Chapter 1: Tools of Geometry

1.1: Points, Lines, and Planes

Vocabulary

A point is a single point that is zero dimensional.

A line segment extends from one point (endpoint) to another point.

A line extends from infinitely in two directions and can be referred by a line segment (two points) on the line.

A ray is a line that extends from one point (endpoint) to infinity in one directions

A plane is flat surface (like a paper) that extends to infinity. It is often portrayed as a parallelogram. Can be referred to by three points on the plane.

Two lines or planes that are parallel never meet or intersect

Two lines or planes that are perpendicular meet or intersect at a \(90^{\circ}\) angle.

Visual Planes, Points, Lines, and Rays

Line segment \(\overline{AB}\) lies on line \(l\) which can be referred to as line \(\overleftrightarrow{AB}\). line \(l\) and line \(y\) meet at point \(E\). Ray \(z\) (or \(\overrightarrow{BE}\)) originates at point \(B\). Everything is on plane \(x\) (or plane \(FCD\)).

Intersecting Planes

Two lines meet at a point while two planes meet at a line.

Plane \(FEH\) meets plane \(ABC\) at \(\overleftrightarrow{IJ}\).

Note

Remember that planes, like lines, extend forever. Therefore planes meet at lines, not at line segments as it might appear from the visual. In this example, plane \(FEH\) meets plane \(ABC\) at \(\overleftrightarrow{IJ}\), not at \(\overline{IJ}\).

1.2: Line Segments and Distance

Vocabulary

A congruent line segment is the same length as another line segment. More information below.

Additive property of length

According to the additive property of length, \(\overline{AB} + \overline{BC} = \overline{AC}\)

Distance Formula

This is used to find the distance between two points on a coordinate plane. This formula is not actually a formula as it states to simply use the Pythagorean theorem to solve (\(rise^2 + run^2 = distance^2\)).

Finding a Line Segment Length with Algebra

Example Question: Find the value of x and RS if S is between R and T if \(RS = 5x, ST = 3x, RT = 48\)

We first find x:

\[\begin{split}\overline{RS} + \overline{ST} &= \overline{RT} \\ 5x + 3x &= 48 \\ 8x &= 48 \\ x &= 6 \\\end{split}\]

We can then find \(\overline{RS}\) with \(x \cdot 5 = 30\).

Congruent Line Segments

In the image, \(\overline{AB}\) is congruent with \(\overline{BC}\) as they are both the same length. In fancy terms thats \(\overline{AB} \cong \overline{BC}\).

1.3: Locating Points and Midpoints

Vocabulary

A midpoint is a point in the middle of a line segment or coordinate plane.

Finding the Midpoint

Formula for finding the midpoint on a line segment:

\[\frac{x_{1} + x_{2}}{2}\]

Tip

A bit like finding an average for a dataset

Formula for finding the midpoint on a line on a coordinate plane:

\[(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})\]

For example for the points (0, 0) and (12, 8):

\[\begin{split}midpoint &= (\frac{0 + 12}{2}, \frac{0 + 8}{2}) \\ midpoint &= (6, 4)\end{split}\]

(6, 4) is the coordinate for the midpoint.

Distance Formula

This is used to find the distance between two points on a coordinate plane. This formula is not actually a formula as it states to simply use the Pythagorean theorem to solve (\(rise^2 + run^2 = distance^2\)). This can also be expressed as \((x_{1} - x_{2}\))

1.4: Angle Measure

Vocabulary

An acute angle is an angle that is less than \(90^{\circ}\)

An obtuse angle is an angle that is less than \(180^{\circ}\) and greater than \(90^{\circ}\)

A right angle is an angle that is \(90^{\circ}\)

An angle is made up of two side or the rays/lines that form the angle

A vertex is a common endpoint of rays/sides

Congruent angles are angles that has the same measure

An angle bisector is a ray that spits an angle into two congruent angles.

Note

You can refer to an angle by a point on both sides and the vertex

Finding the vertex

If we want to find the vertex for \(\angle 1\), we first check what sides make up \(\angle 1\). In this figure, \(\overrightarrow{AC}\) and \(\overrightarrow{AB}\) make up the angle. The common meeting point or vertex is at point \(A\).

Solving x from Angle Bisector

If \(m \angle ABE = 60\) and \(m \angle ABD = 4x + 14\), what is x?

Knowing that \(\angle ABD\) bisects \(\angle ABE\):

\[\begin{split}2(4x + 14) &= 60 \\ 4x + 14 &= 30 \\ 4x &= 16 \\ x &= 4\end{split}\]

1.5: Angle Relationships

Vocabulary

Two complementary angles add up \(90^{\circ}\).

Two supplementary angles add up \(180^{\circ}\).

Congruent angles have the same measure.

Vertical angles are opposite angles formed when two lines intersect, always congruent.

Two adjacent angles share a vertex and one side.

Examples

\(\angle ADB\) and \(\angle BDE\) are complementary angles. \(\angle ADC\) and \(\angle ADE\) are supplementary angles. \(\angle EDG\) and \(\angle ADC\) are congruent angles. \(\angle EDF\) and \(\angle BDC\) are vertical angles. \(\angle FDG\) and \(\angle EDG\) are adjacent angles.

Calculating Tips

For a need to find a missing angle from a pair of the angles mentioned above. For the examples below x is the given angle and y is the angle the you need to find.

Complementary Angles: \(y = 90 - x\)

Supplementary Angles: \(y = 180 - x\)

Congruent Angles: \(y = x\)

Vertical Angles: \(y = x\)

1.6: Two-Dimensional Figures

Vocabulary

A polygon is any closed straight sided figure

A triangle has 3 sides and angles.

A quadrilaterals has 4 sides and angles.

A pentagon has 5 sides and angles.

A hexagon has 6 sides and angles.

A heptagon has 7 sides and angles.

A octagon has 8 sides and angles.

A nonagon has 9 sides and angles.

A decagon has 10 sides and angles.

A dodecagon has 12 sides and angles.

All sides and angles of a regular polygon are the same.

Not all sides and angles of a irregular polygon are the same.

A convex does not have any interior angles greater than \(180^{\circ}\)

A concave has interior angles greater than \(180^{\circ}\)

Perimeter and Area

Area is \(length \cdot width\)

Perimeter is \(length \cdot 2 + width \cdot 2\)

Circumference and Area of a Circle

Circumference is \(2 \pi r\)

Area is \(\pi r^2\)

1.7: Transformations in the Plane

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1.8: Three-Dimensional Figures

Vocabulary

A polyhedron is a three-dimensional figure made up of polygon faces.

An edge is a line segment where two faces meet.

A vertex is where edges meet.

A pyramid is a three-dimensional figure containing of a base of any shape which determines the name and amount of faces. From each angle, a line rises up to meet at a central point.

A triangular pyramid is a pyramid with a triangle base.

A pentagonal pyramid is a pyramid with a pentagon base.

A prism has a pair of parallel bases which can be any shape. Lines connect the bases at their angles.

A cone is a pyramid with a circular base.

A cylinder is a prism with a circular base.

An intersecting part of a three-dimensional shape is when lines that make the figure up meet.

Parallel lines are on the same plane and never meet

Skew lines never intersect but are on the same plane.

A face is a flat surface of a three-dimensional figure.

Examples

In the figure, we can see that planes make up the three-dimensional figure. We can use the same rules as we did in 1.1: Points, Lines, and Planes. For example plane (face) \(ABC\) meets plane (face) \(FAB\) at \(\overleftrightarrow{AB}\) (edge). Edges \(\overline{AB}\) and \(\overline{AD}\) meet at vertex \(A\). \(\overleftrightarrow{AB}\) amd \(\overleftrightarrow{DC}\) are parallel. \(\overleftrightarrow{ED}\) amd \(\overleftrightarrow{AC}\) are skew lines.

Find the Volume in A Prism

To find the volume in a prism multiply the area of the base with the height.

Example: A circle volume is \(\pi r^2 h\)

Find the Volume in A Pyrarid

The regular forumla \(\frac{1}{3} \cdot base \cdot height\)

Find Surface Area of a Cone

The forumla is \(\pi r (r + hypotenuse)\)

Find Surface Area of a Cylinder

The formula is \(2 \pi rh\)

Tip

Its just circumfrence times height.

1.9: Two-Dimensional Representations of Three-Dimensional Figures

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1.10: Precision and Accuracy

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