Properties Of A Triangle

Properties of a Triangle 1. Angle Sum Property: The sum of the measures of the three angles of a triangle is 180o . Let us prove this property Proof:

suur Let PQR be any triangle. Draw a line AB parallel to QR , passing through P. Mark the angles as shown in figure 4-13.

Now, ∠4 = ∠2

(Alternate angles)

∠5 = ∠3

(Alternate angles)

Therefore ∠1 + ∠ + ∠3 = ∠1 + ∠4 + ∠5 = 180o

(Linearity property)

Therefore, Sum of the angles of a triangle is 180o (Two right angles). Hence proved. 2. Exterior Angle Property: In a triangle, the measure of an exterior angle equals the sum of the measures of the remote interior angles.

In figure 4-14, Triangle PQR is shown. The side QR is extended to S. Now, ∠PRS is the exterior angle. With reference to the exterior ∠PRS , ∠PQR and ∠RPQ are remote interior angles (or opposite interior angles). Let us prove this property Proof: Refer to figure 4-14 ∠PRS or ∠4 is an exterior angle.

We have to prove that ∠4 = ∠1 + ∠2 We know that the sum of the angles of a triangle equals 180o . In Triangle PQR, ∠1 + ∠2 + ∠3 = 180o

… (1)

Since ∠3 and ∠4 form a linear pair

∠3 + ∠4 = 180o

… (2)

From (1) and (2), we have ∠3 + ∠4 = ∠1 + ∠2 + ∠3

or ∠4 = ∠1 + ∠2

Hence Proved

Since ∠4 = ∠1 + ∠2 , this implies that ∠4 > ∠1 and ∠4 > ∠2 , Therefore, the exterior angle of a triangle is greater than each remote interior angle. 3. Triangles Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Let us prove this property Here, we have to prove that PQ + PR > QR , PQ + QR > PR and QR + PR > PQ . Extend side QP to S such that PS=PR. Join SR. Proof: In Triangle PQR , we have PS=PR (By construction) Thus Triangle PSR is an isosceles triangle. We know that in an isosceles triangle, angles opposite equal sides are equal.

Therefore ∠PSR = ∠PRS or ∠PRS = ∠PSR

… (1)

(Identity congruence)

∠QRP + ∠PRS > ∠PRS

(The sum of two non-zero numbers is greater than each individual number)

∠QRP + ∠PRS > ∠PSR

(From 1)

or ∠QRS > ∠PSR ( Since ∠QRP and ∠PRS are adjacent angles) or ∠QRS > ∠QSR ( Since ∠PRS and ∠QSR is the same angle) or QS > QR Since [side opposite to greater angle is greater] or QP+PS > QR ( Since QS=QP+PS) or QP+PS>QR ( Since PS=PR by construction0 or PQ+PR>QR ( Since QP and PQ the same side of the triangle taken in different order).

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