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Geometry Semester 2 Finals by Mind Map: Geometry Semester 2 Finals
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Geometry Semester 2 Finals

Vocabulary

Circumcenter

point of intersection of the perpendicular bisectors

Concurrent

lines or line segments that intersect in a single point

Incenter

point of intersection of bisectors

Circumscribe

to draw a geometric figure AROUND another figure so that the two are in contact but do not intersect

Inscribe

to draw a figure INSIDE another so that their boundaries touch but do not intersect

Median

a line segment that connects the midpoint and the vertex of the opposite side

Centroid

the point of intersections of the medians of a triangle

Altitude

a perpendicular line segment to a side of a triangle that passes through the opposite vertex

Orthocenter

points of intersection of the altitudes of a triangle

Midsegment

a line segment whose endpoints are midpoints of an included and an opposite line

Conjecture

an educated guess through observation

Similarity

same shape but NOT the same size

Similar polgyons, all corresponding angles are congruent, all corresponding sides are in proportion

Proportion

an equation in which 2 fractions are set equal to each other

Pre-image

the first set of coordinates of a shape

Image

the set of coordinates after a transformation has been applied

the set of coordinates after a transformation has been applied

Postulates and Theorems

Angle Side Angle

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

CPCTC

Corresponding Parts of Congruent Triangles are Congruent, Since ____ ≅ ____, _____≅______, and _______≅_______, triangle ____ is congruent to triangle ____ by _______. Since corresponding part of congruent triangles are congruent, _________., Since ____ ≅ ____, _____≅______, and _______≅_______, triangle ____ is congruent to triangle ____ by _______. Since corresponding part of congruent triangles are congruent, _________.

Hypotenuse Leg Theorem

If the hypotenuses are congruent and a pair of legs is congruent then the triangles are congruent.

Side Side Side

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

Side Angle Side

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

Triangle Angle and Opposite Side Comparison Theorem

If two sides of a triangle are not congruent, then the larger angle lies opposite the longest side.

Corollary to the Triangle Exterior Angle Theorem

The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.

Triangle Inequality Theorem

The sum of the lengths of any two sides if a triangle is greater than the length of the third side.

Angle Angle Similarity

If two angles from two different triangles are congruent, then the two triangles are similar., by the Triangle Angle Sum Theorem

Triangle Angle Sum Theorem

The sum of all angles in a triangle is 180.

Corresponding Angles Theorem

When there are 2 parallel lines crossed by a transversal, then corresponding angles are congruent.

Pythagorean Theorem

leg₁² + leg₂² = hypotenuse²

Side Splitter Theorem

a+b/c = (a/c) + (b/c)

Base Angle Theorem

in an isosceles triangle, base angles must have the same measure

Vertical Angles Theorem

vertical angles are always congruent.

Proofs

Indirect proofs

A proof in which the statement to be proved is assumed to be false. You then prove that the statement is not possible, confirming it is a false statement.

Coordinate proofs

prove distances are equal

prove slopes are equal, which means they are parallel

slopes multiply to -1, which means they are perpendicular

Properties

Comparison Property of Inequality

'if a=b+c and c>0, then a>b'

Equality

Reflexive, a = a

Symmetric, if a = b, b = a

Transitive, if a = b and b = c, then a = c

Addition, if a = b, then a+c=b+c

Subtraction, if a = b, then a-c=b-c

Subsitution, if a = b, then a may be replaced with b

Congruence

Reflexive, AB≅AB

Symmetric, If AB≅CD, then CD≅AB

Transitive, If AB≅CD and CD≅EF, then AB≅EF

Shapes

Rhombus

4 congruent sides

Parallelogram

2 pairs of parallel sides

a parallelogram has two pairs of congruent sides. If the length of one side is changed, the length of the opposite side will change to the same measure

opposite angles of a parallelogram have the same measure, and if one is changed, the measure of the opposite angle equally changes.

Rectangle

4 right angles, 2 pairs of congruent sides, all adjacent sides perpendicular

in a rectangle, diagonals bisect. Diagonals in a rectangle are congruent.

Trapezoid

exactly one pair of parallel sides, exactly one pair of congruent sides

Square

4 congruent sides, all adjacent sides perpendicular, 4 right angles

Kite

has no parallel sides, 2 congruent sides

opposite sides are NOT congruent

Formulas

distance

√(x₂-x₁)² + (y₂-y₁)²

midpoint

(average x, average y), ((y₂-y₁) / 2), ((x₂-x₁) / 2)

slope

(change in y/change in x), (y₂-y₁) / (x₂-x₁)

Surface Area

Cube, 6a²

Rectangular Prism, 2ab + 2bc + 2ac

Sphere, 4Πr²

Cylinder, 2Πr² + 2Πrh

Square Pyramid, s²+2sl

Cone, Πr² + Πrl

Trigonometry

SOHCAHTOA, Finding measure of a side, Finding hypotenuse, Tangent = opposite/adjacent, Finding adjacent, Sin = opposite/hypotenuse, Finding opposite, Cosine = adjacent/hypotenuse, Finding measure of an angle, Inverse of trigonometry

Trigonometric ratio, ratio you can write for a particular angle in a right angle

Volume

Cylinder, Πr² x h

Right-angled Triangular Prism, ½ (b x h) x l

Triangular Prism, ½ (b x h)

Cube, l x h x w

Area

triangle, ½bh

trapezoid, ½h (b₁+b₂)

rectangle/parallelogram, bh

rhombus/kite, ½d₁d₂

regular polygon, ½ap

circle, Πr²

F+V = E +2, Faces+Vertices = Edges +2

Transformations

Translation

slide, T (6, -7), Add 6 to the x-coordinate, subtract seven from the y-coordinate

reflections, x-axis, the x-coordinates remain of the same value, but the y coordinate becomes it's opposite (1,2)→(1,-2). The image is mirrored over the x-axis., y-axis, the y-coordinates don't change, but the x-coordinates become their opposite value (1,2)→(-1,2). The image is mirrored over the y-axis., y=x, the value of y becomes the value of x, and vice versa (1,3)→(3,1)., x = 1, reflect over x=1, setting 1x as the line of reflection. Do the same for y or a different value.

rotation, you take the image and rotate it a certain amount of degrees. Then you pretend that the turned axes are the axes straight up, and take what the coordinates would be., Clockwise., -90, -180, -270, -360, Counter-clockwise., 90, 180, 270, 360

dilation, multiply the coordinates of the pre-image by the scale factor,

composition, one transformation followed by another, T (5,-1) ºR 90°

DIY's

construct a perpendicular bisector

on the segment, spread compass to a little more than halfway, and draw an arch crossing over the line. Repeat on the other end, and draw a line through where the arches meet.

circumscribe

Put compass at circumcenter, letting it touch one vertice. It should pass through all vertices when you make a circle.

Sum of the measures of a quadrilateral is 360.