Geometry Concepts ANGLES Angles acute right obtuse straight protractor Complementary angles Supplementary angles Adjacent angles Linear pairs

Transcription

1 Gemetry Cncepts ANGLES Angles can be classified as: acute (less than 9 ) right (equal t 9, a square crner) btuse (greater than 9 but less than 18 ) straight (equal t 18, a straight line). Angles are measured with a prtractr. Cmplementary angles are tw angles whse measures add up t 9. Supplementary angles are tw angles whse measures add up t 18 Adjacent angles lie next t each ther, sharing a side and a vertex. Linear pairs are tw angles whse measures add up t 18 they lie alng a straight line. Vertical angles are frmed when tw lines intersect they are the angles that are ppsite each ther. Vertical angles are cngruent. LINES Lines are measured with a ruler. Cmmn metric measures are: Meter: a little mre than a yard Centimeter: a little less than half an inch. There are 1 centimeters in a meter. Millimeter: abut the thickness f a pencil lead. There are 1 millimeters in a meter. There are 1 millimeters in a centimeter. Perpendicular lines intersect t frm a 9 angle. Symbl is Parallel lines never intersect. They are the same distance apart everywhere. Symbl is A bisectr is a line that cuts an angle r line segment in half. SYMMETRY A figure has symmetry if a line can be drawn that divides the figure int mirrr images: ne side f the line is the exact mirrr image f the ther side f the line CONGRUENT Cngruent means same size, same shape. The wrd cngruent is used when referring t shapes: we say tw triangles are cngruent if they are the same size and same shape. The symbl fr cngruent is. (The wrd equal is used when referring t a distance r length we say that tw measures are equal: 4 meters = 4 centimeters) CONGRUENT TRIANGLES Tw triangles can be prven t be cngruent using these pstulates: SSS: Side-Side-Side SAS: Side-Angle-Side (the angle must be between the tw sides) ASA: Angle-Side-Angle (the side must be between the tw angles) AAS: Angle-Angle-Side (tw sides and the angle NOT between them) HL: Hyptenuse-Leg (in a right triangle, the hyptenuse and ne f the legs)

2 TRIANGLES: Triangles are named by: Sides Scalene: N sides equal Issceles: Tw sides equal Equilateral: All sides equal Angels Acute: All angles less than 9 Right: One angle = 9 Obtuse: One angle greater than 9 Equiangular: All angles are equal (all angles are (Equiangular triangles are als equilateral) 6 ) Issceles Triangles: In an issceles triangle, the angles ppsite the equal sides are cngruent, and the sides ppsite the cngruent angles are equal. Sides: In triangles, the sum f the shrter tw sides will always be greater than the third side. Side-Angle Relatinship: In triangles, the shrtest side is always ppsite the smallest angle; the lngest side is always ppsite the largest angle. Cnversely, the smallest angle is always ppsite the shrtest side, and the largest angle is always ppsite the lngest side. Similar Triangles are the same shape, but nt the same size. Similar triangles have the same angle measures, and their sides are prprtinal. Symbl is ~ PARALLEL LINES: When tw parallel lines are cut by a transversal, the angles are designated: Exterir: angles 1,, 7, 8 Interir: angles 3, 4, 5, 6 Alternate interir: 3 & 6; 4 & 5 Alternate exterir: 1 & 8; & 7 Crrespnding: 1 & 5; & 6; 3 & 7; 4 & Parallel Line Pstulates: When tw parallel lines are cut by a transversal: Alternate exterir angles are cngruent Alternate interir angles are cngruent Crrespnding angles are cngruent Interir angles n the same side f the transversal add t 18

3 POLYGONS Plygns are tw-dimensinal clsed figures wh sides are line segments. Plygns are named: 3-sided: triangle 4-sided: quadrilateral 5-sided: pentagn 6-sided: hexagn 7-sided: heptagn 8-sided: ctagn 9-sided: nnagn 1-sided: decagn 1-sided: ddecagn n-sided: n-gn Regular Plygns have all sides cngruent, all angles cngruent Quadrilaterals are 4-sided plygns. Parallelgrams have ppsite sides that are parallel and cngruent Rhmbus have ppsite sides parallel, all sides cngruent Rectangles are parallelgrams with fur 9 angles Squares are rectangles with fur cngruent sides Trapezids have ne pair f parallel sides Kites have tw pair f adjacent cngruent sides Angle measures f a plygn: if the number f sides f the plygn is called n, then: The sum f the interir angle measures is fund by ( n )18 Triangles cntain 18 Quadrilaterals cntain 36 T find the measure if EACH interir angle in a regular plygn: The sum f the exterir angle measures is always 36 ( n ) 18 n

4 Perimeter is the distance arund the utside edge f a plygn. T find the perimeter, add up the lengths f all sides. Area is a measure f hw many squares cver a surface. The label fr area is always units squared Sme cmmn area and perimeter frmulas: (b = base, h = height, l = length, w = width, s = side) Rectangle: A bh r A l w P l w Square: A s P 4s Parallelgram: A bh where h b Triangle: A 1 bh r bh where h b b1 b Trapezid: A h where h b CIRLCES (r = radius, frm center t utside edge; d = diameter, edge t edge ging thrugh center) Circle: A r where C d r VOLUME Vlume is the measure f the space inside a 3-dimensinal figure: hw many cubes are inside a figure. The label fr vlume is always units cubed. The general vlume frmula is V = Bh, where B is the area f the base f the figure. Mre specifically, the vlumes f sme cmmn shapes are: Rectangular prism (bx): V l wh Cube: 3 V s Cylinder: V r h Sphere: V 4 r 3 3 Surface Area: SA 4 r

5 Right Triangle Gemetry The study f Right Triangles is an intricate part f the study f mathematics. Right Triangles are ften fund in architecture and every day shapes. Every right triangle has ne 9 angle. The side ppsite the right angle is called the hyptenuse the hyptenuse is always the lngest side f a right triangle. The ther tw sides f the right triangle are called the legs. leg = a leg = b leg = b Hyptenuse = c Pythagrean s Therem gives the relatinship between sides f a right triangle: a b c Use Pythagrean s Therem t find a missing side f a right triangle if tw f the sides are given: *If the tw legs are given, add their squares, then take the f their sum t get the hyptenuse. *If the hyptenuse and ne f the legs are given, subtract the squares, then take the difference t get the remaining leg. f their Special Right Triangle: the Issceles Right Triangle Tw f the sides are the same length (cngruent), therefre the angles ppsite the cngruent sides are als cngruent, making them each 45. If the cngruent sides are given, the hyptenuse is fund by multiplying ne f the cngruent sides times. x x If the hyptenuse is given, then the cngruent sides are fund by dividing the hyptenuse by. x Special Right Triangle: x 3 x x The hyptenuse is always twice as lng as the shrt side. The medium-length side is fund by taking the shrt side times 3. If the medium-length side is given, divide its length by 3 t btain the shrt side, then duble the shrt side t btain the hyptenuse.