Planimetry: Basic Facts for EGE

Comprehensive vocabulary flashcards covering the fundamental facts of planimetry including triangle properties, parallel lines, similarity, and right-triangle theorems for exam preparation.

Sum of angles in a triangle

The sum of the interior angles of any triangle is equal to $180^{\circ}$.

Exterior angle of a triangle

An angle that is equal to the sum of the two interior angles of the triangle that are not adjacent to it.

First criterion for triangle equality (SAS)

If two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another triangle, the triangles are equal.

Second criterion for triangle equality (ASA)

If a side and its two adjacent angles of one triangle are respectively equal to a side and its two adjacent angles of another triangle, the triangles are equal.

Third criterion for triangle equality (SSS)

If three sides of one triangle are respectively equal to the three sides of another triangle, the triangles are equal.

Isosceles triangle

A triangle with two equal sides, called lateral sides; the third side is called the base.

Base angles of an isosceles triangle

The angles opposite the lateral sides which are always equal to each other.

Properties of the base altitude in an isosceles triangle

The bisector, median, and altitude drawn to the base of an isosceles triangle all coincide.

Circumcenter of a triangle

The center of the circle circumscribed around a triangle, located at the intersection of the perpendicular bisectors of the sides.

Alternate interior angles

Angles formed when a transversal intersects two parallel lines; these angles are equal.

Corresponding angles

Angles in the same relative position at each intersection where a straight line crosses two others; if the lines are parallel, these angles are equal.

Consecutive interior angles

Angles on the same side of the transversal and between the two parallel lines; their sum is equal to $180^{\circ}$.

Generalized Thales' Theorem

States that parallel transversals create proportional segments on the lines they intersect.

Converse Generalized Thales' Theorem

If lines intersecting two other lines cut off proportional segments starting from the vertex, then those lines are parallel.

Criteria for triangle similarity

Triangles are similar if they have: 1) Two proportional sides and equal included angles, 2) Two equal angles (AA), or 3) Three proportional sides.

Midsegment of a triangle

A segment connecting the midpoints of two sides of a triangle; it is parallel to the third side and half its length.

Area property of the midsegment

The midsegment cuts off a triangle that is similar to the original, and the three midsegments divide the triangle into four triangles of equal area.

Median of a triangle (Area)

A segment that divides the triangle into two triangles of equal area (equal-magnitude triangles).

Centroid division ratio

The point where the medians of a triangle intersect divides each median in a ratio of $2:1$, counting from the vertex.

Median of a right triangle

The median drawn from the vertex of the right angle is equal to half of the hypotenuse.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: $a^2 + b^2 = c^2$.

Altitude in a right triangle

The altitude from the right angle divides the triangle into two triangles similar to the original; its square equals the product of the hypotenuse segments it creates.

Incircle in a right triangle

The vertex of the right angle, the center of the inscribed circle, and the points of tangency on the legs form a square.

30-60-90 Triangle property

The leg opposite the $30^{\circ}$ angle is equal to exactly half of the hypotenuse.