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Download Core III Unit 4 – Useful Definitions, Postulates, and Theorems.
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Transcript
Core III Unit 4 – Useful Definitions, Postulates, and Theorems. Inductive Reasoning: Reasoning by which a conclusion is based on several past observations. Deductive Reasoning: Proving statements by reasoning from accepted postulates, definitions, theorems, and given information. Counterexample: An example to prove an if-then statement false. Supplementary Angles: Two angles whose measures add to 180 degrees. Complementary Angles: Two angles whose measures add to 90 degrees. Linear Pair: Two adjacent angles whose measures add to 180 degrees (their outer sides create a straight angle). Vertical Angles: Angles created by the intersection of two lines. The angles are directly opposite each other and are equal in measure. Parallel Lines: Lines in a plane that do not intersect. Transversal: A line that intersects two or more lines. Bisector: A line, segment or ray that divides an angle or line segment into two equal parts. Corresponding Angles: Angles that are in the same relative position. Alternate Interior Angles: Two non-adjacent interior angles on opposite sides of the transversal. Same-Side Interior Angles: Two interior angles on the same side of the transversal. Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent. Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Same Side Interior Angles Theorem: If two parallel lines are cut by a transversal, then same side interior angles are supplementary. Converse: When you switch the hypothesis and conclusion in an If - Then Statement. (converse of If p, then q: If q, then p) Converse of the Corresponding Angles Postulate: If two lines are cut by a transversal so that corresponding angles have equal measure, then the lines are parallel. Converse of the Alternate Interior Angles Theorem: If two lines are cut by a transversal so that alternate interior angles have equal measure, then the lines are parallel. Converse of the Same-Side Interior Angles Postulate: If two lines are cut by a transversal so that same-side interior angles are supplementary, then the lines are parallel. Interior Angles of a Triangle: The interior angles of any triangle sum to 180º. Similarity: Similar plane shapes: corresponding angles are congruent (have the same measure) and the lengths of corresponding sides are proportional. Scale factor: The proportional factor between corresponding sides in similar figures. Side-Angle-Side Similarity Theorem: If an angle of one triangle has the same measure as an angle of a second triangle, and if the lengths of the corresponding sides including these angles are related by a scale factor k, then the two triangles are similar. Side-Side-Side Similarity Theorem: If the lengths of three sides of one triangle are related by a scale factor k to the lengths of the three sides of another triangle, then the triangles are similar. Angle-Angle Similarity Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Congruence: SSS Postulate: If each side of one triangle is congruent to the corresponding side of another triangle, then the triangles are congruent SAS Postulate: If two sides and the angle between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent ASA Postulate: If two angles and the side between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent. AAS Theorem: If two angles and a side not between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent. Parallelograms: A quadrilateral is a parallelogram if and only if both pairs of opposite sides parallel. If both pairs of opposite sides in a quadrilateral are congruent, then it is a parallelogram. If both pairs of opposite angles in a quadrilateral are congruent, then it is a parallelogram. If both pairs of adjacent angles in a quadrilateral are supplementary, then it is a parallelogram. If both diagonals bisect each other in a quadrilateral, then it is a parallelogram. Rectangles: - Are parallelograms Four right angles Congruent diagonals Rhombuses: - Are parallelograms Four congruent sides Diagonals are perpendicular Diagonals bisect its angles Squares: - Are parallelograms Are rhombuses Are rectangles Four right angles ERROR: undefined OFFENDING COMMAND: STACK: