Download Topics for Midterm 2011accel

Geometry – accel.
TOPICS FOR GEOMETRY MIDTERM
Basic Terms – Chapter 1 & 2
 bisect
 collinear
 coplanar
 length vs. distance
 angle
 line
 ray
 segment
 midpoint
 vertex
 angle measure
 adjacent angles
 vertical angles
 angle bisector
 congruent
 obtuse angle
 acute angle
 right angle
 exterior angles
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Dec. 2011
complementary angles
supplementary angles
parallel lines
perpendicular lines
transversal
skew lines
alternate interior angles
corresponding angles
same side interior angles
CPCTC
isosceles triangle
median
altitude
perpendicular bisector
centroid
orthocenter
circumcenter
Formulas – Chapter 3
 sum of interior angles in any n-gon: (𝑛 − 2)180
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measure of each interior angle in a regular n-gon:
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sum of exterior angles in any n-gon = 360
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measure of each exterior angle in a regular n-gon:
(𝑛−2)180
𝑛
360
𝑛
Parallel Lines – Chapter 3
 properties of parallel lines
 proving lines parallel
o corresponding angles
o alternate interior angles
o same side interior angles
o two lines perpendicular to the same line are parallel
o two lines parallel to the same line are parallel to each other
Polygons – Chapter 3
 regular polygons
 angles of a triangle
 exterior angle theorem (for triangles)
 angles in a polygon
o formulas for interior / exterior angles
 tessellations
o definition, regular / semi-regular tessellations
o determining whether a given set of regular polygons will tessellate
o determining the missing polygon in a tessellation
Inequalities for One Triangle – Section 6-4
 each side must be less than the sum of the other two sides
o determining whether a set of 3 sides could form a triangle
o determining the largest/smallest length for the third side, given 2 sides
 largest side opposite largest angle; smallest side opposite smallest angle
Congruent Triangles- Chapter 4
 5 ways to prove triangles congruent:
o
SSS, SAS, ASA, AAS
o
HL – must be a right triangle
o
NOT SSA!
 CPCTC: Using congruent triangles to prove segments or angles congruent
 Isosceles Triangle Theorem and its converse
 Perpendicular Bisector Theorem
 Special Lines of Triangles:
o Median
 goes from vertex to midpoint of opposite side
 are concurrent at centroid
 distance from vertex to centroid is twice the distance from centroid to
midpoint of side
o Altitude
 goes from vertex, perpendicular to line containing opposite side
 acute triangle: altitudes are all in interior of triangle
 obtuse triangle: two altitudes are outside of the triangle
 right triangle: two sides of the triangle are altitudes
 altitudes are concurrent at orthocenter
o Perpendicular Bisector
 perpendicular to segment (side of triangle) at its midpoint
 does not necessarily pass through a vertex of the triangle
 perpendicular bisectors concurrent at circumcenter
 circumcenter is center of circle that contains the three vertices of the triangle
Quadrilaterals – Chapter 5
 Properties of Parallelograms
1. both pairs of opposite sides parallel (def.)
2. both pairs of opposite sides congruent
3. both pairs of opposite angles congruent
4. consecutive angles supplementary
5. diagonals bisect each other
 Proving a Quadrilateral is a Parallelogram
1. both pairs of opposite sides parallel (def.)
2. both pairs of opposite sides congruent
3. both pairs of opposite angles congruent
4. diagonals bisect each other
5. one pair of opposite sides congruent and parallel
 Rectangles – all properties of parallelogram, plus:
1. all angles are right angles (def.)
2. diagonals are congruent
3. proving a quadrilateral is a rectangle
 use definition (4 right angles)
 parallelogram + one right angle
 parallelogram + congruent diagonals
 Rhombuses – all properties of parallelogram, plus:
1. all sides are congruent (def.)
2. diagonals are perpendicular
3. diagonals bisect vertex angles
4. proving a quadrilateral is a rhombus
 use definition (4 congruent sides)
 parallelogram + 2 adjacent sides congruent
 parallelogram + diagonals perpendicular
 Squares – all properties of rectangle, rhombus (and parallelogram)
 Theorems Involving Parallel Lines
1. Triangle Midsegment Theorem & converse
2. If three or more parallel lines cut off congruent segments on one transversal, then
they cut off congruent segments on any other transversal
 Misc. Theorems, proofs
1. The midpoint of the hypotenuse of a right triangle is equidistant from the 3 vertices
2. Mother-Daughter proofs:
 daughter of any quadrilateral is a parallelogram
 daughter of a rectangle is a rhombus
 daughter of a rhombus is a rectangle
 Trapezoids
1. Trapezoid is a quadrilateral with exactly one pair of opposite sides parallel
 terms: bases, legs, base angles, median (midsegment)
2. Isosceles trapezoid
 base angles congruent
 diagonals congruent
3. Median of a trapezoid
 parallel to the bases
 length is average of the lengths of the bases
 Coordinate Geometry
1. Using coordinate geometry methods to prove a quadrilateral is a special type
Coordinate Geometry – Chapter 13
 formulas
o distance formula:
𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
o equation of circle: (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2 center = (𝑎, 𝑏), radius = 𝑟
o midpoint formula:
(
𝑥1 +𝑥2 𝑦1 +𝑦2
2
𝑦 −𝑦
o slope formula: 𝑚 = 𝑥2 −𝑥1
2
1
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2
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simplifying radicals
writing equation of a circle
o given: center & radius
o given: center & point on circle
o given: endpoints of diameter
determining center & radius from equation of circle
writing equation of a line
o slope-intercept equation of a line: 𝑦 = 𝑚𝑥 + 𝑏
o point-slope equation of a line: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
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o standard form equation of a line: 𝐴𝑥 + 𝐵𝑦 = 𝐶
 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠
 𝐴≥0
 𝐴 𝑎𝑛𝑑 𝐵 𝑛𝑜𝑡 𝑏𝑜𝑡ℎ 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑧𝑒𝑟𝑜
distance from a point to a line
using point-slope or slope-intercept forms
converting to standard form
horizontal lines: 𝑦 = #
vertical lines: 𝑥 = #
parallel & perpendicular lines
o slopes
o writing standard form equations
finding x- and y-intercepts
points of intersection – systems of equations