Triangles
Intro
Triangles are polygons with three sides.
You can spot triangles throughout nature and their shape is used in buildings, bridges and all sorts
of other places. One of the reasons why triangles are used so much is because they are one of the
strongest shapes and can support heavy loads.
Classifying Triangles
Triangles are classified by either their sides or angles; he tables below show the different types.
Table 1: Classifying by Sides
Triangle Name
Description
Scalene
No sides congruent.
Isosceles
At least two sides congruent.
Equilateral
All sides congruent.
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Example
Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles in solving problems.
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Table 2: Classifying by Angles
Triangle Name
Description
Right
Exactly one right angle.
Equiangular
All interior angles are congruent; they each
measure exactly 60°.
Acute
All interior angles are less than 90°.
Example
44°
70°
Obtuse
66°
Exactly one interior angle measures more
than 90°.
136°
Anatomy of a Triangle
Triangles are named with three distinct letters. Take a look at the figure below:
B
Vertex – where two
sides meet.
Side – segment which
connects two vertices.
Interior Angle
Exterior Angle
A
C
This triangle can be named: ΔABC, ΔBCA, ΔCAB, ΔACB or ΔBAC (Note: The “Δ” is the symbol for
triangle).
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Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles in solving problems.
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Area of Triangles
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The area of a triangle can be given by: bh , where b is the base of the triangle and h is the height.
Consider the following examples:
Notice that our height, 7 cm, is drawn. Remember, the height of a
figure must be perpendicular (form a 90° angle), therefore we can
not use the slanted side (9 cm) as our height.
9 cm
7 cm
To find the area of this triangle, we will substitute our given values
into our formula:
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(12)(7) = 42 cm2.
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12 cm
When you are working with a right triangle, the height and base are
interchangeable since those two sides form a right angle. Take a
look at the following example:
Notice that our height is also the side of our triangle. To find the
area, we will substitute our given values into our formula:
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(8)(11) = 44 mm2.
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Let’s look at how our height is defined in an obtuse triangle.
As you can see, our height is drawn as an extension of the
triangle. Obtuse triangles can have the height drawn
outside or inside the triangle, just remember the height
will always form a 90° angle.
15 mm
8 mm
11 mm
9 in
5 in
In this example, we will again substitute our given values into our
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formula: (5)(4) = 10 in2.
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Triangle Sum Theorem
All triangles have a unique characteristic: The sum of the
interior angles of any triangle is always 180°.
This is because if we tore each of the three angles from a
triangle and connected them at one point, they would form a
straight line (which is always 180°).
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Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals and circles in solving problems.
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