CHAPTER 6 – TRIANGLES
CONCEPT NOTES
Two polygons are said to be similar if they have
1. Same shape
2. Same or different size (All congruent shapes are similar but all similar shapes
are not
congruent.)
Two polygons are similar if following two criteria are met-
1. Their corresponding angles are equal, and
2. Their corresponding sides are in the same ratio (or proportion)
Mathematical Expression for Similarity
In Δ ABC and Δ DEF, if
(i) ∠ A = ∠ D, ∠ B = ∠ E, ∠ C = ∠ F and
(ii) AB
DE
=
BC
EF
=
CA
FD
,
then Δ ABC ~ Δ DEF.
Criteria of Similarity
Side-Side-Side (SSS) Similarity Criterion – When the corresponding sides of any
two
triangles are in the same ratio, then their corresponding angles will be equal, and
the triangle
will be considered similar triangles.
Angle Angle Angle (AAA) Similarity Criterion – When the corresponding angles of
any two
triangles are equal, then their corresponding side will be in the same ratio, and
the triangles
are considered to be similar.
Angle-Angle (AA) Similarity Criterion – When two angles of one triangle are
respectively
equal to the two angles of the other triangle, then the two triangles are
considered similar.
Side-Angle-Side (SAS) Similarity Criterion – When one angle of a triangle is equal
to one
angle of another triangle and the sides including these angles are in the same
ratio
(proportional), then the triangles are said to be similar.
Basic Proportionality Theorem (BPT)
The basic proportionality theorem states that “If a line is drawn parallel to one
side of a triangle
intersecting the other two sides in distinct points, then the other two sides are
divided in the same
ratio.”
Thus, according to the Basic proportionality theorem, if PQ ∥ BC then AP
PB
=
AQ
CONCEPT NOTES
Two polygons are said to be similar if they have
1. Same shape
2. Same or different size (All congruent shapes are similar but all similar shapes
are not
congruent.)
Two polygons are similar if following two criteria are met-
1. Their corresponding angles are equal, and
2. Their corresponding sides are in the same ratio (or proportion)
Mathematical Expression for Similarity
In Δ ABC and Δ DEF, if
(i) ∠ A = ∠ D, ∠ B = ∠ E, ∠ C = ∠ F and
(ii) AB
DE
=
BC
EF
=
CA
FD
,
then Δ ABC ~ Δ DEF.
Criteria of Similarity
Side-Side-Side (SSS) Similarity Criterion – When the corresponding sides of any
two
triangles are in the same ratio, then their corresponding angles will be equal, and
the triangle
will be considered similar triangles.
Angle Angle Angle (AAA) Similarity Criterion – When the corresponding angles of
any two
triangles are equal, then their corresponding side will be in the same ratio, and
the triangles
are considered to be similar.
Angle-Angle (AA) Similarity Criterion – When two angles of one triangle are
respectively
equal to the two angles of the other triangle, then the two triangles are
considered similar.
Side-Angle-Side (SAS) Similarity Criterion – When one angle of a triangle is equal
to one
angle of another triangle and the sides including these angles are in the same
ratio
(proportional), then the triangles are said to be similar.
Basic Proportionality Theorem (BPT)
The basic proportionality theorem states that “If a line is drawn parallel to one
side of a triangle
intersecting the other two sides in distinct points, then the other two sides are
divided in the same
ratio.”
Thus, according to the Basic proportionality theorem, if PQ ∥ BC then AP
PB
=
AQ
