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Geometry Review
Everything from Vocab to tricks & tips, ALL UNITS
| Term | Definition |
|---|---|
| Basic Rigid Motion | Transformations that preserve segment lengths and angle measures |
| Composition of Transformations | Doing multiple transformations in order, E.g. thing1 ∘ thing2 |
| Isometry | A transformation that preserves distance |
| Direct Isometry | Isometry preserving orientation |
| Opposite Isometry | Isometry changing orientation |
| Image | Transformation's result figure |
| Pre-Image | Figure before transformation |
| Regular Polygon | Polygon of all equal side lengths and angle measures |
| Formula for Angle of Rotational Symmetry | 360 ÷ # of regular polygon’s sides |
| Reflecting over y-axis (surround subscript with [brackets] ) | r[y-axis] → P’(-x,y) |
| Reflecting over x-axis | r[x-axis] → P’(x,-y) |
| Reflecting over y=x | r[y=x] → P’(y,x) |
| Reflecting over y=-x | r[y=-x] → P’(-y,-x) |
| Reflecting over origin | r[origin] → P’(-x,-y) |
| Translation formula | T[a,b] → P’(x+a,y+b) |
| Rotation about the origin | R[origin,90°] → P’(-y,x) |
| Congruent | Two figures are congruent if there exists a finite composition of basic rigid motions that maps one figure to the other. |
| Parallel | Two lines are parallel if they lie IN THE SAME PLANE and do not intersect. |
| Corresponding Parts of Congruent Figures are Congruent (CPCFC) | If two SIDES of a triangle are congruent or equal in measure then the sides opposite those ANGLES are congruent or equal in measure. (Vice versa) |
| Angle Angle Side triangle congruence criteria | Given two triangles ABC and A'B'C'. If AB=A'B' (Side), m∠B=m∠B' (Angle), and m∠C=m∠C' (Angle), then the triangles are congruent. This means, if in two triangles, two pairs of angles and the pair of sides in between those two pairs of angles is congruent then the triangles are congruent. |
| Angle Side Angle triangle congruence criteria | Given two triangles ABC and A'B'C'. If m∠CAB=m∠C'A'B' (Angle), AB=A'B' (Side), and m∠CBA=m∠C'B'A' (Angle), then the triangles are congruent. This means, if in two triangles, two pairs of angles and the pair of sides in between is congruent then the triangles are congruent. |
| Side Angle Side triangle congruence criteria | Given two triangles ABC and A'B'C'. If AB=A'B' (Side), m∠A=m∠A' (Angle), and AC=A'C' (Side), then the triangles are congruent. This means, if in two triangles, two pairs of sides and the pair of angles in between is congruent then the triangles are congruent. |
| Hypotenuse-Leg triangle congruence criteria | Given two RIGHT triangles ABC and A'B'C' with right angles ∠B and ∠B'. If AB=A'B' (Leg) and AC=A'C' (Hypotenuse), then the triangles are congruent. |
| Side Side Side triangle congruence criteria | Given two triangles ABC and A'B'C'. If AB=A'B' (Side), AC=A'C' (Side), and BC=B'C' (Side) then the triangles are congruent. |
| Skew lines | Two lines in 3D space are skew lines if they do not lie in the same plane. In that case, they do not intersect and are not parallel. |
| Centroid | The point of concurrency of the three medians of a triangle. AKA the center of gravity |
| Concurrent | When three or more lines meet at a single point |
| Median of a triangle | A line segment connecting a vertex to the median of the opposite side |
| Median or midsegment of a trapezoid | A line segment connecting the midpoints of the legs of a trapezoid |
| Midsegment | A line segment that joins the midpoints of two sides of a triangle or trapezoid |
| Parallelogram | A quadrilateral with both opposite sides parallel |
| Rectangle | A parallelogram with 4 right angles |
| Rhombus | A parallelogram with 4 congruent sides |
| Square | A quadrilateral that is both a rectangle and rhombus |
| Kite | A quadrilateral with two pairs of adjacent sides congruent and NO opposite sides congruent |
| Trapezoid | A quadrilateral with at least one pair of opposite sides parallel |
| Isosceles trapezoid | a trapezoid with congruent legs |
| Parallelogram properties - parallel sides | Both pairs of opposite sides are parallel |
| Parallelogram properties - divide | A diagonal divides it into two congruent triangles |
| Parallelogram properties - equal sides | Both pairs of opposite sides are congruent |
| Parallelogram properties - opposite angles | Both pairs of opposite angles are congruent |
| Parallelogram properties - diagonal | Diagonals bisect each other |
| Parallelogram properties - consecutives | Consecutive angles are supplementary |
| Figure | A set of points in a plane |
| Linear pair | Two adjacent angles that form a straight line. Linear pairs are supplementary. |
| Circle | The set of all points in the plane that are a distance r from a center point C |
| Line segment | The set consisting of all the points on a line between two specified points |
| Complementary angles, Right Angle | TWO angles whose measures add up to 90° (AKA Right angle) |
| Supplementary angles, Straight Angle | TWO angles whose measures add up to 180° (AKA Straight angle) |
| Zero Angle | Ray that measures 0° |
| Midpoint | A point on a segment that divides the segment into two congruent segments |
| Collinear | When three or more points are contained on the same line |
| Perpendicular bisector | A segment that passes through the midpoint and forms right angles with the given line segment |
| Adjacent angles | Two angles that share a common side |
| Auxiliary line | Adding or extending segments, lines, or rays in a figure |
| Isosceles triangle | A triangle with at least 2 equal side lengths. If |
| Vertical angles | Two angles whose sides form opposite rays. Vertical angles are congruent. |
| Congruence in parallel | If transversals cut parallels, then corresponding, alternate interior, and/or alternate exteriors are congruent. |
| Supplementary in parallel | if transversal cuts parallels, then same side interiors are supplementary. |
| 360 sum | the sum of the measures of all angles formed by three or more rays with the same vertex is 360 |
| 90 sum | When one angle of a triangle is a right angle, the sum of the other two angles are 90 |
| 180 sum | angles that form straight line adds to 180 |
| 180 triangle sum | angles of triangles add to 180 |
| Parallel auxiliary line | There is only one line through a given point not on the given line that can be drawn parallel to the given line. |
| Extended auxiliary line | Auxiliary lines can be drawn through any two given points. |
| Partition postulate | A whole is equal to the sum of parts |
| Substitution axiom | A quantity may be substituted for its equal. After reflexive axiom |
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