Download ASSIGNMENT 9 – SIMILAR TRIANGLES

ASSIGNMENT 9 – SIMILAR TRIANGLES
Definition: Two triangles are similar, if and only if there is some way to match the
vertices of one triangle to those of the other such that the corresponding sides are in the
same ratio and corresponding angles are congruent. One triangle is a scaled- up version
of the other. The scaling factor is the constant of proportionality between the
corresponding sides.
Theorem 1: Suppose a line is parallel to one side of a triangle and intersects the other
two sides in different points. Then, this line divides the intersected sides into proportional
segments
A
D
E
l
B
( Given l is parallel to BC, we claim:
Exercise 1: Prove
C
BD CE

)
AD AE
AB AC

AD AE
Exercise 2: State and prove the converse of Theorem 1
Theorem 2: (AAA similarity condition) If in two triangles there is a correspondence in
which the three angles of one triangle are congruent to the three angles of the other
triangle, then the triangles are similar.
Theorem 2: (SAS similarity condition) If in two triangles there is a correspondence in
which two sides of triangle are proportional to two sides of the other triangles and the
included angles are congruent, then the triangles are similar.
Theorem 3: (SSS similarity condition). State and prove the SSS similarity condition.
Exercise 3: State and prove Menelaus’ Theorem (Exercise 2.5.7 in text)
Exercise 4: State and prove Blaise Pascal’s Theorem (Exercise 2.5.8 in text)