Internal angle sum of triangle

In summary: ABC is less than the sum of the measures of the exterior angles, which is less than 540 degrees minus the sum of the internal angles. This can be derived by considering three triangles and using their respective angle measures.
  • #1
bonfire09
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Problem: Let A, B, C be three non-collinear points. Let D, E, F be points on the respective interiors of segments BC, AC and AB. Let θ, φ and ψ be the measures of the respective angles ∠BFC, ∠CDA and ∠AEB. Prove IAS(ABC) < θ +φ + ψ < 540 - IAS(ABC).(IAS means internal angle sum). Now I am supposed to use the external angle inequality which is the measure of an exterior angle of a triangle is greater than that of either opposite interior angle. Not sure how to do it. I've been struggling for hours with it. Oh i forgot to mention this is still in absolute geometry so we can't use that the the angles of a triangle add up to 180*.
 

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  • #2
consider the following triangles,

$\Delta ABE$,
$$\psi =180-\left[{A}+({B}-r)\right]\qquad (1)$$
$\Delta ADC$,
$$\varphi = 180-\left[{C}+({A}-q)\right]\qquad (2)$$
$\Delta ABE$,
$$\theta = 180-\left[{B}+({C}-p)\right]\qquad (3)$$

Adding (1),(2),(3) together,

$$\psi+\varphi+\theta=540-(A+B+C)-\underbrace{[(A-q)+(B-r)+(C-p)]}_{>0}$$
you can derive one inequality from this

then,
from $\Delta ABD$
$$B+q=\varphi \qquad (4)$$
from $\Delta AFC$
$$A+p=\theta\qquad (5)$$
from $\Delta BCE$
$$C+r=\psi\qquad (6)$$

adding (4),(5),(6) together,
$$(A+B+C)+\underbrace{(p+q+r)}_{>0}=\varphi+\theta+\psi$$

from this you can get the other inequiality
 

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Related to Internal angle sum of triangle

1. What is the formula for calculating the internal angle sum of a triangle?

The formula for calculating the internal angle sum of a triangle is 180 degrees. This means that if you add up all three interior angles of a triangle, the sum will always be 180 degrees.

2. How is the internal angle sum of a triangle related to its sides?

The internal angle sum of a triangle is related to its sides through the Triangle Sum Theorem, which states that the sum of the measures of the interior angles of a triangle is always 180 degrees. This theorem is true for all types of triangles, whether they are equilateral, isosceles, or scalene.

3. Can the internal angle sum of a triangle ever be greater than 180 degrees?

No, the internal angle sum of a triangle can never be greater than 180 degrees. This is because a triangle is a closed shape, meaning that all three of its interior angles must add up to 180 degrees. If the internal angle sum is greater than 180 degrees, then the shape is no longer a triangle.

4. How does the internal angle sum of a triangle change if one of the angles is changed?

The internal angle sum of a triangle will always remain 180 degrees, regardless of any changes to the individual angles. This means that if one angle is increased, another angle must decrease in order to maintain a total of 180 degrees. Similarly, if one angle is decreased, another angle must increase.

5. Why is the internal angle sum of a triangle important in geometry?

The internal angle sum of a triangle is important in geometry because it is a fundamental property of triangles. It allows us to solve for unknown angles and sides, and is used in many geometric proofs and constructions. Additionally, the concept of the internal angle sum can be extended to other shapes and polygons, making it an essential concept in higher level mathematics.

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