TRIANGLES
Triangle : A triangle is defined as a plane figure bounded by three sides. It has three angles.
Types of Triangles (On the basis of sides) :
- Scalene Triangle : If a triangle is having all sides unequal it is known as scalene
- Isosceles Triangle : A triangle having two sides ang two angles equal is called isosceles
- Equilateral Triangle : A triangle having all the sides and all the angles equal is called an equilateral
On the Basis of Angles
- Acute triangle : A triangle which is having all angles acute is known as an acute angled
- Right angle triangle : A triangle having one angle as 900 is known as right angle
- Obtuse triangle : A triangle having one angle as obtuse is called obtuse angle
Properties of Triangles :
- The total sum of all the three angles of a triangle is 1800.
- Since the sum of the three angles of a triangle is 1800, therefore a triangle can have only one right
- In a 900 triangle (right triangle), the sum of the remaining two acute angles is 900 e. either 450 or 300 and 600.
- A triangle can have only one obtuse
- In an isosceles triangle the angles opposite to equal sides are also
Exterior Angle of a triangle :
If any one side of a triangle is produced, the exterior angle created is equal to the sum of two interior opposite angles.
Consider a triangle ALC in which side. LC is produced to D. In this case the
<ACD is exterior angle.
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Exterior angle <ACD is equal to the aum of <BAC and <ABC.
Median of a triangle : A line from the vertex of a triangle to the middle point of the opposite side is known as the median. AD is the median. There are three medians in a triangle from the three vertices.
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Altitudes of a triangle : The perpendicular line from the vertex of a triangle to the opposite side is known as altitude. AD is the altitude. There are three altitudes of a triangle from three vertices.
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- The line joining the two middle points of two sides in a triangle is parallel to the third It is equal to the half of the third side e.g.
DE || MN
and DE = 1/2 MN.
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Pythagoras Theorem : Samos Pythagoras was a Greek mathematician who gave a very important theorem of a right angled triangle. This theorem states that the square of hypotenuse in a right angled triangle is equal to the sum of the squares of the other two sides e.g.
AN2 = AM2 + MN2
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Converse of Pythagoras Theorem :
In a triangle, if the square of one side is equal to the sum of the squares of remaining two sides, then the triangle is right angled.
Theorem on Acute triangle : In an acute triangle ABC, in which B is acute and AN is perpendicular to BC then
AC2 = AB2 + BC2 – 2 BC C BN
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Obtuse triangle Theorem : In an obtuse triangle ABC in which B > 90 and AN is perpendicular as shown than
AC2 = AB2 + BC2 + 2Bc C BN
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Congruent triangles : Congruent figures are those which have the same size and shapes. The symbol @ is used to show congruency.
Two triangles are congruent under following conditions.
- SAS (Side-angle-side) congruence : Two triangles are said to be congruent if any two sides of one triangle and the included angle are equal to the corresponding sides and the included angle of the other
- ASA (angle-side-angle) congruence : Two triangles are said to be congru- ent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other
- AAS (angle-angle-side) congruence : Any two angles and a non-included side of one triangle are equal to the two angles and side of another triangle, then the two triangles are said to be
- SSS (side-side-side) congruence : If three sides of one triangle are equal to the corresponding three sides of the other triangle, the two triangles are said to be
- If the hypotenuse and one side to one triangle are equal to the hypotenuse and the corresponding side of the other The two right triangles are congruent.
Important results about congruent trinagles : (1) Two congruent triangles are definitely equiangular but equilangular triangls need not be congruent. There is no condition like AAA for congruence of triangles.
(2) If two triangles are congruent, then they are similar. Its converse is not true. In triangles ABC and DEF.
AB AC BC
(3) —– = —– =—— = a constant = k (say)
DE DF EF
When k = 1, we get congruent figures. The ratio of corresponding sides is K.
The corresponding sides are parallel to each other. The ratio of the areas of two similar figures is k2.
In solids, the ratio of the volumes of two similar solids will be k3.
- The order of letters in the two triangles also indicates the correspondence between the vertices of the two tirangles. Then from the manner the two triangles are named,, we can easily infer the six equalities between the corresponding parts of the two congruent (Abbreviated as C.P.C.T.)
- Congruent triangle cover each other on
- In congruent triangles corresponding sides lie opposite to equal angles and corresponding angles lie opposite to equal
Inequality relations in a triangle :
- In a triangle the greater side has the greater angle opposite to
- In a triangle if two angles are unequal, the greater angle has the greater side opposite to
- The sum of any two sides of a triangle is greater than the third
- The difference of any two sides of a triangle is less than the third
(1) Similarity of two triangles : Figures having same shape but not necessarily the same size are called similar figures. The symbol – stands for ‘is similar to’.
Two triangles are said to be similar if,
- Their corresponding angles are equal, and
- The lengths of their corresponding sides are
(2) Condition for similarity of two triangles :
(SAS) Similarity : If a pair of corresponding angles are equal and the sides including them are proportional, then the triangles are similar.
(AA) Similarity : If two pairs of corresponding angles are equal, then triangles are similar.
(SSS) Similarity : If three pairs of corresponding sides are proportional then the two triangles are similar.
(3) Characteristic Properties of Similar Triangles :
- AAA Similarity or AA
- SSS
- SAS
- Similar triangles not necessarily be To establish the similarity of two Ds, it is sufficient to satisfy one condition. The Ds are similar if.
- Their corresponding angles are
- Their corresponding sides are
Basic proportionality theorem : When a line is drawn parallel to one side of a triangle intersecting the other two sides are divided in the same ratio.
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AG AH MG HN Then —– = —– or —– = on
MG HN AN AH
AM AN
adding 1 to bothe sides we get —— = ——
MG AH
A perpendicular drawn from the vertex at 900 of a right-angled triangle divides the triangle into two triangles similar to each other and also to the original triangle.
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then DMNE = DAME = DAMN.
RESULTS ON RATIO OF THE AREAS OF TWO SIMILAR TRIANGLES
Theorem 1 : The ratio of the areas of two similar triangles is equal to the ratio of the square of any two corresponding sides.
Theorem 2 : The ratio of the areas of two similar triangles is equal to the Squares of the corresponding altitudes.
Theorem 3 : The ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding medians.
Theorem 4 : The ratio of areas of two similar triangles is equal to the ratio of the squares of corresponding angle bisector lines.
Theorem 5 : If the areas of two similar triangles are equal, then the triangles are congruent.
The line joining the mid-point of any two sides of a triangle, is parallel to the third side and half of it.
The line drawn through the mid-point of one triangle, parallel to another side, intersects the third side at its mid-point.
If we have l, m, and n three parallel lines and the intercepts made by them on one transversal are equal, then the intercepts on any other transversal are also equal.
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—————————– Here. AB = BC = DL = LF.
