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Transcript
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 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 measure of each interior angle in a regular n-gon: sum of exterior angles in any n-gon = 360 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 , 2 ) 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 ) 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