Bryce's math notes! -Lines connected by | denote they are in a group. Search for [#post] and this will bring you to the start of each postulate. Search for [#thrm] to find theorems. Table of contents: [#dict] Dictionary [#gc] Geometric Concepts [#tr] Transversals [#pa] Parallelism [#dict] (Brief) Dictionary Congruent: Have the same shape and size. Postulate: A thing suggested or assumed as true as the basis for reasoning, discussion, or belief. Theorem: A statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements. Theorems are reversable. [#gc] Geometric Concepts This lesson contains definitions, words, and symbols that are commonly used in geometry. It is of utmost importance that these are learned. Points are an undefined term. A point is a location with no size whatsoever. A point is literally a dot, usually named a letter such as X, Y, or Z. | Two points can form a line segment. Diagram: o A line segment is two points (which are called the endpoints) and all the points on a line between those endpoints. It can be compared to a stick, as it has a beginning and an end. Diagram: o----o A line is also an undefined term. A line is straight and infinitely long, it has no endpoints. | Given two points, there is only one line through the points. Diagram: <-----> A ray has one endpoint and extends infinitely in another direction. Think of a laser beam. Diagram: o-----> A plane is also an undefined term. It's a flat 2D surface that has infinite length and width, but no thickness. | Three points not in a line always determine a plane. | A plane is drawn like a parallelogram and has no boundaries. Remember! Line segments and rays are both parts of a line. Parallel lines are two or more lines who never intersect. | Segments and rays can also be parallel. Perpendicular lines are lines that intersect to form four right angles. Intersecting lines intersect, but are not parallel nor perpendicular. | Intersecting lines form two pairs of vertical angles. Vertical angles are congruent. Think butterfly wings. An angle consists of two rays that have a common endpoint. Diagram: =========== ^ / / o-----> =========== | A vertex is the common endpoint of two rays that form an angle. | A right angle's measure is 90 degrees, all right angles are congruent. | An acute angle's measure is less than 90 degrees but greater than 0 degrees. | An obtuse angle's measure is greater than 90 degrees but less than 180 degrees. | A symbol such as <3 is referring to the angle. | A symbol such as m<3 is referring to the measurement of the angle. Example: m<3 = 45 means the measure of angle 3 is 45 degrees. Adjacent angles have the same vertex and one common ray. They are next door neighbors. Adjacent angles cannot overlap. Straight angles: If the rays of an angle point in opposite directions, the line formed is sometimes called a straight angle. Complementary angles are two angles that add up to 90 degrees. Supplementary angles are two angles that add up to 180 degrees. Congruency: Two objects are congruent if all their measurements are equal. | If two figures are congruent, they could be placed atop each other and their points would match perfectly. | Lengths aren't congruent, they're equal. A transversal is a line the intersects two other lines. | When a transversal intersects parallel lines, the corresponding angles formed at congruent. See notes on transversals. Skew lines are two lines that do not intersect but aren't parallel either, they don't lie on the same plane. END OF GEOMETRIC CONCEPTS [#tr] Transversals Parallel lines never intersect and lie within the same plane. This symbol is used to represent parallel lines: || Skew lines never intersect and lie within seperate planes. Segments and rays are parallel if the lines that contain them are parallel. Planes are parallel if they never intersect. Examples: A ceiling and a floor or the walls of a hallway. A transversal is a line that crosses two other coplanar lines at different points. | Parallel lines have interior and exterior spaces. Interior is within the two lines and exterior is outside the lines. Imagine the lines make a "pole". | When a transversal intersects two parallel lines, inside and outside angles are created. | Angles on the same side of a transversal are coined "same-side" angles. Angles on opposite sides of a transversal are called alternate/opposite side angles. | The same-side or alternate label is combined with the exterior or interior label to form the names. For example: alternate-exterior angle X, same-side interior angle C. | Alternate interior and exterior angles are... diagonal from each other. | Same-side interior and exterior angles locations are self-explanatory. | Same-side interior angles are often called consecutive interior angles, which of course, means the same thing. | Below is a diagram explaining corresponding angle pairs. ================= 1|2 1|2 <--|---------|--> 3|4 3|4 ================= | Vertical angles are congruent. | Below is a diagram explaining vertical angles. ======= \ 2 / \ / 1 X 1 / \ / 2 \ ======== | BE AWARE: Angle pairs exist even when the figure is offbeat! | Linear angles create lines, you know, 180 degrees. | Below is a diagram explaining a linear pair of angles. ================= 1|1 2|2 <--|---------|--> 3|3 4|4 ================= END OF TRANSVERSALS [#pa] Parallelism Remember that postulates are accepted as true. [#post] If a transversal cuts through two parallel lines, then the corresponding angles are congruent. [#thrm] If a transversal cuts through two parallel lines, the alternate exterior angles are congruent. [#thrm] If a transversal cuts through two parallel lines, the alternate interior angles are congruent. [#thrm] If a transversal cuts through two parallel lines, the same-side interior angles are supplementary. [#thrm] If a transversal cuts through two parallel lines, the same-side exterior angles are supplementary. [#thrm] If a transversal is perpendicular to one of two parallel lines, it is also perpendicular to the other. Any two angles in a transversal are either congruent or supplementary. Two lines are parallel if: | Corresponding angles are congruent. | Alternate exterior angles are congruent. | Alternate interior angles are congruent. | Same-side interior angles are supplementary. | Same-side exterior angles are supplementary. END OF PARALLELISM [#]