Download Core III Unit 4 – Useful Definitions, Postulates, and Theorems.

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
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