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Properties of Triangles
EQ: How can I determine if a set of attributes will create one
unique triangle, many triangles, or no triangle at all?
What is the
triangle
inequality
theorem?
The Triangle Inequality Theorem states that the sum of any two
sides must be greater than the length of the third side.
a+b>c
Example 1:
Side lengths 3, 4, & 5 can create one unique triangle because:
 3+4>5
 4+5>3
 3+5>4
Example 2:
Side lengths 3, 4, & 9 cannot create a triangle because:
 3+4<9
Sample Problem 1:
If the two smaller sides of a triangle have lengths 6 cm and 8 cm,
the third side must be less than ________ cm.
Sample Problem 2:
If two sides of a triangle have lengths 5 cm and 8 cm, the third
side must be greater than ____________ and less than
___________.
What is the
The triangle angle-sum theorem states that all three angles of
triangle angle- any triangle must have a sum of 180°.
sum theorem?
Example 1: 30°, 40°, & 110° angles can form triangles because
their sum is 180°
Example 2: 45°, 50°, & 90° do not form a triangle because 45 +
50 + 90 > 180.
Similar triangles will have congruent angles; which means that
any set of angles that add up to 180 degrees can create more
than one triangle with those measurements.
Example:
The triangles above have the same angle measurements because
they are similar triangles.
Angle-Angle-Angle (AAA) - If you are given 3 angle
measurements only, you can create many triangles, as long as
the 3 angle measurements total up to 180.
Which
combinations
of known
sides and
angles will
create a
unique
triangle?
If you know the length of two adjacent sides and their included
angle, then you can only make one unique triangle.
(SAS - Side-Angle-Side)
This is because the only way to complete the triangle is to
connect the two given sides.
Example:
Likewise, If you know two angle measurements and the side
length that connects them, then you can only make one unique
triangle. (ASA - Angle-Side-Angle)
This is because the only way to complete the triangle is to extend
the two angles until they intersect.
Example:
If you are given three side lengths that meet the requirements of
the triangle inequality theorem (a + b > c), then they will create
only one unique triangle (SSS - Side-Side-Side)
Example:
Given side lengths 3 cm, 4 cm, & 5 cm there is only one way to
put those together to form a triangle.