MATH471 (Abstract Geometry) Exam 1 Review

acute angle

an angle between 0 and 90 degrees

right angle

an angle that is exactly 90 degrees

obtuse angle

an angle that is between 90 and 180 degrees

straight angle

an angle that is exactly 180 degrees

complementary angles

angles whose sum equals 90 degrees

supplementary angles

angles whose sum equals 180 degrees

scalene triangle

a triangle with sides of 3 different lengths

right triangle

a triangle that contains 1 90 degree angle

isosceles triangle

a triangle which has 2 congruent sides; 2 congruent base angles

equilateral/equiangular

a triangle where all 3 sides are congruent and all 3 angles are congruent

acute triangle

a triangle where each of the interior angles are less than 90 degrees

obtuse triangle

a triangle where 1 of the 3 interior angles is greater than 90 degrees

median

the line segment joining a vertex of a triangle to the midpoint of the opposite side

altitude

the perpendicular line drawn from one of the triangle’s vertices to its opposite side

angle bisector

the line that divides an interior angle into 2 equal angles

incenter

the point of intersection of all the 3 interior angles bisectors of the triangle

centroid

the intersection of the 3 median of the triangle

circumcenter

the intersection of the 3 perpendicular bisectors of a triangle’s sides

orthocenter

the point of intersection where the 3 altitudes of a triangle meet

parallelogram

a quadrilateral with opposite sides that are parallel

rectangle

a quadrilateral with 4 right angles

rhombus

a quadrilateral in which all 4 congruent sides

square

a quadrilateral with 4 congruent sides and 4 right angles

kite

a convex quadrilateral which has 2 distinct pairs of adjacent congruent sides, but not all sides are mutually congruent

trapezoid

a quadrilateral with exactly one pair parallel sides

isosceles trapezoid

a quadrilateral with exactly one pair of parallel sides, and opposite sides (non-parallel) that are congruent

chord

a line segment with both endpoints on a circle

secant

a line intersecting a circle in 2 distinct points

tangent

a line intersecting a circle in a single point

central angle

an angle with vertex at the center of the circle

inscribed angle

angle ABC if AC and BC are 2 distinct chords of circle O with common endpoint C

arc

a portion of the circumference of a circle that connects 2 distinct points on the circle

isometry

a transformation that preserves distance

Euclid’s 1st Postulate

a straight line may be drawn from any point to any other point

Euclid’s 2nd Postulate

a finite straight line may be extended continuously in a straight line

Euclid’s 3rd Postulate

a circle may be described with any center and radius

Euclid’s 4th Postulate

all right angles are equal to each other

Euclid’s 5th Postulate (The Parallel Postulate)

if a straight line meets 2 other straight lines so as to make the 2 interior angles on 1 side of it together less than 2 right angles, the other straight lines, if extended indefinitely, will meet on that side on which the angles are less than 2 right angles

Playfair’s Postulate (Euclidean Parallel Postulate)

for any line m in the plane and point P not on m, there exists a unique line n containing P that is parallel to m

Neutral Parallel Postulate

for any line m in the plane and point P not on m, there exists at least one line n containing P that is parallel to m

collinear

a set of points that all lie on the same line

midpoint

the point M between points A and B such that AM = MB

SAS

if 2 sides of a triangle are congruent to 2 sides of another triangle and the angle formed by the congruent sides is also congruent, then the 2 triangles are congruent

Exterior Angle Theorem

any measure of an exterior angle of a triangle is greater than either of the non-adjacent interior angles of the triangle

Saccheri-Legendre Theorem

The sum of the interior angles of a triangle is less than or equal to 180 degrees

defect

Triangle: d(ABC) = 180 - S(ABC) [From SL Theorem, d(ABC) is greater than or equal to 0]

Exterior Angle Theorem (In the Neutral Plane)

the measure of an exterior angle of a triangle is greater than the sum of the measure of the 2 non-adjacent interior angles

ASA

if 2 angles of 1 triangle are congruent to 2 corresponding angles of another triangle, and the pair of corresponding sides common to the 2 angles are also congruent, then the 2 triangles are congruent

AAS

if 2 angles of 1 triangle are congruent to 2 corresponding angles of another triangle, and 1 of the 2 pairs of corresponding sides not common to the 2 angles are also congruent, then the 2 triangles are congruent

Isosceles Triangle Theorem

2 sides of a triangle are congruent if and only if the 2 angles opposite the congruent sides are congruent

Hinge Theorem

if 2 angles of a triangle are not congruent, then the side opposite the larger of the 2 angles is longer than the side opposite the smaller of the 2 angles

Converse of the Hinge Theorem

if 2 sides of a triangle are not congruent, then the angle opposite the longer of the 2 sides is larger than the angle opposite the shorter of the 2 sides

The Triangle Inequality

the sum of the lengths of any 2 sides of a triangle is greater than the length of the third side

SSS

if all 3 sides of 1 triangle are congruent to the corresponding 3 sides of a second triangle, then the 2 triangles are congruent

Criteria for Triangle Congruence

SAS, ASA, AAS (SAA), SSS

HL

if 2 right triangles are such that the hypotenuse of each and a corresponding second side (a leg) of each are congruent, then the 2 triangles are congruent

Alternate Interior Angle Theorem

if 2 lines m and n are intersected by a transversal t with a pair of congruent alternate interior angles, then m is parallel to n

convex quadrilateral

the vertex of each angle lies in the interior of its opposite angle

Quadrilateral Congruence

SASAS, ASASA, SASSS

Saccheri Quadrilateral

a quadrilateral with 2 consecutive right angles, say at A and B, with AD congruent to BC

Lambert Quadrilateral

a quadrilateral with 3 right angles

if ABCD is a convex quadrilateral, then S(ABCD)…

less than or equal to 360 degrees

Euclid’s 5th Postulate is equivalent to….

Playfair’s postulate <=> CAIAT <=> S(ABC) = 180 degrees <=> rectangle exists

Converse of the Alternate Interior Angle Theorem (CAIAT)

if 2 lines m and n are intersected by a transversal t and m is parallel to n, then the alternate interior angles are congruent

Euclidean Exterior Angle Theorem

the measure of an exterior angle of a triangle is equal to the sum of the measures of the 2 non-adjacent interior angles

area of a parallelogram

base x altitude

area of a triangle

½ base x altitude

Heron’s Formula (area of a triangle)

llABCll = sqrt(s(s-a)(s-b)(s-c)

s = ½ (a+b+c)

area of trapezoid

½ h(BC+AD)

cyclic quadrilateral

vertices all lie on a circle; opposite angles are supplementary

Triangle Similarity

SAS, AAA, SSS

Inscribed Angle Theorem

the measure of an inscribed angle is ½ the measure of its intercepted arc

Ptolemy’s Theorem

the product of the lengths of the diagonals of a cyclic quadrilateral is equal to the sum of the products of the lengths of its opposite sides

Chord-Chord Angle Theorem

if 2 chords intersect at an interior point of a circle, then the measure of an angle at the point of intersection is the average of the measure of its intercepted arc and the arc intercepted by its vertical angle

Chord-Chord Segment Theorem

if 2 chords AB and CD of circle O intersect at a point P interior to circle O, then AP x BP = CP x DP

Tangent-Chord Theorem

the angle formed by a tangent to a circle and a chord with one endpoint at the point of tangency is ½ the measure of its intercepted arc

Tangent-Secant Angle Theorem

if a secant line and a tangent line to a circle intersect at an exterior point, then the measure if the angle with vertex at the point of intersection is ½ the positive difference in the measures of its 2 intercepted arcs

Tangent-Secant Segment Theorem

if a secant line AB and a tangent line PT intersect at the exterior point P, with T the point of tangency to circle O and P-A-B, then PB x PA = (PT)²

Tangent-Tangent Theorem

if 2 tangent lines PT’ and PT intersect at the exterior point O, with T, T’ the points of tangency to circle O, then

1.) PT = PT”

2.) m angle TPT” = 180 - m arc TT”

Secant-Secant Segment Theorem

AB x AC = AD x AE

orthogonal

if 2 circles intersect at 2 points and the tangent lines to each circle at a point of intersection are perpendicular to each other

A’ is the ____ of A

image

A is the _____ of A’

pre-image

invariant/fixed point

if F(A) = A

invariant figure

for every point A in F, F(A) = F

Common Invariant Properties

angles, area, betweenness, collinearity, congruence, conic, distance, orientation, parallel lines, ratios, ratios along lines

direct isometry

preserves orientation

indirect isometry

reverses orientation

Direct Isometries

Translation & Rotation

Indirect Isometries

Reflection & Glide Reflection

translation

for any point P in the plane and image T_v (P) = P’, PP’ = v

rotation

for any point P ≠ O and image R_{O, θ} (P) = P’, OP = OP' and m∠POP’ = θ

reflection

R_m (P) = P for any point P in m, and if P not in m with R_m (P) = P’, then m is the perpendicular bisector of PP’

glide reflection

T_v ∘ R_m where m is parallel to v