Geometry Transformations, Congruence, and Triangle Properties

What is a reflection in geometry?

A rigid motion across a line of reflection where the preimage and image are congruent.

What does a line of reflection serve as in a reflection?

The perpendicular bisector of the segment connecting any point and its image.

What is a translation in geometry?

A rigid motion that moves every point of a figure the same distance in the same direction.

How can a translation be described?

By a vector, where each point maps to a new position.

What is a rotation in geometry?

A rigid motion defined by a center of rotation and an angle of rotation.

What preserves orientation in rotations?

Rotations preserve orientation when the angle is a multiple of 360 degrees.

What is symmetry in geometry?

A figure possesses symmetry if it can be mapped onto itself using a rigid motion.

What is congruence in geometry?

Two figures are congruent if they have exactly the same size and shape.

What is an isosceles triangle?

A triangle with at least two congruent sides, called legs.

What is the base in an isosceles triangle?

The third side that is not congruent to the legs.

What is the SAS congruence criterion?

Two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle.

What is the SSS congruence criterion?

Two triangles are congruent if three sides of one triangle are congruent to three sides of the other triangle.

What does CPCTC stand for?

Corresponding Parts of Congruent Triangles are Congruent.

What is the ASA congruence criterion?

Two triangles are congruent if two angles and the included side are congruent.

What is the AAS congruence criterion?

Two triangles are congruent if two angles and a non-included side are congruent.

What does the HL theorem state?

Two right triangles are congruent if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of the other.

What is the perpendicular bisector of a segment?

The set of all points equidistant from the endpoints of the segment.

What is the circumcenter of a triangle?

The point where the perpendicular bisectors of the sides of a triangle are concurrent.

What is the incenter of a triangle?

The point where the angle bisectors of a triangle are concurrent.

What is a median in a triangle?

A segment from a vertex to the midpoint of the opposite side.

What is the centroid of a triangle?

The point where the three medians of a triangle intersect.

What is the orthocenter of a triangle?

The point where the three altitudes of a triangle intersect.

What does the Triangle Inequality Theorem state?

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

What is the Hinge Theorem?

When two triangles have two pairs of congruent corresponding sides, the longer third side is opposite the larger included angle.

What is the sum of the measures of the exterior angles of any polygon?

Always 360 degrees, regardless of the number of sides.

What is a kite in geometry?

A quadrilateral with two pairs of adjacent congruent sides.

What is an isosceles trapezoid?

A trapezoid with congruent legs and congruent base angles.

What defines a parallelogram?

A quadrilateral with both pairs of opposite sides parallel.

What are the properties of a parallelogram?

Opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.

How can you prove a quadrilateral is a parallelogram?

By showing that both pairs of opposite sides are congruent, both pairs of opposite angles are congruent, one pair of opposite sides is both parallel and congruent, the diagonals bisect each other, or one angle is supplementary to both its consecutive angles.

What defines a rhombus?

A rhombus is a parallelogram with four congruent sides and diagonals that are perpendicular bisectors of each other.

What characterizes a rectangle?

A rectangle is a parallelogram with four right angles and congruent diagonals.

What is a square?

A square is both a rhombus and a rectangle, having four equal sides, four right angles, and diagonals that are perpendicular and congruent.

What is a dilation in geometry?

A dilation is a transformation that preserves angle measure but not length, defined by a center of dilation and a scale factor.

What is a similarity transformation?

A similarity transformation is a composition of a dilation with one or more rigid motions, resulting in figures that are similar.

What are the criteria for triangle similarity?

AA Similarity (two angles are congruent), SSS Similarity (three sides are proportional), SAS Similarity (two sides proportional and included angle congruent).

What does the Side-Splitter Theorem state?

A segment parallel to one side of a triangle divides the other two sides proportionally.

What is the Pythagorean Theorem?

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).

What are the properties of a 45°-45°-90° triangle?

The legs are congruent, and the hypotenuse is √2 times the length of a leg.

What are the properties of a 30°-60°-90° triangle?

The hypotenuse is twice the shorter leg, and the longer leg is √3 times the shorter leg.

What is the Law of Sines?

For any triangle, the ratio of the length of a side to the sine of its opposite angle is constant (a/sin(A) = b/sin(B) = c/sin(C)).

What is the Law of Cosines?

For any triangle, c² = a² + b² - 2ab*cos(C), relating the lengths of sides to the cosine of one angle.

What is the angle of elevation?

The angle above the horizontal from the observer's line of sight to an object above.

What is the angle of depression?

The angle below the horizontal from the observer's line of sight to an object below.

What does the Distance Formula calculate?

The distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂ - x₁)² + (y₂ - y₁)²).

What is the Midpoint Formula?

The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

What is the equation of a circle in the coordinate plane?

The equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

What is the equation of a parabola with vertex at the origin?

The equation is y = ax², where a determines the direction and width of the parabola.

What is the Triangle Midsegment Theorem?

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

What is the Triangle-Angle-Bisector Theorem?

An angle bisector of a triangle divides the opposite side into segments proportional to the adjacent sides.

What is a geometric mean?

The geometric mean of two positive numbers a and b is √(ab), often used in right triangle relationships.

What is the relationship between altitude and segments in a right triangle?

The length of the altitude is the geometric mean of the two segments of the hypotenuse created by the altitude.

What is the significance of congruent angles in similar triangles?

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

How do you determine if two triangles are similar using SSS?

If the three side lengths of one triangle are proportional to the three side lengths of another triangle, the triangles are similar.

What is the relationship between slopes in coordinate geometry?

Equal slopes indicate parallel lines, while the product of slopes equal to -1 indicates perpendicular lines.