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10.1
Points, Lines, and Planes
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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.
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Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other.
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The union of two sets,
and
, contains all points that are in both sets and is symbolized as
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The intersection of two sets
and
includes only the points common to both sets and is symbolized as
10.2
Angles
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Angles are classified as acute if they measure less than
obtuse if they measure greater than
and less than
right if they measure exactly
and straight if they measure exactly
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If the sum of angles equals
, they are complimentary angles. If the sum of angles equals
, they are supplementary.
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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
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The sum of the interior angles of a triangle equals
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Two triangles are congruent when the corresponding angles have the same measure and the corresponding side lengths are equal.
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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.
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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
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Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number of their sides.
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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.
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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 \(a=\frac{(n-2) 180^{\circ}}{n}\).
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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 \(b=\frac{360^{\circ}}{n}\).
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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
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A tessellation is a particular pattern composed of shapes, usually polygons, that repeat and cover the plane with no gaps or overlaps.
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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
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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.
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The area of a parallelogram is found using the formula \(A=b h\), where \(b\) is the base and \(h\) is the height.
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The area of a rectangle is found using the formula \(A=l w\), where \(l\) is the length and \(w\) is the width.
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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.
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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.
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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
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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.
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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
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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.
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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.
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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.
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The primary trigonometric functions are \(\sin \theta=\frac{o p p}{h y p}, \cos \theta=\frac{a d j}{h y p}\), and \(\tan \theta=\frac{o p p}{a d j}\).
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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.