Geometry Tutorial Part - 2

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Geometry Tutorial Part - 2

Author: Sureshbala

PARALLEL LINES

When a straight line cuts two or more parallel lines, then the cutting line are called the TRANSVERSAL. When a straight line XY cuts two parallel lines PQ and RS [as shown Fig. 1.05], the following are the relationships between various angles that are formed. [M and N are the points of intersection of XY with PQ and RS respectively]

(a) Alternate angles are equal, i.e.

$\angle PMN = \angle MNS$ and $\angle QMN = \angle MNR$

(b) Corresponding angles are equal, i.e.

$\angle XMQ = \angle MNS ; \angle QMN =\angle SNY$
$\angle XMP = \angle MNR ; \angle PMN =\angle RNY$

(c) Sum of interior angles on the same side of the cutting line is equal to $180^{\circ}$ , i.e

$\angle QMN+ \angle MNS =180^{\circ}$ and $\angle PMN +\angle MNR=180^{\circ}$

(d) Sum of exterior angles on the same side of the transversal is equal to $180^{\circ}$ , i.e

$\angle XMQ+ \angle SNY =180^{\circ}$ and $\angle XMP +\angle RNY=180^{\circ}$

If three or more parallel lines make intercepts on a transversal in a certain proportion, then they make intercepts in the same proportion on any other transversal as well. In Fig, 1.06, the lines AB, CD and EF are parallel and the transversal XY cuts them at the points P, Q and R. If we now take a second transversal, UV, cutting the three parallel lines at the points J, K and L, then we have PQ/QR = JK/KL.

If three or more parallel lines make equal intercepts on one transversal, they make equal intercepts on any other transversal as well.

TRIANGLES

Sum of the three angles of a triangle is $180^{\circ}$

The exterior angle or triangle (at each vertex) is equal to the sum of the two opposite interior angles. (Exterior angle is the angle formed at any vertex, by one side and the extended portion of the second side at that vertex).

A line perpendicular to a side and passing through the midpoint of the side is said to be the perpendicular bisector of the side. It is not necessary that the perpendicular bisector of a side should pass through the opposite vertex in a triangle in general.

The perpendicular drawn to a side from the opposite vertex is called the altitude to that side.

The line joining the midpoint of a side with the opposite vertex is called the median drawn to that side. A median divides the triangle into two equal halves as far as the area is concerned.

An equilateral triangle is one in which all the sides are equal (and hence, all angles are equal, i.e., each of the angles is equal to $60^{\circ}$ ). An isosceles triangle is one in which two sides are equal (and hence, the angles opposite to them are equal). A scalene triangle is one in which no two sides are equal.

In an isosceles triangle, the unequal side is called the BASE. The angle where the two equal sides meet is called the VERTICAL ANGLE. In an isosceles triangle, the perpendicular drawn to the base from the vertex opposite the base (i.e., the altitude drawn to the base) bisects the base as well as the vertical angle. That is, the altitude drawn to the base will also be the perpendicular bisector of the base as well as the angular bisector of the vertical angle. It will also be the median drawn to the base.

In an equilateral triangle, the perpendicular bisector, the medium and the altitude drawn to a particular side coincide and that will also be the angular bisector of the opposite vertex. If a is the side of an equilateral triangle, then its altitude is equal to $\sqrt {3} a/2$

Sum of any two sides of a triangle is greater than the third side; difference of any two sides of a triangle is less than the third side.

If the sides are arranged in the ascending order of their measurement, then the angles opposite the sides (in the same order) will also be in ascending order (i.e. greater angle has greater side opposite to it); if the sides are arranged in descending order of their measurement, the angles opposite the sides in the same order will also be in descending order (i.e., smaller angle has smaller side opposite to it).

There can be only one right angle or only one obtuse angle in any triangle. There can also not be one right angle and an obtuse angle both present at the same time in a triangle. Hypotenuse is the side opposite the right angle in a right-angled triangle. In a right-angled triangle, hypotenuse is the largest side. In an obtuse angled triangle, the side opposite the obtuse angle is the largest side.

In a right-angled triangle, the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the squares on the other two sides. In Fig. 1.08, AC^2=AB^2+BC^2

In an acute angled triangle, the square of the side opposite the acute angle is less than the sum of the squares of the other two sides by a quantity equal to twice the product of one of these two sides and the projection of the second side on the first side. In Fig. 1.09, AC^2=AB^2+BC^2-2BC.BD

In an obtuse angled triangle, the square of the side opposite the obtuse angle is greater than the sum of the squares of the other two sides by a quantity equal to twice the product of one of the sides containing the obtuse angle and the projection of the second side on the first side. In Fig. 1.10, AC^2=AB^2+BC^2=2BC.BD

In triangle, the internal bisector of an angle bisects the opposite side in the ratio of the other two sides. In triangle ABC, if AD is the angular bisector of angle A, then

BD/DC = AB/AC. This is called the angular Bisector Theorem (refer to Fig. 1.11)

In $\triangle ABC$ if AD is the median from A to side BC (meeting BC at its mid point D), then 2(AD^2+BD^2)=AB^2+AC^2 . This is called the Apollonius Theorem. This will be helpful in calculating the lengths of the three medians given the lengths of the three sides of the triangle (refer to Fig. 1.12).