Relationship Between Triangles and Straight Angles

[darkchild]

My girlfriend is re-learning geometry. She has noticed that the sum of the angles of a triangle is equal to the measure of a straight angle and half the measure of a full revolution. She wants me to ask if there is any special relationship between a triangle and a straight angle or between a triangle and a circle that accounts for these similarities. I have no idea what to tell her, as I've never even thought of this sort of question before.

 

[LCKurtz]

It follows from the postulates of Euclidean geometry. It isn't true in non-Euclidean geometries. If you draw a triangle on the surface of a sphere the sum of its interior angles is greater than 180°. Specifically it follows from the postulate that only one line through a given point can be drawn parallel to a given line and the equality of alternate interior angles when two parallel lines are cut by a transversal.

 

[gsal]

Hey, darkchild, I am not sure but I get the feeling that you are asking a rather basic question and I feel like LCKurtz's answer might have implicitly answered your question and gone beyond it, so I am not sure is clear enough...so, allow me to explicitly mention a couple of things:

First, let's agree that we are talking about what most of us call triangles...those 3-sided figures that one can draw on a board or piece of paper, in other words, they are 2 dimensional, flat surfaces.

So, if you take the 3 angles inside a triangle and you add them up...they always add up to 180 degrees.

So, if you have a rectangular triangle where you know on of the angles is 90 deg, then necessarily the other two angles add up to 90, too, of course.

 

[HallsofIvy]

Start with any triangle and, at one vertex, draw a line parallel to the opposite side. The two sides of the triangle at that vertex form three angles with the parallel line. One of those angles, the middle one, is an angle in the triangle. The other two are "congruent" to the two base angles of the triangle because they are "alternate interior angles" on line intersecting both parallel lines. Thus, the measures of the three angles in the triangle add to the same as the three angles at the parallel line.

 

[CellCoree]

I can confirm that there is indeed a special relationship between triangles and straight angles. This is known as the Angle Sum Theorem, which states that the sum of the angles in any triangle is always equal to 180 degrees, or half the measure of a straight angle. This theorem can be proven using basic geometric principles and is a fundamental concept in geometry.

Furthermore, there is also a relationship between triangles and circles that accounts for this similarity. This is known as the Inscribed Angle Theorem, which states that an inscribed angle in a circle is half the measure of the central angle that intercepts the same arc. In simpler terms, this means that if you draw a triangle within a circle, the angle formed by the two sides of the triangle at the center of the circle will be half the measure of the angle formed by the two sides of the triangle at the circumference of the circle. This theorem can also be proven using geometric principles and is another fundamental concept in geometry.

In summary, the Angle Sum Theorem and the Inscribed Angle Theorem explain the relationship between triangles and straight angles, as well as between triangles and circles. These theorems are important in understanding the properties and relationships of geometric shapes and are essential for solving problems in geometry. I hope this explanation helps your girlfriend in her re-learning of geometry.