Geometry: Triangle Properties, Theorems, and Inequalities

Vocabulary flashcards covering triangle angle sum theorems, exterior angle relationships, isosceles and equilateral triangle properties, triangle inequalities, and perpendicular and angle bisectors.

Triangle Inequality Theorem

The theorem stating that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side ($a + b > c$, $b + c > a$, and $a + c > b$).

Range of the Third Side

For a valid triangle with two known side lengths, the length of the third side must be strictly greater than their positive difference and strictly less than their sum ($|a - b| < x < a + b$).

Opposite Side-Angle Relationship Theorem

If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side (the longer the side, the bigger the opposite angle).

Opposite Angle-Side Relationship Theorem

If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle (the bigger the angle, the longer the opposite side).

Corollary to the Triangle Angles Sum Theorem

The acute angles of a right triangle are complementary ($m\angle 1 + m\angle 2 = 90^\circ$), and the longest side in a right triangle is the hypotenuse.

Interior Angles of a Triangle

The three angles formed on the inside (interior) of a triangle at each vertex.

Triangle Angles Sum Theorem

The theorem stating that the sum of the measures of the three interior angles in any triangle is always $180^\circ$.

Exterior Angle of a Triangle

An angle formed outside a triangle by extending one of its sides, which forms a linear pair with its adjacent interior angle.

Opposite Interior Angles

The two interior angles of a triangle that are not adjacent (remote) to a given exterior angle.

Exterior Angle Theorem

The theorem stating that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two opposite interior angles.

Equilateral Triangle

A triangle having three congruent sides of equal length.

Isosceles Triangle

A triangle having at least two congruent sides of equal length.

Scalene Triangle

A triangle having no congruent sides, meaning all three side lengths are different.

Legs of an Isosceles Triangle

The two congruent sides of equal length in an isosceles triangle.

Base of an Isosceles Triangle

The third side of an isosceles triangle that does not match the length of the two congruent legs.

Base Angles of an Isosceles Triangle

The two angles located where the legs meet the base of an isosceles triangle, which are always congruent in measure.

Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Converse of the Base Angles Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Corollary to the Base Angles Theorem

If a triangle is equilateral, then it is also equiangular (all three interior angles are equal).

Corollary to the Converse of the Base Angles Theorem

If a triangle is equiangular, then it is also equilateral (all three side lengths are equal).

Perpendicular Bisector

A line, ray, or segment that intersects a segment at its midpoint at a $90^\circ$ angle, dividing it into two congruent halves.

Equidistant

Being at an equal distance from two or more specified points, segments, or lines.

Perpendicular Bisector Theorem

If a point lies on the perpendicular bisector of a segment, then it is equidistant from the two endpoints of the segment.

Converse of the Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of that segment.

Angle Bisector Theorem

If a point lies on the bisector of an angle, then the point is equidistant from the two sides of the angle.

Converse of the Angle Bisector Theorem

If a point is on the interior of an angle and equidistant from the sides of the angle, then the point lies on the angle bisector.

Incenter

The point of concurrency of a triangle where the three interior angle bisectors intersect, representing the center of the inscribed circle.