SAT II Math: Looking at Lines and Angles
Plane geometry is the study of lines and shapes in two dimensions. Imagine a tool that could prove that the Earth is round and that the planets move around the sun in predictable orbits. Those are some of the wonders of geometry. It has been extremely important in the history of mathematical development. Using geometry, we can make models of the physical world and apply mathematical concepts to them. We make hypotheses and predictions about the real world and use geometry to proves that's really how the world goes 'round. Geometry starts with the basics, in this case plane geometry, and builds on that foundation to construct ever-increasing complex models to more accurately portray the real world.
The SAT II Math test spends about 20 percent of the Level IC test on plane geometry and measurement. Although the Level IIC portion does not test you on plane geometry per se, you're still expected to know the basic principles of plane geometry in order to work the more advanced level coordinate and solid (3-dimensional) geometry and to succeed on that test.
Getting the skinny: Some basic definitions
The first thing you need to do in understanding geometry is to get to know the various terms for geometric shapes and forms. While you aren't tested on the definitions, it's important to understand their meaning to solve problems on the SAT II Math test. Here are the more common terms that will pop up at one time or anther on the test:
- Plane: A perfectly flat surface that has no thickness and extends forever in two directions.
- Line: A straight path of points that extends forever in two directions. A line does not have any width or thickness. Because a point is very, very tiny, a line is very, very thin. Arrows are used to show that the line goes on forever. The word line is often used to indicate a line segment or a ray.
- Line segment: The set of points on a line between any two points on the line, basically just a piece of a line from one point to another that contains all the points in between.
- Ray: A ray is like half of a line; it starts at an endpoint and extends forever in one direction. You can think of a ray as just like a ray extending from the sun (the endpoint) and shining as far as it can go. While the sun's rays may eventually run out of energy on their path, a ray in geometry keeps going and going.
- Midpoint: The point halfway between two points on a line segment. If a point along a line segment is the same distance from each of the two ends of the line segment, that point is the midpoint on the line segment.
- Bisect: To cut something exactly in half, such as a line segment cutting another line segment or an angle or a polygon into two equal parts. A bisector is a line that divides the line segment, angle, or polygon into two equal parts.
- Intersect: Just like it sounds, it simply means to cross; that is, when one line or line segment crosses another line or line segment.
- Collinear: A set of points that lie on the same line.
- Vertical: Lines that run straight up and down.
- Horizontal: Lines that run straight across from right to left (or from left to right if you're holding your paper upside down).
- Parallel: Lines that run in the same direction always remaining the same distance apart. Parallel lines will never intersect with one another.
- Perpendicular: When two lines intersect to form a square corner. The intersection of two perpendicular lines forms a right angle or a 90° angle.
- Angle: The intersection of two rays sharing a common endpoint. The common endpoint is called the vertex. The size of an angle depends on how much one side rotates away from the other side. An angle is usually measured in degrees or radians.
- Acute angle: Any angle measuring less than 90°. Like an acute or sharp pain, the acute angle has a sharp point.
- Right or perpendicular angle: An angle measuring exactly 90°. It makes up a square corner.
- Obtuse angle: An angle that measures more than 90° but less than 180°. While an acute angle can be quite sharp, an obtuse angle couldn't poke a hole in butter. An obtuse angle is actually quite dull or blunt.
- Straight angle: An angle that measures exactly 180° is straight. A straight angle appears to be a straight line or line segment.
- Complementary angles: Angles that, when added together, total 90°. Together, they form a right angle, so just remember that it's the "right" thing to do to give an angle a complement.
- Supplementary angles: Angles whose measurements total 180° are supplementary. They form a straight line. Just remember that vitamin supplements can keep you on the straight and narrow.
- Congruent: Objects that are equal in size and shape are congruent. Two line segments having the same length are congruent. Two angles having the same measure are congruent. Two congruent triangles have their corresponding sides all the same length, and their corresponding angles are all the same measurement.tabmarktabmark
Fishing for answers: Some rules for lines and angles
The rules for lines and angles are direct applications that arise from the basic definitions you've just studied.
When two lines intersect, the opposite angles are always congruent or equal, and the adjacent angles are always supplementary. Opposite angles are also known as vertical angles. Adjacent angles have a common side, so they are right next to each other.
When parallel lines are crossed by a third line that is not perpendicular to them, the resulting small and large angles share certain properties. Each of the small angles is equal to each other. The large angles are also equal to each other. The measurement of any small angle added to that of any large angle will equal 180°.