Triangles

We know that a closed figure formed by three
intersecting lines is called a triangle(‘Tri’ means ‘three’).A
triangle has three sides, three angles and three vertices.
For e.g.-in Triangle ABC, denoted as ∆ABC AB,BC,CA are
the three sides, ∠A,∠B,∠C are three angles and A,B,C are
three vertices.
A
B C

• A plane figure with three straight sides and
three angles.
• A triangle is a closed figure with three sides.
• Any of various flat, three-sided drawing and
drafting guides, used especially to draw
straight lines at specific angles.

A closed figure with three sides.

Acute triangles are triangles in
which the measures of all three
angles are less than 90 degrees.
Obtuse triangles are triangles in
which the measure of one angle is
greater than 90 degrees.
Right triangles are triangles in
which the measure of one angle
equals 90 degrees.

Equilateral triangles are triangles in
which all three sides are the same
length
Isosceles triangles are triangles in
which two of the sides are the same
length.
Scalene triangles are triangles in
which none of the sides are the
same length.

Let us take ∆ABC and ∆XYZ such that
corresponding angles are equal and corresponding sides
are equal:
A
B C
X
Y Z
Corresponding
Parts
∠A=∠X
∠B=∠Y
∠C=∠Z
AB=XY
BC=YZ
AC=XZ

Now we see that sides of ∆ABC coincides with
sides of ∆XYZ.
A
B C
X
Y Z
Two triangles are congruent, if all the sides and all the
angles of one triangle are equal to the corresponding
sides and angles of the other triangle.
Here, ∆ABC ≅ ∆XYZ

A corresponds to X
B corresponds to Y
C corresponds to Z
For any two congruent triangles the
corresponding parts are equal and are termed as:
CPCT – Corresponding Parts of Congruent Triangles

• Two triangles are congruent if two sides and
the included angle of one triangle are equal to
the two sides and the included angle of other
triangle.
A
B C
P
Q R
S(1) AC = PQ
A(2) ∠C = ∠R
S(3) BC = QR
Now If,
Then ∆ABC ≅ ∆PQR (by SAS congruence)

• Two triangles are congruent if two angles and
the included side of one triangle are equal to two
angles and the included side of other triangle.
A
B C
D
E F
Now If,
A(1) ∠BAC = ∠EDF
S(2) AC = DF
A(3) ∠ACB = ∠DFE
Then ∆ABC ≅ ∆DEF (by ASA congruence)

•Two triangles are congruent if any two pairs of
angle and one pair of corresponding sides are
equal.
A
B C P
Q
R
Now If,
A(1) ∠BAC = ∠QPR
A(2) ∠CBA = ∠RQP
S(3) BC = QR
Then ∆ABC ≅ ∆PQR (by AAS
congruence)

• If three sides of one triangle are equal to
the three sides of another triangle, then the
two triangles are congruent.
Now If, S(1) AB = PQ
S(2) BC = QR
S(3) CA = RP
A
B C
P
Q R
Then ∆ABC ≅ ∆PQR (by SSS congruence)

• If in two right-angled triangles the hypotenuse
and one side of one triangle are equal to the
hypotenuse and one side of the other triangle, then
the two triangles are congruent.
Now If, R(1) ∠ABC = ∠DEF = 90°
H(2) AC = DF
S(3) BC = EF
A
B C
D
E F
Then ∆ABC ≅ ∆DEF (by RHS congruence)

• If ADE is any triangle and BC is drawn
parallel to DE, then AB/BD = AC/CE

• If ADE is any triangle and BC is drawn parallel to
DE, then AB/BD = AC/CE
• To show this is true, draw the line BF parallel to AE
to complete a parallelogram BCEF:
 Triangles ABC and BDF have exactly the same
angles and so are similar

• If two similar triangles have sides in the ratio x:y,
then their areas are in the ratio x2:y2
Example:
• These two triangles are similar with sides in the ratio 2:1
(the sides of one are twice as long as the other):
• What can we say about their areas?

• The answer is simple if we just draw in three more
lines:
• We can see that the small triangle fits into the big
triangle four times.
• So when the lengths are twice as long, the area
is four times as big
• So the ratio of their areas is 4:1
• We can also write 4:1 as 22:1

1. Two figures are congruent, if they are of the same
shape and size.
2. If two sides and the included angle of one triangle is
equal to the two sides and the included angle then
the two triangles are congruent(by SAS).
3. If two angles and the included side of one triangle
are equal to the two angles and the included side of
other triangle then the two triangles are congruent(
by ASA).

4. If two angles and the one side of one triangle is
equal to the two angles and the corresponding side
of other triangle then the two triangles are
congruent(by AAS).
5. If three sides of a triangle is equal to the three
sides of other triangle then the two triangles are
congruent(by SSS).
6. If in two right-angled triangle, hypotenuse one side
of the triangle are equal to the hypotenuse and one
side of the other triangle then the two triangle are
congruent.(by RHS)

7. Angles opposite to equal sides of a triangle are equal.
8. Sides opposite to equal angles of a triangle are equal.
9. Each angle of equilateral triangle are 60°
10. In a triangle, angles opposite to the longer side is larger
11. In a triangle, side opposite to the larger angle is longer.
12. Sum of any two sides of triangle is greater than the
third side

Triangles

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Triangles