# A (straight) line has length but no width or thickness. A line is understood to extend indefinitely to both sides. beginning or end.

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1 Points, Lines, and Planes Point is a position in space. point has no length or width or thickness. point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. (straight) line has length but no width or thickness. line is understood to extend indefinitely to both sides. beginning or end. It does not have a line consists of infinitely many points. The four points,,, are all on the same line. Postulate: Two points determine a line. We name a line by using any two points on the line, so the above line can be named as any of the following: ny three or more points that are on the same line are called colinear points. In the above, points,,, are all colinear. Ray is part of a line that has a beginning point, and extends indefinitely to one direction. ray is named by using its beginning point with another point it contains. In the above, ray is the same ray as or. ut ray is not the same ray as. (line) segment is a finite part of a line between two points, called its end points. segment has a finite length. In the above, segment is not the same as segment Segment ddition Postulate: In a line segment, if points,, are colinear and point is between point and point, then

2 + = You may look at the plus sign, +, as adding the length of the segments as numbers. Or you may look at it as putting the two segments together. If two segments have the same length, we say that they are congruent to each other, and use the symbol, =, to denote. Pictorially, congruent segments (and other figures) can be put on top of each other (super-impose) and fit perfectly. If two different lines intersect, then their intersection is a point, we call that point the point of intersection of the two lines. If is a line segment and M is a point on that makes M = M, then M is the midpoint of. If there is another segment (or line) that contains point M, that line is a segment bisector of. M In the above, segment M = M (notice that we use a tick-mark on each of the two segments to indicate that the two segments are congruent to each other), so M is the mid-point of. Since is a line segment that contains the mid-point, M, of, so is a segment bisector, or simply a bisector of. The two line segments and intersect at the point M, so M is the point of intersection of the two segments. surface has length and width, but no thickness. plane surface, or simply a plane, is a surface where any straight line that contains two points on the plane lies entirely on the plane. plane extends indefinitely to all directions. (Think of a plane as a piece of flat paper that is infinitely large, have no thickness) ngles n angle is formed by two rays with a common end point. ach ray form a side of the angle and the commond end point is the vertex of the angle. The symbol for an angle is n angle can be named by naming just the point of the vertex, if there is only one angle with that vertex.

3 n angle can also be named using three points, where the middle point is the vertex and the first and last point is any point on each of the sides. We may also put a letter or number in the angle and name the angle with using the alphabet or number. 1 The above angle can be named as just, or we can name it as or. We can also name it as 1 The size of an angle is determined by how much one side of the angle must be rotated about the vertex to meet the other side of the angle. (Think of the size of an angle as how much the angle opens.) We may also assign numbers to the size of an angle. commonly used measurement of size of angles is degrees. We use a little circle to represent an angle measurement in degrees. For example, is 33 degrees, and we write m = 33 to denote this fact. 33 If two angles have the same size (same amount of opening, same degree measure), we say that the two angles are congruent angles. F In the picture above, = F. Notice that we use a tick mark on the angles to indicate that they are congruent. Geometrically, this means that the two angles can be placed on top of each other and fit perfectly. efinition: djacent ngles are two angles that share a common side with and have the same vertex.

4 In the above, and are adjacent angles, they share a common side and have the same vertex,. efinition: Vertical ngles are two non-adjacent angles formed by two intersecting lines. Vertical angles also share the same vertex. In the picture above, segment intersects at point, so and are vertical angles. and are also vertical angles. ngle ddition Postulate: The sum of the measure of two adjacent angles is equal to the measure of the angle formed by the non-common sides of the two adjacent angles. In the above, m + m = m. Right ngles: If two lines intersect and make the two adjacent angles equal to each other, then each of the equal angle is a right angle. The two lines that intersect this way is said to be perpendicular to each other.

5 In the picture above, intersects at point in such a way that makes =. Therefore both and are both right angles. Notice that we use an angled marking to indicate a right angle. Since they intersect to form right angles, segments and are perpendicular to each other. We write n acute angle is one which is less than a right angle. an obtuse angle is one that is greater than a right angle. In the above, is acute, is right, and is obtuse. In degree measure, a right angle has a measurement of 90. Notice that two adjacent right angles form a straight line. Therefore, a straight line (straight angle) has a measurement of 180 efinition: Two angles are complementary if, when placed adjacent to each other with one side in common, their non-common sides form a right angle. Numerically, we say that two angles are complementary if the sum of their degree measurement equals 90 In the above picture, and are complementary because they form a right angle. xample: 35 angle is complementary to a 55 angle. efinition: Two angles are supplementary if, when placed adjacent to each other with one side in common, their non-common sides form a straight line. Numerically, we say that two angles are supplementary if the sum of their degree measure equals 180

6 In the above, and are supplementary. Their non-common sides form the straight line. xample: 76 angle is supplementary to a 104 angle. ngle isector: If a line cuts an angle into two equal smaller angles, the line is said to bisect the angle and is an angle bisector of the angle. In the picture above, =, so is the angle bisector of Just like with segments, we use tick-marks on the angles to indicate that they are congruent to each other. When we look at two geometrical figures as being the same, there are two interpretations. Geometrically, we say the geometrical objects are congruent, =. This means that the two geometrical objects have the exact same shape and form and can be placed on top of each other (super-imposted) perfectly. lgebraically, we can assign a numerical value to some of the aspects of a geometrical figure, such as the length of a line segment or a measurement to an angle. We can then interpret two geometrical objects being equal to mean that they have the same numerical measurement. For example, we say that m = m F to mean that the two angles have the same measurement (as numbers). Under all circumstances, if two geometrical figures are congruent geometrically then their characteristics will also be equal numerically, but you should understand that there is a difference between being congruent (geometrically) and being equal (numerically).

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Heron s Formula Lesson Summary: Students will investigate the Heron s formula for finding the area of a triangle. The lab has students find the area using three different methods: Heron s, the basic formula,

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

### 13.1 Lines, Rays, and Angles

? Name Geometry and Measurement 4.6. 13.1 Lines, Rays, and ngles Essential Question How can you identify and draw points, lines, line segments, rays, and angles? MHEMIL PROEE 4.1., 4.1.E Unlock the Problem

### Notes on Congruence 1

ongruence-1 Notes on ongruence 1 xiom 1 (-1). If and are distinct points and if is any point, then for each ray r emanating from there is a unique point on r such that =. xiom 2 (-2). If = and = F, then