Congruence

G.GCO.1Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects.G.GCO.10Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following:G.GCO.10aOpposite sides of a parallelogram are congruent.G.GCO.10bOpposite angles of a parallelogram are congruent.G.GCO.10cDiagonals of a parallelogram bisect each other.G.GCO.10dRectangles are parallelograms with congruent diagonals.G.GCO.10eA parallelograms is a rhombus if and only if the diagonals are perpendicular.G.GCO.11Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships.G.GCO.2Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions.G.GCO.3Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations.G.GCO.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.G.GCO.5Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations, and describe a sequence of transformations that maps a figure onto its image.G.GCO.6Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other.G.GCO.7Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions.G.GCO.8Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following:G.GCO.8aVertical angles are congruent.G.GCO.8bWhen a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary.G.GCO.8cAny point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment.G.GCO.8dPerpendicular lines form four right angles.G.GCO.9Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following:G.GCO.9aMeasures of interior angles of a triangle sum to 180°.G.GCO.9bBase angles of isosceles triangles are congruent.G.GCO.9cThe segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.G.GCO.9dThe medians of a triangle meet at a point.
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