# Geometry Semester 2 Final Study Guide

Parallelogram

A quadrilateral in which both pairs of opposite sides are parallelogram.

Rhombus

A parallelogram which has four congruent sides.

Rectangle

A parallelogram which has four right angles.

Square

A parallelogram which has four congruent sides and four right angles.

Kite

A quadrilateral with two pairs of congruent sides and no congruent opposite sides; diagonals are perpendicular; symmetry; longer diagonal line of symmetry.

Trapezoid

A quadrilateral with exactly one pairs of parallel sides.

Isosceles Trapezoid

A trapezoid in which the nonparallel opposite sides (leg) are congruent; base angles are congruent; diagonals are congruent.

Consecutive Angles

Angels that are opposite of each other.

Base Angles of Trapezoid

The two angles that share a base.

Mid-segment of a Trapezoid

Line segment connecting the midpoints of the legs; runs parallel to the bases of the trapezoid; its length is the average of the base length.

Properties of Parallelograms/Prove a Quadrilateral is Parallelogram

Both pairs of opposite sides are parallel; both pairs of opposite sides are congruent; both pairs of opposite angles are congruent; consecutive angles are supplementary; the diagonals bisect each other; show that one pair of sides are simultaneously congruent and parallel.

Theorem 6-4

If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on very transversal/

Properties of a Rhombus

All the properties of a parallelogram; the diagonals bisect the vertex angles; the diagonals are perpendicular.

Properties of a Rectangle

All the properties of a parallelogram; the diagonals are congruent(distance formula).

Properties of a Square

All the properties of a parallelogram, rhombus and rectangle.

Circle

The set of all the points equidistant from the given point.

Radius

Segment from center to vertex.

Diameter

A segment that contains the center of the circle and has both endpoints on the circle.

Central Angle

Formed by two consecutive radii with the center as its vertex; 360/n (n= number of sides).

Semicircle

Arc whose measure equals 180 degrees; named using three letters.

Minor Arc

Arc who measure is between 0 and 180 degrees; named using three letters.

Major Arc

Arc whose measure i s between 180 and 360 degrees; named using three letters.

Adjacent Arc

Arcs in the same circle with any one point in common.

Circumference

Distance around a circle; C=(pi)(d) or 2(pi)(r).

Arc Length

Part of the circle’s circumference.

Congruent Arc

Arcs that have the same measure and length.

Sector of a Circle

Fraction of a circle, formed by two radii and arc; named same as central angle.

Segment of a Circle

Formed by arc and segment connecting end points.

Pythagorean Theorem

(aa)+(bb)=(cc)

Area of a Rectangle

A=bh

Area of a Parallelogram

A=bh

Area of a Triangle

A=1/2bh

A= 1/2anSin

A= 1/2anSin

Area of a Trapezoid

A=1/2h(b1 + b2)

Area of a Rhombus of a Kite

A= 1/2(d1*d2)

Area of a Regular Polygon

A= 1/2ap

Formula for Arc Length

length of arc AB= measure of arc AB/360 *2(pi)(r)

Area of a Circle

A=(pi)(r2)

Area of a Sector of a Circle

measure of arc AB/360 * (pi)(r2)

Heron’s Formula

Find are if given all sides (SSS); are of a triangle; =the square root of(s(s-a)(s-b)(s-c) where s= a+b+c/2.

45-45-90 Triangle

short leg=x; long leg=x hypotenuse=xroot2

30-60-90 Triangle

short leg=x: long leg= xroot3; hypotenuse=2x

Ratio

a comparison of two quantities measured in the same unit; will always be reduced to lowest terms and will have no units.

Proportions

a statement that two ratios are equal; have three or more ratios are equal then they can be written as an extended proportion.

Scale Drawing

A drawing that is either an enlargement or a reduction of an object.

Scale Factor

The amount of enlargement or reduction is referred to as.

Quadratic Formula

X= -b + – the square root of b2 -4ac/ 2a

Similar Polygons

All corresponding angles must be congruent; all corresponding sides must be proportional.

Angle-Angle Similarity Postulate

If two angles of one triangle are congruent to two angles in another triangle, then the triangles are similar.

Side-Angle-Side Similarity Postulate

If an angle of one triangle is congruent to one angle in another and the sides including the angles are proportional, then the triangles are similar.

Side-Side-Side Similarity Theorem

If three sides of one triangle are proportional to corresponding sides in another triangle, then the triangles are similar.

Indirect Measurements

Uses similar triangles and proportions to find distances that would otherwise be difficult to find.

Geometric Means

A proportion in which the “means” in the proportion have the same value.

Theorem 8-3

The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

Side-Splitter Theorem

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

Corollary to the Side Slitter Theorem

If three parallel ones intersect two transversals then the segments intercepted on the transversal are proportional.

Triangle-Angle-Bisectors Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other sides of the triangle.

Similarity Ratio (Scale Factor):

Ratio of corresponding parts of similar figures; comparison of LINEAR elements( lengths).

SOH-CAH-TOA

Sine=opposite/hypotenuse; cosine= adjacent/hypotenuse; tangent=opposite/adjacent

Trig inverse

Used to find angle measures

Angle of Elevation

Is measured upward from the horizontal to line of sight.

Angle of Depression

Is measure downward from the horizontal to line of sight.

Polyhedrons

3 dimensional figure whose faces are polygons.

Faces

Planes

Edges:

Lines/segments

Vertices

Points (corners)

Euler’s Formula

F+V=E+2

Platonic Sides

Regular polyhedrons. They are regular because all of their faces are congruent regular polygons.

Tetrahedron

4 faces

Hexahedron

5 faces

Octahedron

8 faces

Dodecahedron

12 faces

Icosahedron

20 faces

Net

2 dimensional pattern that folds into a 3 dimensional figure.

Prism

Polyhedron with 2, congruent, parallel faces and regular faces( lateral faces); named by the shape of its base.

Lateral Area

Area of all lateral faces.

Surface Area

Area of all faces( lateral faces plus both bases).

Right Cylinder

A cylinder where the height is a side of the cylinder.

Oblique Cylinder

A cylinder where the height is outside of the cylinder.

Lateral Area of a Cylinder

2(pi)(r)(h)

Surface Area of a Cylinder

2(pi)(r)(h) + 2(pi)(r2)

Volume of a Cylinder

(pi)(r2)(h)

Slant Height

The height that comes off of the base of a pyramid of cone that goes to the vertex.

Lateral Area of a Pyramid

n(1/2sl)

Surface Area of a Pyramid

n(1/2sl) + s2

Volume of a Pyramid

1/3Bh

Lateral Area of Cone

(pi)(r)(l)

Surface Area of a Cone

(pi)(r)(l) + (pi)(r2)

Volume of a Cone

1/3/(pi)(r2)(h)

Sphere

The set of points in space equidistant from a given point(the center).

Surface Area of a Sphere

4(pi)(r2)

Volume of a Sphere

4(pi)(r3)/3

Tangent Line

Line that intersects a circle in exactly one point (called the point of tangency)

Theorem 11-1

A tangent line is perpendicular to a radius to the point of tangency

Theorem 11-3

Two segments tangent to a circle from a point outside the circle are congruent.In addition to the two segments being congruent, line AB is congruent to line CB.

Chord

A segment whose end points are on a circle.

Theorem 11-4

Within a circle or congruent circles, congruent angles have congruent chords; within a circle or congruent circles congruent chords have congruent arcs; within a circle or congruent circles, congruent arcs have congruent central angles.

Theorem 11-5

Within a circle or congruent circles, chord equidistant from the center are congruent and congruent chords are equidistant from the center.

Theorem 11-6

In a circle, a diameter or radius is perpendicular to a chord if and only if it bisects the chord and its arc.

Inscribed Angles

An angle in a circle whose vertex is on the circle and its sides are chords of the circle. An inscribed angle is never a central angle.

Theorem 11-9

The measure of a inscribed angle is half the measure of the intercepted arc. Two inscribed angles that intercept tc are congruent. An angle inscribed in a semicircle is a right angle. The opposite angle of a quadrilateral inscribed a circle are supplementary.

Theorem 11-10

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Inscribed and Circumscribed

A circle is inscribed in a quadrilateral and a quadrilateral is circumscribed around the circle.

Secant Lines

A line that intersects a circle at exactly two points.

Theorem 11-11

The measure of a vertical angle formed by two secant line sis half of the sum of the two intercepted arcs; the measure of the angle formed by two secant lines is half of the difference of the intercepted arcs; the measure of the angle formed by a secant line and a tangent line is half of the difference of the intercepted arcs; the measure of the angle formed by two tangent lines is half of the difference of the major arc and the minor arc. M<1 = 1/2(mAB- CD)

Theorem 11-12

When P is in the interior of the circle of then a*b=c*d; when P lies outside of the circle then w(w+x)=y(y+z); when P is outside the circle and the one segment is tangent the circle t^2 = y(y+z)

Distance Formula

D = square root of (y2-y1)^2 + (x2-x1)^2

Standard form of a Circle

The equation of a circle with the center (h,k) and the radius r is (x-h)^2 + (y-k)^2 =r^2

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