© Boardworks 2012 1 of 11 Triangles. © Boardworks 2012 2 of 11 Information.
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Transcript of © Boardworks 2012 1 of 11 Triangles. © Boardworks 2012 2 of 11 Information.
© Boardworks 20121 of 11
TrianglesTriangles
© Boardworks 20122 of 11
Information
© Boardworks 20123 of 11
Triangles
A triangle is a polygon made up of three connected line segments that form a closed figure.
AB, BC and AC are the line segments that make up the triangle.
Triangles are named using the triangle symbol, △, and its vertices, e.g., △ABC.
The interior is the set of points inside the triangle.
The exterior is the set of points outside the triangle.
interior
exterior
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Classifying triangles by angle
What is special about the angles in each of the triangles?
acute trianglethree acute
angles
obtuse triangleone obtuse
angle
right triangleone right angle
equiangular triangle
three congruent angles
© Boardworks 20125 of 11
Classifying triangles by side length
What is special about the sides of each triangle?
equilateral triangle
three congruent sides
isosceles triangle
two congruent sides
scalene triangle
no congruent sides
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Classifying triangles
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Angle relationships in triangles
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Triangle sum theorem
Triangle sum theorem: all three angle measures in a triangle add up to 180°.
m∠1 + m∠2 + m∠3 = 180°
Prove the triangle sum theorem.m∠1 = m∠4 and m∠3 = m∠5
m∠4 + m∠2 + m∠5 = 180°
alternate interior angle theorem:
angle addition postulate:
substituting:
angles along a line sum to 180°
since m∠1 = m∠4 and m∠3 = m∠5m∠1 + m∠2 + m∠3 = 180°
© Boardworks 20129 of 11
Using the triangle sum theorem
What are the angle measures in an equiangular triangle?The measures of the angles in a triangle add up to 180°. An equiangular triangle has three congruent angles.
x° = m∠A = m∠B = m∠Cm∠A + m∠B + m∠C = 180°
x° = 180° ÷ 3 = 60°x° + x° + x° = 3x° = 180°
equiangular triangle:
triangle sum theorem:
substituting:
solving:
What is the sum of the measures on the non-right angles in a right triangle?
m∠A + m∠B + m∠C = 180°triangle sum theorem:
substituting:
solving:
m∠A + m∠B + 90° = 180°
m∠A + m∠B = 180 ° – 90° = 90°They are complementary angles.
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Exterior angle theorem
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Third angles theorem