To prove: XA⊥YZ and ∠AXY = ∠AXZ
Proof:
Statements Reasons
1. In ∆XYA and ∆XZA
i. XY = XZ (S) --------> Given.
ii. AY = AZ (S) --------> Given.
iii. XA = XA (S) -------> Common side.
2. ∆XYA ≅ ∆XZA --------> By SSS axiom.
3. ∠XAY = ∠XAZ -------> Corresponding angles of congruent triangles.
4. XA⊥YZ -------> Being adjacent angles equal (statement 3).
5. ∠AXY = ∠AXZ -------> Corresponding angles of congruent triangles.
Proved.

27. “A line segment joining the mid-points of any two sides of a triangle is parallel
to the third side and is equal to half of its length.”
Experimental Verification
Three triangles ABC with X and Y mid-points of AB and AC respectively of
different shapes and sizes are drawn with the help of scale, pencil and compass.
Table,
Conclusion: Hence it is experimentally verified that a line segment joining the mid-
points of any two sides of a triangle is parallel to third side and is equal to half of its
length.

28. Theoretical Proof:
Given: ABC is a triangle in which AX = XB and AY =YC.
To prove: XY∥BC and XY = BC/2
Construction: CZ is drawn such that it is parallel to BX and XY is produced to meet at Z.
Proof:
Statements Reasons
1. In ∆AXY and ∆YCZ
i. ∠XYA = ∠ZYC (A) --------> VOA.
ii. AY = YC (S) -------> Given.
iii. ∠XAY = ∠ZCY (A) --------> Alternate angles.
2. ∆AXY ≅ ∆YCZ --------> By ASA axiom.
3. AX = CZ --------> Corresponding sides of congruent triangles.
4. AX = BX --------> Given.
5. CZ = BX -------> From statements 3 and 4.
6. CZ ∥ BX --------> By construction.
7. XBCZ is a parallelogram. ------> Being opposite sides equal and parallel (5. And 6.)
8. XY = YZ i.e. XY = XZ/2 ------> Corresponding sides of congruent triangles.
9. XZ∥BC and XZ = BC -------> Opposite sides of a parallelogram.
10. XY∥BC and XY = BC/2 -------> From Statements 9 and 8.
Proved.