Geometry (US High School)
US High School Geometry aligned with Common Core (CCSS.MATH.CONTENT.HSG): congruence and rigid motions, similarity and dilations, right-triangle trigonometry, circles, coordinate geometry, geometric measurement, modeling, and probability.
Ämne: Matematik · Nivå: Gymnasium (16–19) · 507 kort
Innehåll
- A point is an exact location in space. It has no length, width, or thickness and is represented by a dot and labeled with a capital letter.
- A line is a straight one-dimensional figure that extends infinitely in both directions. Two points determine a unique line.
- A plane is a flat two-dimensional surface extending infinitely in all directions. Three non-collinear points determine a unique plane.
- A line segment is a portion of a line bounded by two endpoints. It has a definite length, unlike a line.
- A ray is part of a line that starts at one endpoint and extends infinitely in one direction.
- Collinear points are three or more points that lie on the same line. Any two points are always collinear.
- Coplanar points are points that lie in the same plane. Any three points are always coplanar.
- An angle is formed by two rays sharing a common endpoint called the vertex. The rays are the sides of the angle.
- An acute angle measures between 0° and 90° (exclusive). A right angle measures exactly 90°.
- An obtuse angle measures between 90° and 180° (exclusive). A straight angle measures exactly 180°.
- A reflex angle measures greater than 180° but less than 360°.
- Complementary angles are two angles whose measures sum to 90°. Supplementary angles sum to 180°.
- Vertical angles are formed by two intersecting lines and are always congruent (equal in measure).
- A linear pair consists of two adjacent angles whose non-common sides form a straight line. Their measures sum to 180°.
- Parallel lines are coplanar lines that never intersect and maintain a constant distance apart. They have equal slopes.
- Perpendicular lines intersect at a 90° angle. Their slopes are negative reciprocals (m₁ × m₂ = -1).
- Skew lines are non-coplanar lines that do not intersect and are not parallel. They exist only in three dimensions.
- A transversal is a line that crosses two or more other lines, creating 8 angles at the intersections.
- When parallel lines are cut by a transversal, corresponding angles are congruent (in the same position at each intersection).
- When parallel lines are cut by a transversal, alternate interior angles are congruent (between the lines, on opposite sides of the transversal).
- When parallel lines are cut by a transversal, co-interior angles (same-side interior) are supplementary, summing to 180°.
- A triangle is a three-sided polygon. The sum of its interior angles is always 180°.
- An equilateral triangle has three congruent sides and three congruent 60° angles. It is also equiangular.
- An isosceles triangle has at least two congruent sides. The angles opposite the congruent sides are also congruent (Base Angles Theorem).
- A scalene triangle has no congruent sides and no congruent angles. All three sides have different lengths.
- A right triangle contains exactly one 90° angle. The side opposite this angle is the hypotenuse, the longest side.
- The Pythagorean Theorem states a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse.
- The converse of the Pythagorean Theorem: if a² + b² = c² in a triangle, then the triangle is a right triangle.
- Common Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). Multiples of these are also valid.
- In a 45-45-90 triangle, the legs are congruent and the hypotenuse equals leg × √2.
- In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2 (short leg : long leg : hypotenuse).
- The Triangle Inequality Theorem: the sum of any two sides of a triangle must be greater than the third side.
- The Exterior Angle Theorem: an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.
- A median of a triangle is a segment from a vertex to the midpoint of the opposite side. Every triangle has three medians.
- The centroid is the intersection point of the three medians of a triangle. It divides each median in a 2:1 ratio from the vertex.
- An altitude is a perpendicular segment from a vertex to the line containing the opposite side. The orthocenter is where altitudes meet.
- The circumcenter is the intersection of the perpendicular bisectors of a triangle's sides. It is equidistant from all three vertices.
- The incenter is the intersection of the three angle bisectors of a triangle. It is equidistant from all three sides.
- The Midsegment Theorem: a segment joining midpoints of two sides of a triangle is parallel to the third side and half its length.
- Two triangles are congruent if they have the same size and shape. Corresponding sides and angles are equal.
- SSS Congruence Postulate: if three sides of one triangle equal three sides of another, the triangles are congruent.
- SAS Congruence Postulate: two sides and the included angle of one triangle congruent to corresponding parts proves congruence.
- ASA Congruence Postulate: two angles and the included side congruent between two triangles proves the triangles are congruent.
- AAS Congruence Theorem: two angles and a non-included side congruent between two triangles proves congruence.
- HL Congruence Theorem (for right triangles only): if the hypotenuse and a leg are congruent, the right triangles are congruent.
- SSA (Side-Side-Angle) does NOT prove congruence in general. It is sometimes called the "ambiguous case" of triangles.
- AAA (Angle-Angle-Angle) does NOT prove congruence; it proves similarity. Triangles can have the same angles but different sizes.
- CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent" — used after proving triangles congruent.
- Two figures are similar if they have the same shape but not necessarily the same size. Corresponding angles are equal, sides proportional.
- AA Similarity Postulate: if two angles of one triangle are congruent to two angles of another, the triangles are similar.