1) Triangles are three-sided polygons formed by three line segments. The sum of the three interior angles is always 180 degrees. 2) There are certain properties that apply to all triangles, such as being rigid flat shapes that satisfy the Triangle Inequality. 3) Triangles can be categorized based on their sides and angles. Congruent triangles are identical in shape and size, while similar triangles have the same shape but may differ in size. The properties of similar triangles can be used to solve proportional relationships.

1. Some Basic Facts of Triangles

2. Some Basic Facts of Triangles
A triangle is a three–sided polygon,
formed by three line segments.
The standard way of labeling a triangle
is to label a side and it's opposite angle
with the same letter as shown.
Basic Facts of Triangles
* All triangles are rigid. 1 1
* All triangles are flat, i.e. every triangle lies in a plane.
A shape made from 4 or more sides
might be bent and protrudes into space
as shown.
Each side of a triangle ties down
the other two sides so they can't
move. Shapes with 4 or more
sides may be squashed as shown.
1
1
1
1
1
1
1
1

3. Some Basic Facts of Triangles
Two sides of any triangle can’t be too
short such that they can't meet as shown.
This translates into an inequality about the lengths of the sides.
* (The Triangle Inequality)
Let x, y, and z be the lengths of the sides of a triangle.
The sum of the lengths of any two sides is more than the length
of third side, i.e. x + y > z, x + z > y, and y + z > x.
Example A.
Hence the length of the third side must be between 8 and 38.
2315The third side must be
less than 15 + 23 = 38,
but must be
more than 23 – 15 = 8.
more than 8
23
less than 38
15
Example A.
a. The lengths of two sides of a triangle are 15 and 23.
The length of the third side must be between what values?

4. Some Basic Facts of Triangles
Since triangles are flat, we can joint the three angles as shown.
a c
a
b

5. Some Basic Facts of Triangles
Since triangles are flat, we can joint the three angles as shown.
a
b
c
a c
c

6. Some Basic Facts of Triangles
Since triangles are flat, we can joint the three angles as shown.
a
b
c
a c
c c

7. Some Basic Facts of Triangles
Since triangles are flat, we can joint the three angles as shown.
So the angles of any triangle may be joint into a straight line.
a
b
c
a c
c c

8. Some Basic Facts of Triangles
Since triangles are flat, we can joint the three angles as shown.
So the angles of any triangle may be joint into a straight line.
* The sum of all three angles is 1800.
a
b
c
a c
c c

9. Fact About Similar Triangles
congruent triangles
(the same)
Two triangles that are exactly
the same so they may be
stacked on top of each other
perfectly are said to be
congruent.
A triangle that has two equal sidesA triangle that has two equal sides
is called an isosceles triangle.
* An isosceles triangle
may be cut into two
congruent right triangles
that are the mirror images
of each other.
isosceles triangles

10. Fact About Similar Triangles
congruent triangles
(the same)
Two triangles that are exactly
the same so they may be
stacked on top of each other
perfectly are said to be
congruent.
A triangle that has two equal sidesA triangle that has two equal sides
is called an isosceles triangle.
* An isosceles triangle
may be cut into two
congruent right triangles
that are the mirror images
of each other.
isosceles triangles

11. Cross–Multiplication for Solving Proportional Equations
If A = C then AD = BC.
B D ,
Example B. Solve for x.
Cross
x 2 2 x
a. 3 = 5 c. (x + 1) = 3
B D,
are said to be in proportion or proportional.
Review on Proportions
Two equal ratios A:B = C:D, or two fractions that A = C
CrossCross
6 = 5x
=b.
5
2 (x + 2)
3 (x – 5)
6 =x
x 2 2 x
x(x + 1) = 6
x2 + x = 6
x2 + x – 6 = 0
(x + 3) (x – 2) = 0
So the solutions are
x = –3 and x = 2.
2(x – 5) = 3(x + 2)
2x – 10 = 3x + 6
–10 – 6 = 3x – 2x
–16 = x
Cross
Cross

12. Fact About Similar Triangles
CA B
Two triangles are similar (the same shape)
if the corresponding sides of triangles are proportional.
So given the following triangles,
they are similar means that
a = b = c
that is, all three fractions are the same.
From this triple–equal–ratios,
a
=
A a
= A
b B, c C ,
b
= B
C , c
Example C. Given that the following
triangles are similar, what is x?
5
4
3
10
8
etc..
x
We must have that x = 10 = 2 so x = 6.
3 5
From this triple–equal–ratios,
we get the proportions:

13. Examples of Similar Triangles
Similar triangles appear in the following situations.
I. Given that DE is parallel to AB,
then ΔABC is similar toΔDEC.
II. Given a right triangle as shown,
with CD perpendicular to AB,
A
B
C
D

14. Examples of Similar Triangles
Similar triangles appear in the following situations.
I. Given that DE is parallel to AB,
then ΔABC is similar toΔDEC.
II. Given a right triangle as shown,
with CD perpendicular to AB,
then the triangles
ΔABC, ΔACD, andΔBCD
are similar.
A
B
C
D

15. Examples of Similar Triangles
Example D. Find x, assuming the
sides are parallel.
By similar triangles,
x + 6 = 5
6 3
3
x + 6 = 6*5 = 10
So x = 4So x = 4
Example E. Find x, assuming
the sides are parallel.
By similar triangles,
x 4
x + 5 = 10 4(x + 5) = 10x
4x + 20 = 10x
20 = 6x
x = 20/6 = 10/3

16. 5
x
6
5
9
x
5
x
6
6 4
14
x x
6 x–3
14
x
6 4
x + 6
a. b. c.
d. e. f.
Ex. (Similar Triangles) Solve for the variables.
5
9
5
x+8 x+3
4
5
x
y
z 1
2
x
y
z
1 2
x
y
z
g. h. i.