Definitions [2]
Define a triangle.
A plane figure bounded by three lines in a plane is called a triangle. The figure below represents a ΔABC, with AB, AC andBC as the three line segments.
A triangle (denoted by the symbol △) is the simplest closed shape in geometry. It is a two-dimensional figure made by connecting three points that do not lie on the same straight line (non-collinear).
Theorems and Laws [1]
In the adjoining figure, QX and RX are the bisectors of the angles Q and R respectively of the triangle PQR.
If XS ⊥ QR and XT ⊥ PQ;
Prove that:
- ΔXTQ ≅ ΔXSQ.
- PX bisects angle P.
Given: A(ΔPQR) in which QX is the bisector of ∠Q and RX is the bisector of ∠R.
XS ⊥ QR and XT ⊥ PQ.
We need to prove that:
- ΔXTQ ≅ ΔXSQ.
- PX bisects angle P.
Construction: Draw XZ ⊥ PR and join PX.
i. In ΔXTQ and ΔXSQ,
∠QTX = ∠QSX = 90° ...[XS ⊥ QR and XT ⊥ PQ]
∠TQX = ∠SQX ...[QX is bisector of ∠Q]
QX = QX ...[Common]
∴ By Angle-Side-Angle Criterion of congruence,
ΔXTQ ≅ ΔXSQ
ii. The corresponding parts of the congruent triangles are congruent.
∴ XT = XS ...[c.p.c.t.]
In ΔXSR and ΔXRZ
∠XSR = ∠XZR = 90° ...[XS ⊥ QR and ∠XSR = 90°]
∠XRS = ∠ZRX ...[RX is bisector of ∠R]
RX = RX ....[Common]
∴ By Angle-Angle-Side criterion of congruence,
ΔXSR ≅ ΔXRZ
The corresponding parts of the congruent triangles are congruent.
∴ XS = XT ...[c.p.c.t.]
From (1) and (2)
XT = XZ
In ΔXTP and ΔPZX
∠XTP = ∠XZP = 90° ....[Given]
XP = XP ....[Common]
XT = XZ
∴ By Right angle-Hypotenuse-side criterion of congruence,
ΔXTP ≅ ΔPZX
The corresponding parts of the congruent triangles are
congruent.
∴ ∠TPX = ∠ZPX ...[c.p.c.t.]
∴ PX bisects ∠P.
