Download Math 366 Lecture Notes Section 12.4 – Similar Triangles and Similar

Section 12-4
Math 366 Lecture Notes
Section 12.4 – Similar Triangles and Similar Figures
Two figures that have the same shape but not necessarily the same size are similar. The ratio of
the corresponding side lengths is the scale factor.
Similar Triangles
∆ABC is similar to ∆DEF (∆ABC ∼ ∆DEF) iff ∠ A ≅ ∠ D, ∠ B ≅ ∠ E, ∠ C ≅ ∠ F, and
AB AC BC
=
=
.
DE DF EF
B
A
E
D
C
F
Are all equilateral triangles similar?
Theorem 12-10
SSS Similarity for Triangles
If corresponding sides of two triangles are proportional, then the triangles are similar.
Theorem 12-11
SAS Similarity for Triangles
Given two triangles, if two sides are proportional and the included angles are congruent, then the
triangles are similar.
Theorem 12-12
Angle, Angle (AA) Similarity for Triangles
If two angles in one triangle are congruent, respectively, to two angles of a second triangle, then
the triangles are similar.
1
Section 12-4
In the figure below, given BD=12, solve for x.
D
x
A
8
E
C
5
B
Properties of Proportion
B
B
x
a
z
b
c y
m
y
w
A
e
C
d
A
f
g
h
C
Theorem 12-13
If a line parallel to one side of a triangle intersects the other sides, then it divides those sides into
proportional segments.
Theorem 12-14
If a line divides two sides of a triangle into proportional segments, then the line is parallel to the
third side.
Theorem 12-15
If parallel lines cut off congruent segments on one transversal, then they cut off congruent
segments on any transversal.
Construction Separating a Segment into Congruent Parts
Separate AB into three congruent parts.
A
B
2
Section 12-4
1. Draw any ray, AC , such that A, B, and C are noncollinear.
2. Mark off the given number of congruent segments (of any size) on AC . In this case, we
use three congruent segments.
3. Connect B to A3, the endpoint of the last of the three congruent segments.
4. Through A2 and A1, construct parallels to BA3 .
Note: There is no construction to trisect an angle.
Midsegments of Triangles and Quadrilaterals
A midsegment is a segment that connects midpoints of two adjacent sides of a triangle or
quadrilateral.
Theorem 12-16 The Midsegment Theorem
The midsegment is parallel to the third side of the triangle and half as long.
B
D
E
A
C
Theorem 12-17
If a line bisects one side of a triangle and is parallel to a second side, then it bisects the third side
and therefore is a midsegment.
B
D
C
A
E
3
Section 12-4
Indirect Measurements
Similar triangles have long been used to make indirect measurement (such as determining the
height of the Great Pyramid of Egypt) by using ratios involving shadows.
B
B’
A’
C’
D
A
4
C