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Key Concepts

10.1 Points, Lines, and Planes

  • Modern-day geometry began in approximately 300 BCE with Euclid’s Elements, where he defined the principles associated with the line, the point, and the plane.
  • Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other.
  • The union of two sets, AA and BB, contains all points that are in both sets and is symbolized as AB.AB.
  • The intersection of two sets AA and BB includes only the points common to both sets and is symbolized as AB.AB.

10.2 Angles

  • Angles are classified as acute if they measure less than 90,90, obtuse if they measure greater than 9090 and less than 180,180, right if they measure exactly 90,90, and straight if they measure exactly 180.180.
  • If the sum of angles equals 9090, they are complimentary angles. If the sum of angles equals 180180, they are supplementary.
  • A transversal crossing two parallel lines form a series of equal angles: alternate interior angles, alternate exterior angles, vertical angles, and corresponding angles

10.3 Triangles

  • The sum of the interior angles of a triangle equals 180.180.
  • Two triangles are congruent when the corresponding angles have the same measure and the corresponding side lengths are equal.
  • The congruence theorems include the following: SAS, two sides and the included angle of one triangle are congruent to the same in a second triangle; ASA, two angles and the included side of one triangle are congruent to the same in a second triangle; SSS, all three side lengths of one triangle are congruent to the same in a second triangle; AAS, two angles and the non-included side of one triangle are congruent to the same in a second triangle.
  • Two shapes are similar when the proportions between corresponding angles, sides or features of two shapes are equal, regardless of size.

10.4 Polygons, Perimeter, and Circumference

  • Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number of their sides.
  • The perimeter of a polygon is the measure of the outline of the shape. We determine a shape’s perimeter by calculating the sum of the lengths of its sides.
  • The sum of the interior angles of a regular polygon with nn sides is found using the formula S=(n2)180.S=(n2)180. The measure of a single interior angle of a regular polygon with nn sides is determined using the formula a=(n2)180n.a=(n2)180n.
  • The sum of the exterior angles of a regular polygon is 360.360. The measure of a single exterior angle of a regular polygon with nn sides is found using the formula b=360n.b=360n.
  • The circumference of a circle is C=2πr,C=2πr, where rr is the radius, or C=πd,C=πd, and dd is the diameter.

10.5 Tessellations

  • A tessellation is a particular pattern composed of shapes, usually polygons, that repeat and cover the plane with no gaps or overlaps.
  • Properties of tessellations include rigid motions of the shapes called transformations. Transformations refer to translations, rotations, reflections, and glide reflections. Shapes are transformed in such a way to create a pattern.

10.6 Area

  • The area AA of a triangle is found with the formula A=12bh,A=12bh, where bb is the base and hh is the height.
  • The area of a parallelogram is found using the formula A=bh,A=bh, where bb is the base and hh is the height.
  • The area of a rectangle is found using the formula A=lw,A=lw, where ll is the length and ww is the width.
  • The area of a trapezoid is found using the formula A=12h(b1+b2),A=12h(b1+b2), where hh is the height, b1b1 is the length of one base, and b2b2 is the length of the other base.
  • The area of a rhombus is found using the formula A=d1d22,A=d1d22, where d1d1 is the length of one diagonal and d2d2 is the length of the other diagonal.
  • The area of a regular polygon is found using the formula A=12ap,A=12ap, where aa is the apothem and pp is the perimeter.
  • The area of a circle is found using the formula A=πr2,A=πr2, where rr is the radius.

10.7 Volume and Surface Area

  • A right prism is a three-dimensional object that has a regular polygonal face and congruent base such that that lateral sides form a 9090 angle with the base and top. The surface area SASA of a right prism is found using the formula SA=2B+ph,SA=2B+ph, where BB is the area of the base, pp is the perimeter of the base, and hh is the height. The volume VV of a right prism is found using the formula V=Bh,V=Bh, where BB is the area of the base and hh is the height.
  • A right cylinder is a three-dimensional object with a circle as the top and a congruent circle is the base, and the side forms a 9090 angle to the base and top. The surface area of a right cylinder is found using the formula SA=2πr2+2πrh,SA=2πr2+2πrh, where rr is the radius and hh is the height. The volume is found using the formula V=πr2h,V=πr2h, where rr is the radius and hh is the height.

10.8 Right Triangle Trigonometry

  • The Pythagorean Theorem is applied to right triangles and is used to find the measure of the legs and the hypotenuse according the formula a2+b2=c2,a2+b2=c2, where c is the hypotenuse.
  • To find the measure of the sides of a special angle, such as a 30-60-9030-60-90 triangle, use the ratio x:x3:2x,x:x3:2x, where each of the three sides is associated with the opposite angle and 2xx is associated with the hypotenuse, opposite the 9090 angle.
  • To find the measure of the sides of the second special triangle, the 45-45-9045-45-90 triangle, use the ratio x:x:x2,x:x:x2, where each of the three sides is associated with the opposite angle and x2x2 is associated with the hypotenuse, opposite the 9090 angle.
  • The primary trigonometric functions are sinθ=opphyp,sinθ=opphyp, cosθ=adjhyp,cosθ=adjhyp, and tanθ=oppadj.tanθ=oppadj.
  • Trigonometric functions can be used to find either the length of a side or the measure of an angle in a right triangle, and in applications such as the angle of elevation or the angle of depression formed using right triangles.
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