Geometry unit 11 ANGLE RELATIONSHIPS AND PARALLELS

A line that intersects two or more given lines.

Transversal

The union of three segments determined by three non-collinear points.

Triangle

An angle whose measure is greater than 90° but less than 180°.

Obtuse angle

Two lines that never intersect.

Parallel lines

A polygon with all angles equal and all sides equal.

Regular polygon

An angle whose measure is equal to 90°.

Right angle

Two lines that are neither parallel nor perpendicular.

Skew lines

An angle formed by one side of a triangle and an extension of another side.

Exterior angle of a triangle

An angle whose measure is less than 90°.

Acute angle

Two angles in the same plane that have a common vertex and a common side but no interior points in common.

Adjacent angles

Two lines that intersect and form four equal angles.

Perpendicular lines

If two lines are cut by a transversal so that corresponding angles are equal, then the lines are parallel.

Postulate 10

Every angle corresponds with a unique real number greater than zero and less than 180.

Postulate 6

If →QA lies between →OB and → OC , then m∠BOA + m∠AOC = m∠BOC. (angle addition theorem)

Theorem 3-1

If two parallel planes are cut by a third plane, then the lines of intersection are parallel.

Theorem 3-10

The union of two non-collinear rays that have a common endpoint.

Angle

A line introduced in a figure to aid a proof.

Auxiliary line

Two angles whose measures sum to 90°.

Complementary angles

The set of rays on the same side of a line with a common endpoint in the line can be put into one-to-one correspondence with the real numbers from 0 to 180 inclusive in such a way that:

Postulate 7

one of the two opposite rays lying in the line is paired with zero and the other is paired with 180 and the measure of an angle whose sides are rays of that given set is equal to the absolute value of the difference between the real numbers corresponding to its sides.

Postulate 7-1/7-2

If two parallel lines are cut by a transversal, then the corresponding angles have equal measure.

Postulate 8

Through a point not on a line, one and only one line can be drawn parallel to the line.

Postulate 9

Two angles whose measures sum to 180°.

Supplementary angles

If a transversal is perpendicular to one of two parallels, then it is perpendicular to the other one also.

Theorem 3-11

If two parallel lines are cut by a transversal, then the alternate interior angles are equal.

Theorem 3-12

If two parallel lines are cut by a transversal, then the alternate exterior angles are equal.

Theorem 3-13

In a plane, if two lines are perpendicular to a third line, then they are parallel to each other.

Theorem 3-14

If two lines are cut by a transversal so that alternate interior angles are equal, then the lines are parallel.

Theorem 3-15

If two lines are cut by a transversal so that alternate exterior angles are equal, then the lines are parallel.

Theorem 3-16

The sum of the measures of the angles of a triangle is 180°.

Theorem 3-17

The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles

Theorem 3-18

The sum of the measures of the angles of a quadrilateral is 360°.

Theorem 3-19

If the exterior sides of two adjacent angles are opposite rays, then the angles are supplementary.

Theorem 3-2

If two lines are perpendicular, then they form right angles.

Theorem 3-3

If two adjacent angles have their exterior sides in perpendicular lines, then the angles are complementary.

Theorem 3-4

If two angles are supplementary to the same angle or to equal angles, then they are equal to each other.

Theorem 3-5

If two angles are complementary to the same angle or to equal angles, then they are equal to each other.

Theorem 3-6

If two lines intersect, the vertical angles formed are equal.

Theorem 3-7

All right angles are equal.

Theorem 3-8

If two lines meet and form right angles, then the lines are perpendicular.

Theorem 3-9

Two angles whose sides form two pairs of opposite rays.

Vertical angles