Triangles. A triangle is a polygon with three sides.
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Transcript of Triangles. A triangle is a polygon with three sides.
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TrianglesTriangles
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A triangle is a polygon with three sides.
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Classifying Triangles
For example,
A
B
C
We name a triangle using its vertices.
∆ABC
∆BAC
∆CAB ∆CBA
∆BCA
∆ACB
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Opposite Sides and Angles
A
B
C
We say that is opposite .ABC
What is opposite ?C
What is opposite of ?CA
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Triangles can be classified by their
Sides
• Scalene
• Isosceles
• Equilateral
Angles
• Acute
• Right
• Obtuse
• Equiangular
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Equilateral TriangleA triangle in which all 3 sides are equal
m AC = 7.00 cm
m BC = 7.00 cmm AB = 7.00 cm
A
B
C
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Isosceles TriangleA triangle in which at least 2 sides are equal
m AC = 7.90 cm
m BC = 5.51 cmm AB = 5.51 cm
A
B
C
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Scalene TriangleA triangle in which all 3 sides are different lengths
m AC = 8.21 cm
m BC = 6.60 cmm AB = 4.65 cm
A
B
C
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Acute Triangle
A triangle in which all 3 angles are less than 90˚
mCBA = 78°
mBCA = 58°
mBAC = 44°
A
B
C
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Right TriangleA triangle in which exactly one angle is 90˚
mCBA = 90°
mBCA = 55°
mBAC = 35°
A
B
C
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Obtuse TriangleA triangle in which exactly one angle is greater than 90˚and
less than 180˚
mCBA = 119°
mBCA = 40°
mBAC = 21°
A
B
C
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Equiangular Triangle
A triangle in which all 3 angles are the same measure.
mCBA = 60°
mBCA = 60°mBAC = 60°
A
B
C
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Angles When the sides of a polygon are extended, other anglesare formed. The inside/original angles are the interior angles.The adjacent/outside angles that form linear pairswith the interior angles are the exterior angles.
Interior angles
Exterior angles
3
1
2
5
4
6
<4, <5, <6
<1, <2, <3
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TRIANGLE INVESTIGATION
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Triangle Sum TheoremThe sum of the interior angles in a triangle is
180˚.
60 80
40
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Find the value of x.
Example:
x3x
2x
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EXTERIOR TRIANGLE INVESTIGATION
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Exterior Angle TheoremThe measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
Exterior AngleRemote Interior Angles
A
BC D
m ACD m A m B
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Find the value of x.
Example:
x
70
(2x+10)
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A corollary to a theorem is a statement thatcan be proven easily using another theorem.
Corollary
Definition:
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Third Angle CorollaryIf two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent.
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Equiangular Corollary
Each angle in an equiangular triangle is 60˚.
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Right Angle Corollary
There can be at most one right or obtuse angle in a triangle.
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Acute Corollary
Acute angles in a right triangle are complementary.