Line Segments and Angles Lesson 2 Line Segments and Angles

Measuring Line Segments The instrument used to measure a line segment is a scaled straightedge like a ruler or meter stick. Units used for the length of a line segment include inches (in), feet (ft), centimeters (cm), and meters (m). We usually place the “zero point” of the ruler at one endpoint and read off the measurement at the other endpoint.

Rulers

We denote the length of by So, if the line segment below measures 5 inches, then we write We never write A B

Congruent Line Segments In geometry, two figures are said to be congruent if one can be placed exactly on top of the other for a perfect match. The symbol for congruence is Two line segments are congruent if and only if they have the same length. So, The two line segments below are congruent.

Segment Addition If three points A, B, and C all lie on the same line, we call the points collinear. If A, B, and C are collinear and B is between A and C, we write A-B-C. If A-B-C, then AB+BC=AC. This is known as segment addition and is illustrated in the figure below. A B C

Example In the figure, suppose RS = 7 and RT = 10. What is ST? We know that RS + ST = RT. So, subtracting RS from both sides gives: S T

Midpoints Consider on the right. The midpoint of this segment is a point M such that CM = MD. M is a good letter to use for a midpoint, but any letter can be used. M D

Example A In the figure, it is given that B is the midpoint of and D is the midpoint of It is also given that AC = 13 and DE = 4.5. Find BD. Note that BC is half of AC. So, BC = 0.5(13) = 6.5. Note that CD equals DE. So, CD = 4.5. Using segment addition, we find that BD = BC + CD = 6.5 + 4.5 = 11. B C D E

Example In the figure T is the midpoint of If PT = 2(x – 5) and TQ = 5x – 28, then find PQ. We set PT and TQ equal and solve for x: T Q

Example continued

Measuring Angles Angles are measured using a protractor, which looks like a half-circle with markings around its edges. Angles are measured in units called degrees (sometimes minutes and seconds are used too). 45 degrees, for example, is symbolized like this: Every angle measures more than 0 degrees and less than or equal to 180 degrees.

A Protractor

The smaller the opening between the two sides of an angle, the smaller the angle measurement. The largest angle measurement (180 degrees) occurs when the two sides of the angle are pointing in opposite directions. To denote the measure of an angle we write an “m” in front of the symbol for the angle.

Here are some common angles and their measurements. 1 2 4 3

Congruent Angles Remember: two geometric figures are congruent if one can be placed exactly on top of the other for a perfect match. So, two angles are congruent if and only if they have the same measure. So, The angles below are congruent.

Types of Angles An acute angle is an angle that measures less than 90 degrees. A right angle is an angle that measures exactly 90 degrees. An obtuse angle is an angle that measures more than 90 degrees. acute right obtuse

A straight angle is an angle that measures 180 degrees A straight angle is an angle that measures 180 degrees. (It is the same as a line.) When drawing a right angle we often mark its opening as in the picture below. right angle straight angle

Adjacent Angles Two angles are called adjacent angles if they share a vertex and a common side (but neither is inside the opening of the other). Angles 1 and 2 are adjacent: 1 2

Angle Addition If are adjacent as in the figure below, then C A D B

Example In the figure, is three times and Find Let Then By angle addition, A M T H

Angle Bisectors Consider below. The angle bisector of this angle is the ray such that In other words, it is the ray that divides the angle into two congruent angles. A B C D

Complementary Angles Two angles are complementary if their measures add up to If two angles are complementary, then each angle is called the complement of the other. If two adjacent angles together form a right angle as below, then they are complementary. A B C 1 2

Example Find the complement of Call the complement x. Then

Example Two angles are complementary. The angle measures are in the ratio 7:8. Find the measure of each angle. The angle measures can be represented by 7x and 8x. Then

Supplementary Angles Two angles are supplementary if their measures add up to If two angles are supplementary each angle is the supplement of the other. If two adjacent angles together form a straight angle as below, then they are supplementary. 1 2

Example Find the supplement of Call the supplement x. Then

Example One angle is more than twice another angle. If the two angles are supplementary, find the measure of the smaller angle. Let x represent the measure of the smaller angle. Then represents the measure of the larger angle. Then

Perpendicular Lines Two lines are perpendicular if they intersect to form a right angle. See the diagram. Suppose angle 2 is the right angle. Then since angles 1 and 2 are supplementary, angle 1 is a right angle too. Similarly, angles 3 and 4 are right angles. So, perpendicular lines intersect to form four right angles. 2 1 3 4

The symbol for perpendicularity is So, if lines m and n are perpendicular, then we write The perpendicular bisector of a line segment is the line that is perpendicular to the segment and that passes through its midpoint. m m perpendicular bisector n A B

Vertical Angles Vertical angles are two angles that are formed from two intersecting lines. They share a vertex but they do not share a side. Angles 1 and 2 below are vertical. Angles 3 and 4 below are vertical. 3 1 2 4

The key fact about vertical angles is that they are congruent. For example, let’s explain why angles 1 and 3 below are congruent. Since angles 1 and 2 form a straight angle, they are supplementary. So, Likewise, angles 2 and 3 are supplementary. So, So, angles 1 and 3 have the same measure and they’re congruent. 1 2 3