Mathematics Knowledge Study Guide for the ASVAB

Page 3

Geometry: Part 1

Geometry deals with the properties of points, lines, angles, surfaces, and solids.

Basic Terms

In geometry, we start with what are called undefined terms, meaning that we try our best to describe what they mean, but they don’t have rigorous mathematical definitions. The undefined terms are point, line, and plane. From those, we can build a vocabulary of precisely defined terms.

Points

A point in geometry can be thought of as an infinitely small dot that shows an exact location in space. Because we can’t see an infinitely small dot, we represent it with a dot we can see, and often name it with a capital letter, such as point A. All geometric figures are made of points.

Lines, Etc.

In geometry we imagine that there are points and figures that exist as they are defined below. Our brains are not great at imagining these things, so we need help. To this end, we draw lines, points, angles, etc., on something physical, like paper, to help us visualize the ideal geometry that is in our head. Clearly, we can’t actually draw infinitely long lines that are infinitely thin, but we imagine them being in our figures. In short, true geometry is idealized in our brain and is approximated by the physical figures we draw.

Line—A line is a set of points that lie in a straight path and extend infinitely in opposite directions. It has no thickness, though obviously the line we draw to represent a line does have thickness. It is often symbolized by naming two points on it with a double headed arrow over them: \(\overleftrightarrow{AB}\).

5-line.png

Line segment—A line segment consists of two points and all points between them. It is named by naming its two endpoints with a line over them: \(\overline{AB}\).

6-line-segment.png

Ray—A ray is a part of a line that extends from an endpoint infinitely far in one direction. Rays are named with their endpoint first. The ray below would be named \(\overrightarrow{AB}\).

6-b-ray.png

Midpoint—A midpoint is a point that divides a line segment into two congruent segments. P is a midpoint of \(\overline{PA} \cong \overline{PB}\).

7-line-midpoint.png

Bisector—A bisector of a segment is a line, ray, or segment that intersects the segment at its midpoint: \(\overleftrightarrow{CD}\) bisects \(\overleftrightarrow{AB}\).

8-line-bisector.png

Collinear points—Collinear points are points that lie on the same line. Note that the line doesn’t have to be actually shown in the figure. A, B, and C are collinear points.

9-colinear-points.png

Plane—”Plane” is term without a formal definition, but we think of it as a flat surface that extends without ending. It’s usually named by a single letter, as in plane S, and drawn as shown here.

10-geometric-plane.png

CoplanarPoints that lie in the same plane are said to be coplanar. Below we see plane S with points A and B in the plane. A and B are coplanar.

11-coplanar-points.png

Perpendicular—Perpendicular lines and segments are those that intersect to form a right angle (very often they will form four right angles, but for the definition it is sufficient to say that they form just one): \(\overleftrightarrow{CD} \perp \overleftrightarrow{AB}\).

12-perpendicular-lines.png

ParallelTwo lines that are coplanar and do not intersect (no matter how far they are extended) are parallel lines: \(\overleftrightarrow{CD} \parallel \overleftrightarrow{AB}\).

13-parallel-lines.png

Slope—For any two points on a line in a Cartesian plane, the slope of that line is defined to be the ratio of the vertical distance between the points to the horizontal distance between the points. Most of us just remember rise over run.

The graph below passes through the points \((0, -2)\) and \((3, -6)\). The vertical distance between them is \(-2-(-6) = 4\) the horizontal distance is \(0-3=-3\). The slope is \(\frac{4}{-3}\) or \(-\frac{4}{3}\). A negative slope means the line goes downhill from left to right.

14-graph-of-slope.png

Retrieved from: https://openstax.org/details/books/elementary-algebra-2e

Angles

An angle is defined to be a pair of rays that originate from a common endpoint. The common endpoint (the pointy part) is called the vertex. They are classified according to the size or position of the angle.

Types of Angles

Angles can have a measure of any number of degrees from \(0\) to \(360\), and can be named with three letters, where the middle letter represents the vertex of the angle. They can also be named with a single number, or with a single letter if there is only one vertex at that letter. You will see all three ways below.

Right angle—A right angle has a measure of \(90^\circ\). The angle below, \(\angle CAR\), is a right angle. In math work, you can assume that any angle labeled with a little box, like the one below, is a right angle and measures \(90^\circ\).

15-right-angle.png

Acute angle—An acute angle has a measure that is less than \(90^\circ\). \(\angle FAN\) is an acute angle.

16-acute-angle.png

Obtuse angle—An obtuse angle has a measure that is greater than \(90^\circ\). \(\angle JOE\) is an obtuse angle.

17-obtuse-angle.png

Straight angle—A straight angle is one that has a measure of \(180^\circ\), and looks no different than a straight line. \(\angle SAM\) is a straight angle.

18-straight-angle.png

Full angle—A full angle is one where the starting ray is rotated through \(360^\circ\). It will look exactly like the ray that it started with.

Angle bisector— An angle bisector is a line that divides an angle into two congruent angles.

19-angle-bisector.png

Pairs of Angles

Certain pairs of angles have special names, based on their measures or positions.

Complementary angles—Two angles whose sum is \(90^\circ\) are complementary angles. The figure below shows \(\angle MAN\) and \(\angle NAB\) being adjacent to each other, but they could be anywhere as long as their sum is \(90^\circ\).

20-complementary-angles.png

Supplementary angles—Two angles whose sum is \(180^\circ\) are complementary angles. The figure below shows \(\angle BOT\) being adjacent to \(\angle TOM\), but they could be anywhere as long as their sum is \(180^\circ\).

21-supplementary-angles.png

Vertical angles—Vertical angles are formed by a pair of intersecting lines. Their sides form two pairs of opposite rays. In the figure below, \(\angle 1\) and \(\angle 3\) are a pair of vertical angles, as are \(\angle 2\) and \(\angle 4\).

22-vertical-angles.png

Adjacent angles—Adjacent angles are angles that have a common vertex and a common side, but do not have any interior points in common. In the vertical angles figure above, \(\angle 1\)and \(\angle 2\) are a pair of adjacent angles. (They are also a pair of supplementary angles.)

Linear pair—If two angles are supplementary and adjacent, they are a linear pair. We can use the same figure that we used for supplementary angles.

23-linear-angle-pair.png

Corresponding anglesAngles that are in the same relative position in two different figures are corresponding angles. You could say they match up. In the first figure below, the top two angles are \(\angle C\) and \(\angle D\), so they are a pair of corresponding angles. The lower left angles, \(\angle A\) and \(\angle O\) are another pair of corresponding angles, as are \(\angle T\) and \(\angle G\).

24-corresponding-angles.png

In the figure below, two lines are intersected by a transversal. Again, corresponding angles occupy the same relative positions.

That would make \(\angle 1\) correspond to \(\angle 5\). Also,
\(\angle 2\) corresponds to \(\angle 6\).
\(\angle 3\) corresponds to \(\angle 7\).
\(\angle 4\) corresponds to \(\angle 8\).

It’s worth knowing that if line \(l\) and line \(m\) are parallel, all corresponding angles are congruent.

25-corresponding-angles-with-transversal.png

Congruent angles—Congruent angles are angles that have the same measure.

26-congruent-angles.png

All Study Guides for the ASVAB are now available as downloadable PDFs