Triangles

Triangles
Triangle Classification

A triangle [three sides-three angles] has two names; one identifies the nature of its sides and the other describes its angles.

A triangle with two equal sides is isosceles and no equal side is scalene. If all three angles are less than 90∘, it's acute. If one angle is 90∘, it's a right triangle and if one angle is greater than 90∘, it's obtuse. A triangle cannot contain more than one right or one obtuse angle.

If all three sides are equal, it's equilateral. All equilateral triangles are acute and equiangular. These properties produce seven general triangle classifications.Triangles Types

Angle Sum Theorem

Triangle Angles Sum is 180 Degrees THE SUM OF THE THREE ANGLES OF EVERY TRIANGLE IS 180∘.

Ready for a simple proof? - see following figure.

Only one line passes through A parallel to BC. [Euclid's Parallel Postulate]
Therefore, the yellow angles are equal and the green angles are equal. [Alternate interior angles created by parallels and a transversal]
But [angles on the line], ∠y + ∠z + ∠x = 180∘. [Angles that sit on a line add to 180∘]
Therefore [inside the triangle], ∠y + ∠z + ∠x = 180∘ [The angles of the triangle are the same as those on the line.]
Triangle Angles Sum is 180 Degrees

Since the angle sum is 180∘, each angle in the equilateral-equiangular triangle must be 60∘.
Equilateral Triangle

Pons Asinorum

IF TWO SIDES OF A TRIANGLE ARE EQUAL, THE ANGLES OPPOSITE THEM ARE EQUAL or
THE BASE ANGLES OF AN ISOSCELES TRIANGLE ARE EQUAL.
If AB = AC, then ∠B = ∠C.
Pons Asinorum
Pons Asinorum is latin for "bridge of asses". Lengend claims that the proof of this theorem "in the old days" required a diagram that resembled a bridge, and that the proof was so difficult that only the scholars could understand it and "pass over the bridge", leaving the "asses" behind.

Triangles are the strongest shape because of they can bear weight without geometric distortion.

Bridge
Gable

Pascal's Triangle

Pascal's Triangle
The numbers in Pascal's triangle provide the coefficients in the binomial expansion.
Binomial Expansion

Exterior Angle Theorem

Exterior Angle
An exterior angle of a triangle equals the sum of the two remote angles. Can you see why?

1. A + B = 180, why?
2. [C + D] + B = 180, why?
3. A + B = [C + D] + B, why?
4. therefore, A = C + D, why?

Self Quiz

1. Provide the answers to "why?" in the previous section.
2. Classify a triangle with two equal sides.
3. Question 3

Answers

1.
  1. angles on a line
  2. triangle angle sum
  3. both quantities equal 180
  4. subtract B from both sides

2. Isosceles
3. 50∘

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