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Lesson 2 Line Segments and Angles 1 2 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. 3 Rulers 4 • We denote the length of AB by AB • So, if the line segment below measures 5 inches, then we write • We never write AB 5 in AB 5 in A B 5 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, AB CD if and only if AB CD. • The two line segments below are congruent. 6 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 7 Example R • 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: ST RT RS 10 7 3 S T 8 Midpoints • Consider CD 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. C M D 9 Example • In the figure, it is given that B is the midpoint of AC and D is the midpoint of CE. • 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. A B C D E 10 Example P • In the figure T is the midpoint of PQ. • If PT = 2(x – 5) and TQ = 5x – 28, then find PQ. • We set PT and TQ equal and solve for x: T Q 11 Example continued 2( x 5) 5 x 28 distribute: 2 x 10 5 x 28 subtract 2 x : 10 3 x 28 add 28 : divide by 3: 18 3x 6x Now PT 2(6 5) 2(1) 1 and TQ 5(6) 28 30 28 2. So, PQ PT TQ 2 2 4. 12 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: 45 • Every angle measures more than 0 degrees and less than or equal to 180 degrees. 13 A Protractor 14 • 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. 15 • Here are some common angles and their measurements. m1 45 1 m2 90 2 m3 135 3 4 m4 180 16 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, A B if and only if mA mB. • The angles below are congruent. 17 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 18 • 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 19 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 20 Angle Addition • If ABC and CBD are adjacent as in the figure below, then mABC mCBD mABD C A D B 21 Example • In the figure, mMAH is three times mHAT and mMAT 132. • Find mMAH . A • Let mHAT x. Then M mMAH 3x. • By angle addition, x 3x 132 4 x 132 x 33 So, mMAH 3 33 99. H T 22 Angle Bisectors • Consider ABC below. • The angle bisector of this angle is the ray BD such that mABD mDBC. • In other words, it is the ray that divides the angle into two congruent angles. A D B C 23 Complementary Angles • Two angles are complementary if their measures add up to 90. • 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 1 and 2 are complementary B if ABC is a right angle 24 1 2 C Example • Find the complement of 37. • Call the complement x. • Then 37 x 90 x 90 37 x 53 25 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 7 x 8 x 90 15 x 90 x 6 Then the angle measures are 7 x 42 and 8 x 48. 26 Supplementary Angles • Two angles are supplementary if their measures add up to 180. • 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 1 and 2 are supplementary 27 Example • Find the supplement of 62. • Call the supplement x. • Then 62 x 180 x 180 62 x 118 28 Example • One angle is 30 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 2 x 30 represents the measure of the larger angle. Then x (2 x 30) 180 3x 30 180 combine like terms: subtract 30 : 3x 150 x 50 divide by 3: So the smaller angle measures 50. 29 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. 1 3 4 2 30 • The symbol for perpendicularity is . • So, if lines m and n are perpendicular, then we write m n. • 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 mn bisector n A B 31 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. 1 3 2 4 32 • 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, m1 180 m2. • Likewise, angles 2 and 3 are supplementary. So, m3 180 m2. So, angles 1 and 3 have the same measure and they’re congruent. 1 2 3 33