Contemporary Mathematics

# Key Concepts

## 10.1Points, 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 $A∪B.A∪B.$
• The intersection of two sets $AA$ and $BB$ includes only the points common to both sets and is symbolized as $A∩B.A∩B.$

## 10.2Angles

• Angles are classified as acute if they measure less than $90∘,90∘,$ obtuse if they measure greater than $90∘90∘$ 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 $90∘90∘$, they are complimentary angles. If the sum of angles equals $180∘180∘$, 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

## 10.3Triangles

• 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.

## 10.4Polygons, 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 $S=(n−2)180∘.S=(n−2)180∘.$ The measure of a single interior angle of a regular polygon with $nn$ sides is determined using the formula $a=(n−2)180∘n.a=(n−2)180∘n.$
• 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 $b=360∘n.b=360∘n.$
• 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.

## 10.5Tessellations

• 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.

## 10.6Area

• 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.

## 10.7Volume 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 $90∘90∘$ 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 $90∘90∘$ 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.

## 10.8Right 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 $30∘-60∘-90∘30∘-60∘-90∘$ triangle, use the ratio $x:x3:2x,x:x3:2x,$ where each of the three sides is associated with the opposite angle and 2$xx$ is associated with the hypotenuse, opposite the $90∘90∘$ angle.
• To find the measure of the sides of the second special triangle, the $45∘-45∘-90∘45∘-45∘-90∘$ 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 $90∘90∘$ 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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