Classifying Triangles
We can classify triangles two ways - by their angles or by their sides. All triangles have at least two acute angles, but it’s the third angle that is used to classify the triangle.
Classification of Triangles by Angles
We can classify triangles two ways - by their angles or by their sides. All triangles have at least two acute angles, but it’s the third angle that is used to classify the triangle.
Classification of Triangles by Angles
Classification of Triangles by Side
Example 1
Classify each triangle as acute, equiangular, obtuse or right.
Classify each triangle as acute, equiangular, obtuse or right.
Solution:
a) equiangular
b) obtuse
c) right
d) acute
a) equiangular
b) obtuse
c) right
d) acute
Example 2:
Classify each triangle as equilateral, isosceles or scalene.
a) b)
Classify each triangle as equilateral, isosceles or scalene.
a) b)
Solution:
a) isosceles
b) scalene
a) isosceles
b) scalene
Example 3:
Find \(x\) and the measures of the unknown sides of each triangle.
a) b)
Find \(x\) and the measures of the unknown sides of each triangle.
a) b)
Solution:
a) \(5x - 8 = 3x + 2\) \(2x = 10\) \(x = 5\) \(CA = 5x - 8 = 5(5) - 8 = 25 - 8 = 17\) \(AT = 3x + 2 = 3(5) + 2 = 15 + 2 = 17\) |
b) \(6x - 2 = 3x + 1\) or \(4x = 6x - 2\) or \(4x = 3x + 1\) \(3x = 3\) \(-2x = -2\) \(x = 1\) \(x = 1\) \(x = 1\) \(DO = 4x = 4(1) = 4\) \(OG = 3x + 1 = 3(1) + 1 = 4\) \(DG = 6x - 2 = 6(1) - 2 = 4\) |
Example 4:
Find the measures of the missing angles.
Find the measures of the missing angles.
Solution:
\(m\angle 1 = 180^{\circ} - 37^{\circ} = 143^{\circ}\)
\(m\angle 2 = 180^{\circ} - (143^{\circ} + 22^{\circ}) = 15^{\circ}\)
\(m\angle 3 = 90^{\circ} - 37^{\circ} = 53^{\circ}\)
Example 5:
Find \(x\) and the measure of each angle.
\(m\angle 1 = 180^{\circ} - 37^{\circ} = 143^{\circ}\)
\(m\angle 2 = 180^{\circ} - (143^{\circ} + 22^{\circ}) = 15^{\circ}\)
\(m\angle 3 = 90^{\circ} - 37^{\circ} = 53^{\circ}\)
Example 5:
Find \(x\) and the measure of each angle.
Solution:
\(\begin{align*}\\ (5x - 5 + 4x + 8 + 3x + 9)^{\circ} &= 180^{\circ}\\ 12x + 12 &= 180\\ 12x &= 168\\ x &= 14 \end{align*}\) |
Angle measures: \((5x - 5)^{\circ} = (5(14) - 5)^{\circ} = 65^{\circ}\) \((4x + 8)^{\circ} = (4(14) + 8)^{\circ} = 64 ^{\circ}\) \((3x + 9)^{\circ} = (3(14) + 9)^{\circ} = 51^{\circ}\) |
Exterior Angles
In addition to the three interior angles that a triangle has, it can also have exterior angles formed by one side of the triangle and the extension of an adjacent side. Each exterior angle of a triangle has two remote interior angles that are not adjacent to the exterior angle.
In addition to the three interior angles that a triangle has, it can also have exterior angles formed by one side of the triangle and the extension of an adjacent side. Each exterior angle of a triangle has two remote interior angles that are not adjacent to the exterior angle.
Example 6:
Find the measure of the missing angle.
Find the measure of the missing angle.
Solution:
\(102^{\circ}=m\angle 2 + 27^{\circ}\)
\(75^{\circ} = m\angle 2\)
Example 7:
Find \(m\angle CAB\)
\(102^{\circ}=m\angle 2 + 27^{\circ}\)
\(75^{\circ} = m\angle 2\)
Example 7:
Find \(m\angle CAB\)
Solution:
\(\begin{align*}\\
(3x - 13 + x + 6)^{\circ} &= 151^{\circ}\\
4x - 7 &= 151\\
4x &= 158\\
x &= 39.5\
\end{align*}\)
\(m\angle CAB = (3(39.5) - 13)^{\circ} = 105.5^{\circ}\)
\(\begin{align*}\\
(3x - 13 + x + 6)^{\circ} &= 151^{\circ}\\
4x - 7 &= 151\\
4x &= 158\\
x &= 39.5\
\end{align*}\)
\(m\angle CAB = (3(39.5) - 13)^{\circ} = 105.5^{\circ}\)
Example 8:
Find the value of \(x\).
Find the value of \(x\).
Solution:
\(\begin{align*}\\
65^{\circ} + 90^{\circ} &= 155^{\circ}\\
180^{\circ} - 155^{\circ} &= 25^{\circ}\\
x & = 25^{\circ}
\end{align*}\)
\(\begin{align*}\\
65^{\circ} + 90^{\circ} &= 155^{\circ}\\
180^{\circ} - 155^{\circ} &= 25^{\circ}\\
x & = 25^{\circ}
\end{align*}\)
Quick Check
1) Find \(x\) and the measures of the unknown sides of each triangle.
1) Find \(x\) and the measures of the unknown sides of each triangle.
2) Find the measure of each missing angle.
3) Find the measure of the exterior angle.