Geometry (23-24)

Rules of Construction

1. A straightedge used only to draw segments or lines between two points 2. A compass use to draw a circle or arc of known center and known radius 3. No other tools (except pencil) may be used

Lines will be Parallel if...

1. Any pair of corresponding angles are equal. 2. Any pair of alternate interior angles are equal.

Properties of a Parallelogram

1. Opposite Sides are congruent 2. Opposite Angles are congruent 3. Consecutive Angles are supplementary 4. Diagonals bisect each other

Properties of Rigid Motions

1. Transform lines into lines, segments into segments, and rays into rays 2. Preserve distance and angle measurements

Circumscribed Circle

A circle which passes through all three vertices of a triangle.

Circle

A collection (or set) of points that are a fixed distance (the radius) away from a fixed point (the circle's center).

Transformation

A function (or rule) that for every point, P, in the plane as its input gives or assigns another, single point in the plane, F(P), as its output.

Altitude of a Triangle

A line segment drawn from the vertex of a triangle such that it is perpendicular to the opposite side.

Median of a Triangle

A line segment drawn from the vertex of a triangle to the midpoint of the opposite side.

Segment Bisector

A line, ray, or segment that passes through the midpoint of another segment.

Angle Bisector

A line, segment, or ray that divides an angle into two congruent angles.

Perpendicular Bisector

A line, segment, or ray that passes through the midpoint of another segments and makes only right angles from it.

Axiom

A mathematical statement that is regarded as true without proof (postulate).

Vertical Angles

A pair of opposite congruent angles formed by intersecting lines.

Point

A physical location in space that has no dimensions (length, width, or height). Points are often labeled with capital letters (A, B, C, etc.).

Midpoint

A point that divides a segment into two congruent segments.

Geometric Fact

A point that lies on the perpendicular bisector of a line segment will lie equidistant from its endpoints.

Rectangles

A quadrilateral with four right angles.

Kite

A quadrilateral with two pairs of equal, adjacent sides.

Reflex Angles

A reflex angle has a measure between 180° and 360°. Reflex angles are important in mathematics but will not be important in our initial studies.

Right Angle

A right angle has a measure of 90°. It represents exactly one-quarter of a full rotation (360°). It is perhaps the most important angle in all of mathematics because in indicates two directions that are completely independent of one another.

Straight Angle

A straight angle has a measure of 180°. A straight angle represents exactly half of a full rotation (360°). It would look indistinguishable from a straight line.

Arc

A subset of points that lie on a circle.

Symmetry of a Figure

A transformation that maps a figure onto itself (also known as carrying a figure onto itself).

Fact About Right Angles

All right angles are equal (or congruent).

Acute Angle

An acute angle has a measure between 0° and 90°. Acute comes from the Latin word for sharpening. These angles look sharp or pointy.

Ray

An infinite set (collection) of points that extend forever in one direction from a starting point. Like lines, rays are symbolized by using two points, but only a single arrow (-->AB).

Line

An infinite set (collection) of points that extends forever in two directions. Since lines are uniquely determined by any two points on the line, symbolized by a double arrow over two points (<->AB). Sometimes they are also referred to by lowercase letters (l, m, & n).

Obtuse Angle

An obtuse angle has a measure between 90° and 180°. These are the opposite of acute angles and will look blunt or dull.

Regular Polygon

Any polygon whose sides have all equal lengths and whose angles all have equal measurements.

Trapezoid

Any quadrilateral with at least one pair of parallel sides.

Parallelogram

Any quadrilateral with both pairs of opposite sides being parallel.

Equidistant

At equal distances.

Isosceles Triangle Property #1

Base angles of an isosceles triangle are always congruent.

Proof Reason for Complements

Complements of equal angles are equal

Alternate Interior Angle Pairs

Created when two parallel lines are crossed by a third line (transversal); inside the parallel lines opposite sides of the transversal.

Division P.O.E.

Equals divided by equals result in equals.

Congruent

Having the same size and shape

Addition P.O.E.

If equals are added to equals, the sums are equal.

Subtraction P.O.E

If equals are subtracted from equals, the differences are equal.

Hypotenuse - Leg (HL)

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

Side - Side - Side (S.S.S)

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

S.S.S. (Side, Side, Side) Theorem

If three sides of one triangle are the same length as three sides of another triangle, then these two triangles are congruent.

Angle - Angle - Side (A.A.S.)

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

Angle - Side - Angle (A.S.A)

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Isosceles Triangle Property #2

If two angles of a triangle are congruent, the sides opposite the angles are congruent.

Side - Angle - Side (S.A.S)

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Concurrence

Intersecting at the same point.

Point on a Perpendicular Bisector

Is equidistant from the endpoints of the segment.

Corresponding Angles

Lie on the same side of the transversal and in corresponding positions.

Euclidean Plane

No coordinates.

Collinear

Points that lie on the same line.

Substitution P.O.E.

Quantities equal to each other or the to the same quantity, my be substituted for each other.

Congruence

Sequence of rigid motions that make two figures coincide (lie exactly on top of eachother).

Proof Reason for Supplements

Supplements of equal angles are equal

The Measure of an Angle

The amount of rotation needed to rotate one of the rays about their shared point so that it lies on top of the other ray (m<A, m<BAC).

Angles

The geometric object created by two rays with a common starting point (<A, <BAC) (the vertex is always in the middle).

Distance

The length of a straight line segment that connects two points.

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles (it will be greater than either of its two remote interior angles).

Scale Factor

The multiplicative amount that the picture has been enlarged.

Line Fact #2

The shortest distance between two points consists of the length of the straight line segment that connects them.

= sign

The sign reserved for numerical equations.

Triangle Sum Theorem

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

(If AB + BC = AC) AC > AB and AC>BC

The whole is greater than any of its parts.

AB + BC = AC

The whole is the sum of its parts.

Line Fact #1

Through any two points, only one straight line can be drawn.

Rigid Motions

Transformations that preserve shape and size (three basic ones are rotation, reflection, and translation).

Supplementary Angles

Two angles whose sum is 180 degrees (Straight Angle).

Complementary Angles

Two angles whose sum is 90 degrees (Right Angle).

Perpendicular

Two lines that intersect to form right angles.

Cartesian Plane

With coordinates.

CPCTC

corresponding parts of congruent triangles are congruent.

Circumcenter

the center of a circumscribed circle.