HallsofIvy said:Hey, once you get angels involved anything can happen!
Oh, wait- you mean "angles". Triangles with the same angles are "similar triangles": it follows then that all length's are proportional. That is true not only for the lengths of the sides but all corresponding lengths, such as the height.
The relation between same angles and sides in triangles is known as the Angle-Side-Angle (ASA) congruency. This means that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
The equality of same angles and sides in triangles is explored through the use of geometric proofs and theorems. These proofs use properties of congruent triangles and the relationships between sides and angles in triangles to show that two triangles are congruent.
To prove ASA congruency, the following conditions must be met:
Yes, it is possible for two triangles to have the same angles but different side lengths. This is known as the Angle-Angle-Side (AAS) theorem, where two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle. However, this does not prove congruency between the two triangles.
Understanding the relation between same angles and sides in triangles allows us to accurately measure and construct triangles, as well as solve problems involving triangles. It also serves as a foundation for more advanced concepts in geometry and other fields of mathematics.