**Theorem** is a geometrical statement which is true and can be proved by
using known facts and axioms. Generally, we prove geometrical theorems by two
methods, one method is **experimental
verification** and another method is **theoretical
proof**.

Some of the **triangle
theorems** and their experimental verification and/or theoretical proofs are
given below:

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**Theorem
1:**

**“Among the straight
lines drawn from an external point to a given straight line, the perpendicular
is the shortest one.”**

**Experimental
Verification**

PA, PB, PM and PN are the straight lines drawn from a point P to
the line XY and PA⊥XY. Three such figures are
drawn with the help of scale, pencil and protractor.

Table,

__Conclusion__: It is
experimentally verified that the perpendicular drawn from a point to the
straight lines is the shortest distance from the point to the line.

**Theorem
2:**

**“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.

Table,

__Conclusion__: It is
experimentally verified that the sum of two sides of a triangle is greater than
the third side.

**Theorem
3:**

**“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.

Table,

__Conclusion__: Hence it is
experimentally verified that the side opposite to greater angle is longer than
the side opposite to smaller angle.

**Theorem
4:**

**“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.

Table,

__Conclusion__: Hence it is
experimentally verified that the sum of the angles of a triangle is two right
angles.

**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.

**Theorem
5:**

**“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.

**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.

**Theorem
6:**

**“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.

**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.

**Theorem
7:**

**“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.

**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.

**Theorem
8:**

**“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.

**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.

**Theorem
9:**

**“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.

**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.

**Theorem
10:**

**“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.

**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.

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