2.3 Key theorems and their proofs in Euclidean Geometry
3 min read•july 22, 2024
The Pythagorean Theorem is a cornerstone of geometry, linking side lengths in right triangles. It's not just a formula; it's a gateway to understanding triangle properties, congruence, and similarity. These concepts form the foundation for more complex geometric reasoning.
Parallel lines and circles might seem simple, but they're packed with powerful properties. From angle relationships created by transversals to the unique characteristics of tangents, these ideas are crucial for solving real-world problems and understanding more advanced geometric concepts.
Pythagorean Theorem and Triangle Properties
Proof of Pythagorean Theorem
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States in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides
Expressed algebraically as a2+b2=c2, where c is the hypotenuse length and a and b are the lengths of the other sides
Proof using similar triangles involves:
Drawing the altitude from the right angle to the hypotenuse, dividing the triangle into two smaller similar triangles
Establishing proportions using the properties of similar triangles
Rearranging proportions to obtain equations involving the squares of the sides
Adding equations and simplifying to prove AC2+BC2=AB2
Congruence and similarity in triangles
Triangle congruence occurs when two triangles have exactly the same shape and size
Congruence criteria: SSS (all sides equal), SAS (two sides and included angle equal), ASA (two angles and included side equal), AAS (two angles and non-included side equal)
Triangle similarity occurs when two triangles have the same shape but not necessarily the same size
Similarity criteria: AA (two angles equal), SAS (ratio of two sides equal and included angles equal), SSS (ratios of all three sides equal)
Parallel Lines, Angles, and Circle Properties
Parallel lines and angles
Parallel lines lie in the same plane and do not intersect, even when extended indefinitely
When a transversal intersects two parallel lines:
Corresponding angles are congruent
Alternate interior angles are congruent
Alternate exterior angles are congruent
Consecutive interior angles are supplementary (sum to 180°)
Angle properties:
Supplementary angles sum to 180°
Complementary angles sum to 90°
Vertical angles formed by intersecting lines are congruent
Triangle angle sum is always 180°
Exterior angle of a triangle equals the sum of the two non-adjacent interior angles
Properties of circles and tangents
Circle properties:
Set of all points equidistant from the center in a plane
Radius is the distance from the center to any point on the circle
Chord is a line segment connecting two points on the circle
Diameter is the longest chord, passing through the center, twice the length of the radius
Central angle formed by two radii, its measure equals the intercepted arc measure
Inscribed angle formed by two chords sharing an endpoint, its measure is half the central angle intercepting the same arc
Tangent properties:
Tangent intersects the circle at exactly one point (point of tangency)
Radius drawn to the point of tangency is perpendicular to the tangent line
Two tangent segments from an external point to a circle are congruent
Angle between a tangent and a chord drawn to the point of tangency equals the inscribed angle in the opposite circle segment