Geometry Star Test Study Guide

GEOMETRY STAR TEST STUDY GUIDE

CHAPTER 1

Terms to know:

·  Inductive reasoning : making a conjecture (guess) based on observation of patterns.

·  Deductive reasoning : proving a statement based on facts (definitions, theorems, postulates, …)

·  Counterexample: an example that disproves a statement (just need one).

·  Postulate : a statement that is assumed to be true (also called an “axiom”).

·  Theorem : a statement that must be proven true.

·  Angle types: acute, right, obtuse, straight, adjacent, congruent, vertical, linear pair, complementary, supplementary.

·  Angle Bisector: any figure that divides an angle into 2 congruent angles.

·  Undefined terms : point, line, plane.

Formulas used:

·  Distance :

·  Midpoint : midpt. =

·  Pythagorean Theorem :

CHAPTER 2

Terms to know:

·  Conditional statement : an “if-then” statement.

·  Converse : switch the “if” and the “then” parts of the conditional.

·  Inverse : Negate both the “if” and the “then”.

·  Contrapositive : Switch and negate.

·  Biconditional : an “if and only if” statement. Contains a conditional and its converse.

Important Theorems/Postulates/Properties:

·  Basic Algebraic Properties : Addition, subtraction, multiplication, division properties of equal.

·  Reflexive, symmetric, and transitive props. of equal.

·  Distributive property.

·  Right angle congruence thm. : All right angles are congruent.

·  Linear pair postulate: If two angles form a linear pair then they are supplementary.

·  Vertcial angle thm. : Vertical angles are congruent.

CHAPTER 3

Terms to know:

·  Types of lines : parallel, skew, transversal.

·  Types of angles : corresponding, alternate interior, alternate exterior, consecutive interior. These are formed by 2 lines intersected by a transversal. If the lines are parallel, then the pairs of angles are congruent (AI, AE, and CORR) or supplementary (CON INT).

Coordinate geometry:

·  Slope formula :

·  Parallel lines have equal slopes.

·  Perpendicular lines have negative reciprocal slopes. This means that the product of their slopes equals -1. ()

CHAPTER 4

Terms to know:

·  Types of triangles: Equilateral, isosceles, scalene, acute, equiangular, right, obtuse.

·  Types of angles: Interior and exterior.

Important Theorems:

·  Triangle sum thm. : the sum of the interior angles of a triangle equals 180 degrees.

·  Exterior angle thm. : the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

·  Third angle thm. : If two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent.

·  Triangle congruence thms. : (ways to prove triangles congruent) SSS, SAS, ASA, AAS, HL.

·  Base angles thm. : If two sides of a triangle are congruent, then the angles opposite them are congruent.

·  Converse of Base angles thm : If two angles of a triangle are congruent, then the sides opposite them are congruent.

·  Equilateral triangles are also equiangular.

·  Corresponding parts of congruent triangles are congruent (CPCTC).

PROOFS: Because this test is multiple choice, you will only have to fill in a statement or a reason for a proof already written. Most of the proofs in the test will be from this chapter.

CHAPTER 5

Terms to know:

·  Perpendicular bisector of segment: A figure that passes through a segment at its midpoint and is perpendicular to it.

·  Triangle segments/lines: perp. bisector, angle bisector, median, altitude.

·  Point of concurrency : the point where 3 or more lines intersect.

·  Circumcenter (of a triangle) : the point of concurrency of the perpendicular bisectors of a triangle.

·  Incenter : the point of concurrency of the angle bisectors of a triangle.

·  Centroid : the point of concurrency of the medians of a triangle.

·  Orthocenter : the point of concurrency of the altitudes of a triangle.

·  Midsegment : the segment that connects the midpoints of 2 sides of a triangle.

·  Indirect proof (also called “proof by contradiction”) : You must know the steps, which are:

1) Assume temporarily that the “prove” statement is false (or that the opposite of it is true).

2) Reason logically until you come to contradiction of a known fact.

3) State that since the assumption led you to a contradiction, it must be false, so the “prove” must be true.

Important Theorems:

·  Perp. Bisector thm. : If a point lies on the perp. bis. of a segment, then it is equidistant form the endpoints.

·  Angle Bisector thm. : If a point lies on the angle bis. of an angle, then it is equidistant from the sides of the angle.

·  The circumcenter of a triangle is equidistant from the vertices.

·  The incenter of a triangle is equidistant from the sides.

·  The centroid of a triangle is at a point on each median two-thirds of the distance from the vertex to the midpoint of the opposite side.

·  Midsegment thm. : the midsegment is half the length of the third side and is parallel to it.

·  Triangle Inequality: the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

CHAPTER 6

Terms to know:

·  Polygon

·  Regular polygon

·  Types of polygons: triangles, quadrilaterals, pentagons, …

·  Types of quadrilaterals : parallelograms, rhombuses, rectangles, squares, kites, trapezoids.

·  Midsegment of a trapezoid : a segment connecting the midpoints of the legs.

Important Theorems/Properties:

·  Properties of parallelogram: opposite sides parallel, opposite sides congruent, opposite angles congruent, consecutive angles supplementary, diagonals bisect each other.

·  Properties of rectangle: all parallelogram properties, diagonals congruent, 4 right angles.

·  Properties of rhombus: all par. props., diagonals perpendicular, diags. bisect opposite angles.

·  Properties of trapezoid: one pair of parallel sides.

·  Properties of a kite: 2 pairs of consecutive congruent sides.

·  Midsegment thm. for trapezoid: midsegment is parallel to both bases and one half the sum of the lengths of the bases.

Formulas used:

·  Area of a square:

·  Area of a rectangle :

·  Area of a parallelogram :

·  Area of a triangle :

·  Area of a trapezoid:

·  Area of a kite:

·  Area of a rhombus:

CHAPTER 7

Terms to know:

·  Transformation: an operation that moves (or “maps”) a figure.

·  Preimage: the original figure.

·  Image: the figure after the transformation.

·  Reflection: transforming across or over a line (“flipping”)

·  Rotation : transforming around a point (“turning”)

·  Translation : transforming horizontally or vertically (“sliding:). It looks like .

·  Vector: a quantity that has both direction and magnitude. It looks like .

CHAPTER 8

Terms to know:

·  Ratio: a quotient of 2 quantities.

·  Proportion : the equating of 2 ratios.

·  Similar polygons: 2 polygons that have congruent corresponding angles and proportional sides.

Important Theorems/Properties:

·  Cross Product property: A method for solving proportions. (“product of means equals product of extremes”)

·  Ratio of perimeters of similar polygons : same as the ratio of sides (scale factor).

·  Ways to show triangles similar: AA, SSS, SAS.

CHAPTER 9

Terms to know:

·  Pythagorean Triple: when the sides of a right triangle are positive integers.

Important Theorems/Properties:

·  Pythagorean thm. : Used to find the length of a side of a right triangle when you have 2 side lengths.

·  Converse of the Pythagorean thm. : If , then it is a right triangle.

·  Acute triangle : If , then it is an acute triangle.

·  Obtuse triangle: If , then it is an obtuse triangle.

·  Special right triangles: 30-60-90, 45-45-90.

·  30-60-90: Hypotenuse is twice the short leg, long leg is times the short leg.

·  45-45-90: Hypotenuse is times each leg.

·  Trigonometry: used to find the lengths of sides in a right triangle when Pythagorean thm. or special triangles won’t work.

·  Sine : (SOH)

·  Cosine: (CAH)

·  Tangent : (TOA)

·  For this test we also might need to know these identities: and .

CHAPTER 10

Terms to know:

·  Radius, diameter, chord, secant, tangent, inscribed angle, central angle, intercepted arc, polygon inscribed in a circle, circle that circumscribes a polygon.

Important Theorems/Properties:

We had many theorems in this chapter that might appear on the STAR test, but the most important ones are:

·  If a line is tangent to a circle, then it is perpendicular to the radius at the pt. of tangency.

·  If 2 tangent lines intersect outside a circle, then the segments from that point to the points of tangency are congruent.

·  A central angle is equal to its intercepted arc.

·  An inscribed angle is equal to one half of its intercepted arc.

Coordinate geometry:

·  Equation of a circle: .

CHAPTER 11

Not many of the Geometry Standards come from this chapter. Here are the ideas we need.

·  Sum of interior angles of a polygon: .

·  Sum of the exterior angles of a polygon: S = 360 .

·  Area of an equilateral triangle: .

·  Ratio of areas of similar polygons is the scale factor squared.

·  Circumference of a circle : .

·  Area of a circle: .

CHAPTER 12

Formulas needed:

Surface Area of a Prism: (where B = the area of the base, and P = perimeter of base).

Surface Area of a Cylinder:

Surface Area of a Pyramid:

Surface Area of a Cone:

Surface Area of a Sphere:

Volume of a Prism:

Volume of a Cylinder:

Volume of a Pyramid:

Volume of a Cone:

Volume of a Sphere:

Ratio of Volumes of similar solids is the cube of the scale factor.

CONSTRUCTIONS

You will have to be familiar with the basic constructions. You should know the steps of them and be able to recognize which one is being done if you are given an example.