# The angles of similar triangles are equal always, sometimes, or never?

Angles of similar triangles are ALWAYS equal

We have to start from a definition of similarity. There are different approaches to this. The most logical one I consider to be the definition based on a concept of scaling.

Scaling is a transformation of all points on a plane based on a choice of a scaling center (a fixed point) and a scaling factor (a real number not equal to zero).

Then the definition of similarity is: "two objects are called 'similar' if there exists such a center of scaling and scaling factor that transform one object into an object congruent to another."

Next, we have to prove that a straight line is transformed into a straight line parallel to an original. That causes angles to be transformed into equal angles, which is a subject of this question.

These proofs are presented in the course of advanced mathematics for teenagers at Unizor (follow menu items Geometry - Similarity).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Triangle A has sides of lengths #24 #, #16 #, and #20 #. Triangle B is similar to triangle A and has a side with a length of #16 #. What are the possible lengths of the other two sides of triangle B?
- A triangle has corners points A, B, and C. Side AB has a length of #9 #. The distance between the intersection of point A's angle bisector with side BC and point B is #6 #. If side AC has a length of #10 #, what is the length of side BC?
- A pizza packages its circular pizzas in a cardboard box. The boxes are formed by folding in 1" squares from the corners of the cardboard. If the volume of the box is 324 cubic inches, what are the dimensions of the box, in inches? Draw it?
- Two similar right circular cones have a scale factor of 2:5. What is the ratio of their lateral areas?
- Triangle A has an area of #3 # and two sides of lengths #5 # and #4 #. Triangle B is similar to triangle A and has a side with a length of #14 #. What are the maximum and minimum possible areas of triangle B?

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