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TRIANGLES
In our daily life, we come across many figures with different
shapes and different sizes. If two figures are of same shape
and of same size, then they are called congruent figures. The
concept of two figures being congruent is called Congruence.
Two figures in a plane are congruent, if one figure can be
superimposed over the other covering it exactly. That means,
two triangles are congruent if and only if one can be
superimposed on the other covering it exactly.
If two triangles are equal then three angles of one triangle will
be equal to three corresponding angles of other triangle and
three sides of one triangle will be equal to three corresponding
sides of other triangle. So, in case of two congruent triangle,
six equalities exist.
1. Side Angle Side (SAS)
If two sides and angle between those sides of one triangle are
equal to the two sides and angle between those sides of other
triangle are equal, then the triangles are congruent.
B
C
Q
R
In ΔABC and ΔPQR ,
AB = PQ
∠B= ∠Q
BC = QR
∴ ΔABC ≅ ΔPQR
B
C
Q
R
If ΔABC , is congruent to Δ PQR, then this can be written as
Δ ABC ≅ Δ PQR.
ΔABC ≅ ΔPQR indicates not only the congruence, but also the
correspondence of congruence. So,
ΔABC ≅ PQR
Note
SAS means the equal angle must be between the equal sides,
otherwise the triangle need not be congruent. That means,
Or
SAS ≠ SSA
SAS ≠ ASS
2. Angle Side Angle (ASA)
If two angles and side between those two angles of one
triangle are equal to the two angles and side between those
two angles of other triangle, then the triangles are congruent.
indicates that
∠ A of ΔABC corresponds to ∠ P of ΔPQR ,
∠ B of ΔABC correspond to ∠ Q of ΔPQR
∠ C of ΔABC corresponds to ∠ R of ΔPQR
Side AB of ΔABC corresponds to side PQ of ΔPQR
Side BC of ΔABC corresponds to side QR of ΔPQR
Side AC of ΔABC corresponds to side PR of ΔPQR
In other words, ΔABC ≅ PQR represents the parts of one
triangle and their corresponding equal parts in other triangle.
That means the sequence of corresponding parts is very
important in showing that one triangle is similar to other
triangle. So, it will be wrong to write ΔABC ≅ ΔPQR as
∆ABC ≅ ∆QRP . Because in such case, the sequence of
corresponding parts is wrong.
Conditions For Congruence Of Triangle
We have studied that three vertices and three angles of one
triangle must be equal to three vertices and three angles of
other triangle for the congruence of triangle. But, all these six
conditions are not required to prove that two triangle are
congruent. So, minimum necessary conditions are sufficient to
prove that two triangles congruent.
B
C
Q
R
In ΔABC and ΔPQR ,
∠B= ∠Q
BC = QR
∠C= ∠R
∴ ΔABC ≅ ΔPQR
Note
Sum of three angles of triangle is 180°. So, if two angles of
one triangle are equal to two angles of other triangle, then
third angle of one triangle will automatically be equal to third
angle of other triangle. So, ASA is actually ASAA. That means,
ASA = ASAA
3. Angle Angle Side (ASA)
If two angles and one side of one triangle are equal to the
two angles and one side other triangle, then the triangles are
congruent.
∴ ΔABC ≅ ΔPQR
Note
RHS criteria is applied only in case of right angle triangles. If
the triangles are not right angled triangles, then RHS is not
applicable.
Equality Of Triangle
C
B
R
Q
In ΔABC and ΔPQR ,
∠A= ∠P
∠B= ∠Q
BC = QR
∴ ΔABC ≅ ΔPQR
There is a relation between equal sides and equal angles of a
triangle.
1.
Note
Sum of three angles of triangle is 180°. So, if two angles of
one triangle are equal to two angles of other triangle, then
third angle of one triangle will automatically be equal to third
angle of other triangle. So, AAS is actually AASA. That means,
AAS = AASA
In a triangle, equal sides have equal angles
opposite them.
Or
In a triangle, sides opposite equal angles are equal.
In a triangle, if two sides are equal then the angles opposite
those equal sides will also be equal.
4. Side Side Side (SSS)
If three sides of one triangle are equal to corresponding three
sides of other triangle, then the triangles are congruent.
B
C
B
R
Q
In ΔABC and ΔPQR ,
AB = PQ
BC = QR
CA = RP
∴ ΔABC ≅ ΔPQR
C
In ΔABC ,
∠B= ∠C
So,
AB = AC
2.
In a triangle, equal angles have equal angles
opposite them.
In a triangle, if two angles are equal then the sides opposite
those equal angles will also be equal.
Note
If three sides of two triangles are equal then the triangles are
congruent, but if three angles of two triangles are equal then
the triangle need not be congruent.
AAA is not congruence criterion.
5. Right Angle Hypotenuse Side (RHS)
In two right triangles, if hypotenuse and one side of one
triangle are equal to hypotenuse and one side of other
triangle, then triangles are congruent.
B
C
In ΔABC ,
AB = AC
So,
∠B= ∠C
Inequality Of Triangle
B
C
Q
In right ΔABC and right ΔPQR ,
∠ B = ∠ Q = 90°
AC = PR
BC = QR
R
We come across situations where two quantities are equal,
such as equal lines, equal angles, congruent triangles. It is
called equality. However, there are situations where quantities
are not equal, such as lines with different lengths, angles of
different degrees, dissimilar triangles. It is called the
inequality.
An inequality represents two things: (i) Two quantities are not equal.
(ii) One quantity is greater than or less than the other
quantity.
1.
In a triangle, larger side has larger angle opposite
it.
Or
In a triangle, larger angle has larger side opposite
it.
If two sides of a triangle are not equal, i.e. one side is greater
than the other side, than the larger side has greater angle
opposite it.
Note
If the sum of two sides of a triangle is not greater than the
third side, then such triangle can not be constructed.
3.
In a triangle, any side is greater than the
difference of other two sides.
A triangle has three sides. If we add any two sides, then their
sum will be greater than the third side.
B
B
We know, sum of two sides of triangle is greater than third
side. So,
C
In ∆ ABC,
AC > AB
AB has ∠ C opposite it and AC has ∠ B opposite it.
AC is greater than AB, so angle opposite AC will be greater
than angle opposite BC. That means,
∠B> ∠C
Also,
If two angle of a triangle are not equal, i.e., one angle is
greater than the other angle, then the greater angle has larger
side opposite it.
In ∆ ABC,
AB + BC > AC
⇒ AB > AC – BC
Or
AC + AB > BC
⇒ AC > BC – AB
Or
BC + AC > AB
⇒ BC > AB – AC
5.
B
C
C
In ∆ ABC,
∠ B has side AC opposite it and ∠ C has side AB opposite it.
∠ B greater than ∠ C, so side opposite ∠ C will be larger than
side opposite ∠ B. That means,
Of all the line segments that can be drawn to a
given line from a point not lying on the line, the
perpendicular is the shortest line segment.
If we take a line and a point at any place other than the line.
The shortest distance between the point and the line is the
perpendicular from that point to the line. In other words,
perpendicular from the point to the line is the shortest path
between point and the line.
AC > AB
2.
Sum of two sides of a triangle is greater than third
side.
l
A triangle has three sides. If we add any two sides, then their
sum will be greater than the third side.
A
B
C
D
E
l is a line and P is a point outside that line. PA, PB, PC, PD and
PE are different lines from point P to l. But, PC is
perpendicular, i.e. ∠ P = 90°.
So PC will be the shortest line segment from P to l.
B
In Δ ABC
AB + BC > AC
AB + AC > BC
BC + AC > AB
C
Theorem And Axiom
1.
Two triangles are congruent, if two sides and the
included angle of one triangle are equal to
corresponding two sides and included angle of
other triangle.
2.
Two triangles are congruent, if two angles and the
included side of one triangle are equal to
corresponding two angles and the included side of
other triangle.
3.
Two triangles are congruent, if any two angles and
a side of one triangle are equal to corresponding
two angles and side of other triangle.
4.
Two triangles are congruent, if three sides of one
triangle are equal to corresponding three sides of
other triangle.
5.
Two right angled triangles are congruent, if
hypotenuse and a side of one triangle are equal to
corresponding hypotenuse and a side of other
triangle.
6.
In a triangle, angles opposite equal sides are equal.
7.
In a triangle, sides opposite equal angles are equal.
8.
If two sides of a triangle are unequal, then larger
side has greater angle opposite it.
Or
In a triangle, the greater angle has larger side
opposite it.
9.
The sum of any two sides of a triangle is greater
than the third side.
10. Difference of any two sides of a triangle is less
than the third side.
11. In an isosceles triangle, bisector of vertical angle
bisects the base.
Or
If the bisector of the vertical angle of triangle
bisects the base then the triangle is isosceles.
12. In an isosceles triangle, altitude from the vertices
bisects the base.
Or
If the altitude from vertices of a triangle bisects
the opposite side, then the triangle is isosceles.
13. Of all the line segments that can be drawn to a
given line from a point not lying on the line, the
perpendicular is the shortest line segment.