10.9.1: Key Concepts
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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, A and B, contains all points that are in both sets and is symbolized as A∪B.
- The intersection of two sets A and B includes only the points common to both sets and is symbolized as A∩B.
10.2 Angles
- Angles are classified as acute if they measure less than 90∘, obtuse if they measure greater than 90∘ and less than 180∘, right if they measure exactly 90∘, and straight if they measure exactly 180∘.
- If the sum of angles equals 90∘, they are complimentary angles. If the sum of angles equals 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
10.3 Triangles
- The sum of the interior angles of a triangle equals 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 n sides is found using the formula S=(n-2) 180^{\circ}. The measure of a single interior angle of a regular polygon with n sides is determined using the formula a=\frac{(n-2) 180^{\circ}}{n}.
- The sum of the exterior angles of a regular polygon is 360^{\circ}. The measure of a single exterior angle of a regular polygon with n sides is found using the formula b=\frac{360^{\circ}}{n}.
- The circumference of a circle is C=2 \pi r, where r is the radius, or C=\pi d, and d 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 A of a triangle is found with the formula A=\frac{1}{2} b h, where b is the base and h is the height.
- The area of a parallelogram is found using the formula A=b h, where b is the base and h is the height.
- The area of a rectangle is found using the formula A=l w, where l is the length and w is the width.
- The area of a trapezoid is found using the formula A=\frac{1}{2} h\left(b_1+b_2\right), where h is the height, b_1 is the length of one base, and b_2 is the length of the other base.
- The area of a rhombus is found using the formula A=\frac{d_1 d_2}{2}, where d_1 is the length of one diagonal and d_2 is the length of the other diagonal.The area of a regular polygon is found using the formula A=\frac{1}{2} a p, where a is the apothem and p is the perimeter.
- The area of a circle is found using the formula A=\pi r^2, where r 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 90^{\circ} angle with the base and top. The surface area S A of a right prism is found using the formula S A=2 B+p h, where B is the area of the base, p is the perimeter of the base, and h is the height. The volume V of a right prism is found using the formula V=B h, where B is the area of the base and h 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^{\circ} angle to the base and top. The surface area of a right cylinder is found using the formula S A=2 \pi r^2+2 \pi r h, where r is the radius and h is the height. The volume is found using the formula V=\pi r^2 h, where r is the radius and h 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 a^2+b^2=c^2, where c is the hypotenuse.
- To find the measure of the sides of a special angle, such as a 30^{\circ}-60^{\circ}-90^{\circ} triangle, use the ratio x: x \sqrt{3}: 2 x, where each of the three sides is associated with the opposite angle and 2 x is associated with the hypotenuse, opposite the 90^{\circ} angle.
- To find the measure of the sides of the second special triangle, the 45^{\circ}-45^{\circ}-90^{\circ} triangle, use the ratio x: x: x \sqrt{2}, where each of the three sides is associated with the opposite angle and x \sqrt{2} is associated with the hypotenuse, opposite the 90^{\circ} angle.
- The primary trigonometric functions are \sin \theta=\frac{o p p}{h y p}, \cos \theta=\frac{a d j}{h y p}, and \tan \theta=\frac{o p p}{a d j}.
- 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.