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Chapter 7 Although not very common in the natural world, the idea of parallel lines has great importance for the world that humankind has constructed. For example, when a building is being put up, the beams that support the floors must be parallel. 183 Chapter 7 Focus on Skills ALGEBRA Complete. 1. If a, b, and care all greater than zero and a = b + c, then • is the greatest of the three numbers. 2. If x + a = y and x + b = y, then • 3. If c = d and d = e, then • = • . = • . Solve each equation. 4. 5x - 1 = 2x + 5 6. 6t - 24 = 4t + 14 5. (2x + 5) + (7x - 14) 7. 135 = 71 + z = 180 9. (p - 2)180 = 3,780 8. b + 43 + 82 = 180 GEOMETRY Sketch a figure to match each description. Mark all parallel sides, all congruent sides, and any right angles. 11. a rhombus that is not a rectangle 10. a rectangle that is not a square 13. a parallelogram that is not a rectangle 12. a trapezoid with no congruent sides Draw a figure similar to, but larger than, the one shown and construct the indicated figure. 15. a triangle congruent to ~ABC L1 14. L2 with vertex Q such that L2 = L A Q 16. a perpendicular to k through P ~k B~C 17. a perpendicular to j through M ~j PROBLEM SOLVING 18. In ~RST, mLT = 3 • mLR, mLS = 2 • mLR, and mLR + mLS + mLT= 180°. Find the measure of each angle. 184 Chapter 7 Focus on Skills 7.1 Transversals and Angles Objective: To identify the angles formed when two lines are cut by a transversal. In the- next several lessons, you will discover some properties of the angles that are formed when parallel lines are intersected by other lines. EXPLORING 1. Draw two lines a and b. Draw a third line t that intersects a and b in different points. Label the angles as shown. 2. Classify the angles as acute, right, or obtuse. 3. Must any of the numbered angles be congruent? Why? 4. Must any of the numbered angles be supplementary? Why? A transversal is a line that intersects two or more lines, each at a different point. Line t 1 is a transversal, but t2 is not. In the figure at the right, four of the numbered angles are interior angles inside lines a and b. Four are exterior angles outside lines a and b. Line t is a transversal for lines a and b. L3, L4, L5, and L6 are interior angles. L 1, L2, L7, and L8 are exterior angles. Certain pairs of angles determined by a transversal have special names. Alternate interior angles Corresponding angles • are interior angles • consist of one interior and one exterior angle • are on opposite sides of the transversal • are on the same side of the transversal • have different vertices • have different vertices 7.1 Transversals and Angles 185 Exam pie: Name all interior angles, exterior angles, vertical angles, corresponding angles, and alternate interior angles. Solution: Interior angles: L2, L3, L6, L7 Exterior angles: Ll , L4, L5, L8 Vertical angles: Ll and L7, L2 and L8, L3 and L5 , L4 and L6 Corresponding angles: Ll and L3, L2 and L4, L5 and L7, L6 and L8 Alternate interior angles: L2 and L6, L3 and L7 Look at the figure shown. Which pairs of angles on the same side of the transversal do you think should be called same-side interior angles? same-side exterior angles? Explain your answers. Class Exercises Name each of the following. 1. a transversal 2. four interior angles 3. four exterior angles 4. four pairs of vertical angles 5. four pairs of corresponding angles 6. two pairs of alternate interior angles 7. two pairs of same-side interior angles 8. two pairs of same-side exterior angles Exercises Name each of the following. 1. four interior angles 2. four exterior angles 3. two pairs of alternate interior angles 4. four pairs of corresponding angles S·.J four pairs of vertical angles 186 Chapter 7 Parallel Lines m N arne each of the following. 6. 7. 8. 9. 10. s four interior angles along transversal m four exterior angles along transversal r 9 110 12 111 two pairs of alternate interior angles along transversal s 13 114 16 115 two pairs of alternate interior angles along transversal/ four pairs of corresponding angles along transversal/ 11. two pairs of same-side interior angles along transversal r 12. two pairs of same-side exterior angles along transversal m Classify each pair of angles as (a) alternate interior angles, (b) corresponding angles, (<e)_vertical angles, (d) same-side interior angles, ~adjacent supplementary angles, or ~one of the above. 13. L2 and L8 14. L2 and L6 15. L4 and L5 16. L5 and L7 17. L6 and L7 18. Ll and L3 3 6 19. Ll and L6 20. L2 and L7 4 15 1 I8 2/7 • Name the segments and transversal that determine each pair of alternate interior angles. 21. a. Ll and L2 VY. D C E.,. 22. a. Ll and L4 b. L3 and L4 c. L5andL6 ~3 7 A B d. Ll and L7 ~----...,.....-----, T b.L2 and L3 c. L5 and L8 d. L6 and L7 R v ~ ' "- 1 APPLICATION 23. Linguistics Use a dictionary to find the meanings of correspond and transverse that are related to this lesson. Seeing in Geometry Narne all pairs of corresponding angles determined by AB and CF and transversal RW. 7.1 Transversals and Angles 187 7.2 Parallel Lines and Interior Angles Objective: To use relationships between interior angles and parallel lines. When two lines are parallel and are cut by a transversal, certain relationships exist among the eight angles that are formed. In this lesson and Lesson 7 .3, you will learn about those relationships. EXPLORING 1. Use the two edges of a ruler to draw two parallel lines. Then draw any transversal t. Label the interior angles as shown. 2. Use a protractor to measure L1, L2, L3, and L4. Which pairs of angles are congruent? Which are supplementary? 3. Repeat Steps 1 and 2 using a different pair of parallel lines and a different position for the transversal. Do you get the same results? The EXPLORING activity demonstrates the following. POSTULATE 9 (Alternate Interior Angles Postulate): If two parallelliges are cut by a transversal, then each pair of alternate interior angles are congruent. THEOREM 7.1 (Same-Side Interior Angles Theorem): If two parallel lines are cut by a transversal, then each pair of same-side interior angles are supplementary. 127c Example 1: Find the measure of each numbered angle. a. b. 30° A Solution: a. mL1 =75° mL2 = 105° mL3 = 105° 188 Chapter 7 Parallel Lines 20o D b. Since BC II AD, mL4 = 20°. Since AB II DC, mL5 = 30°. EXPLORING 1. Draw two intersecting lines t and c. Label L1 and point P on line t as shown. 2. Using a protractor and a straightedge, draw line d through point P so that L 1 and L2 are congruent alternate interior angles. 3. What is mL2 + mL3? What is mL 1 + mL3? Why? 4. Describe the relationship between L1 and L3 . 5. What seems to be the relationship between lines c and d? + * 3 c POSTULATE 10: If two lines are cut by a transversal so that one pair of alternate interior angles are congruent, then the lines are parallel. THEOREM 7.2: If two lines are cut by a transversal so that one pair of same-side interior angles are supplementary, then the lines are parallel. Example 2: Tell whether the lines shown are parallel. a.~ 65 ° 65 o c. b. • • m a Solution: a. /II m b. a II b 2. Tell how the measures of each pair of angles are related. b. LB and LC e. LA and LC 911 89° c. m is not parallel to n 1. If Pis a point not on line c, how many lines parallel to c can be drawn through Pin a plane? a. LA and LB d. LA and LD m •n • c. LC and LD f. LB and LD A c lc B 3. RSTW is a parallelogram. LR is a right angle. a. What is mLS? Why? b. What are mLTand mLW? 7.2 Parallel Lines and Interior Angles 189 Class Exercises Name each of the following. 1. an angle congruent to L 1 2. an angle congruent to L4 3. two angles supplementary to L2 4. two angles supplementary to L3 Find the measure of each angle. 5. L1 6. L2 7. L3 9. L5 8. L4 10. L6 11. L7 12. L8 Exercises Name each of the following. 1. 2. 3. 4. an angle congruent to L 1 an angle congruent to L 4 two angles supplementary to L2 two angles supplementary to L3 d Find the measure of each angle. 5. L1 6. L2 7. L ·3 8. L4 9. L5 10. L6 11. L7 12. L8 • ill( 1 · - ,._. ) - .... -, -, • a .. b 13. a. Name two pairs of alternate interior angles. b. N arne two pairs of same-side interior angles. = =, , c. If L1 U , then lll m. d. If mL 1 + m ~ = 180°, then l II m. e. If L2 then l II m. m Find the measure of each angle. 15. EHGF is a parallelogram. 14. PMRQ is a parallelogram. ::~~ c. LR 190 P l 50 Chapter 7 /R Q Parallel Lines a.LHEF ~G b. LHEG c. LHGE E 15o F 16. DABC is a parallelogram. a. LCDA b. L1 c. L2 17. a. L1 A \/ \ D d. L3 b. L2 c. L3 d. L4 c 18. RECT is a rectangle. a. L1 T b. L2 c. L3 R 19. a. L1 C2J 2 b. L2 C lvv - ~~ ---::::> E 1 c. L3 E D Name the segments or lines that must be parallel. 20. Q D 60° 21. A H 22. •n ~m X u -- ,------:----:;JT pv -''-' ' 89° 91 ° 89° 91 ° y ~ Name the lines, if any, that must be parallel if each statement is true. 23. L1 ::: L2 24. mL1 + mL3 = 180° 25. mL3 + mL5 = 180° 26. L2 ::: L4 27. mL2 + mL3 = 180° 28. L4::: L5 = 29. If L4 L6 and L8 be parallel. =L10, name all lines that must 30. If a II band b II c, must line a be parallel to line c? Explain. APPLICATIONS Find the measure of each indicated angle. 31. Algebra LABC and LBCD 32. Algebra E LEFG and LFGH (3x + 20) 0 7.2 Parallel Lines and Interior Angles 191 Find the measure of each indicated angle. 33. Algebra LRST, LPTS, and LSTR 34. Algebra LJ, LK, and LL ,.------.-::--::------:""' T s K:: ,_n . . . _,,. / Are the indicated segments parallel? Explain. 35. Logic AB and DC 36. Logic B R RW and ST (6y- 5) 0 K J ] ~ _, T Historical Note The following statement is equivalent to our Postulates 9 and 10. Given a line land a point P not on l, there is exactly one line through P that is parallel to l. p ~------ · -------------~ Although this statement, known as The Parallel Postulate, may seem obvious, mathematicians tried for more than 2,000 years to prove it. In the early 1800s, two mathematicians, Janos Bolyai of Hungary and Nicolai Lobachevsky of Russia, independently made the same discovery: The assumption that there is more than one line through P that is parallel to l does not lead to a contradiction. In fact, the mathematicians developed a new geometry based on the assumption that an infinite number of lines through P are parallel to l. During the same period, the German mathematician Georg Riemann ' developed another geometry based on the assumption that there are no lines passing through P that are parallel to l. 192 Chapter 7 Parallel Lines 7.3 Parallel Lines and Corresponding Angles Objective: To use facts about parallel lines cut by a transversal and pairs of corresponding angles. In this lesson, you will learn more about how the fact that two lines are parallel relates to the congruence between certain angles formed by a transversal. EXPLORING The measure of L1 =50°. 1. Find the measures of L2, L3, and L4. 2. What is the measure of L6? Why? 3. Find the measures of L5, L7, and L8. 4. Name all pairs of corresponding angles. Compare the measures of corresponding angles. The EXPLORING activity demonstrates the following theorem. THEOREM 7.3 (Corresponding Angles Theorem): Iftwo parallel lines are cut by a transversal, then each pair of corresponding angles are congruent. EXPLORING 1. Enter and run the following LOGO program. To transversal fd 50 bk 100 fd 50 rt 40 fd 50 bk 100 fd 50 fd 20 It 25 fd 50 bk 100 fd 50 end )$ill JJ:S:Im:l~ 2. Are the first and third lines created by the turtle parallel? If not, what must you change to make the turtle draw the third line parallel to the first line? Make the change( s). Run the new program. 7.3 Parallel Lines and Corresponding Angles 193 The EXPLORING activity demonstrates the following theorem. THEOREM 7.4: If two lines are cut by a transversal so that one pair of corresponding angles are congruent, then the lines are parallel. Example: a. Name the lines, if any, that must be parallel. b. Find the measure of L1. Solution: a. Since corresponding angles 2 and 3 are both right angles, e II f (Notice that since mL3 :;:. 89°, lines c and dare not parallel.) b. Since e II f, mLl = 89°. d c 89° 6 e 131 f A statement that follows directly from a theorem is sometimes called a corollary. Part (a) of the Example demonstrates the following corollary of Theorem 7 .4. COROLLARY: If two coplanar lines are perpendicular to a third line, then the two lines are parallel to each other. 1. Describe four methods of showing that two lines are parallel. 2. Explain in your own words the difference between the Corresponding Angles Theorem and Theorem 7.4. Class Exercises Name each of the following. 1. four pairs of corresponding angles 2. three angles congruent to L2 3. three angles congruent to L5 Find the measure of each numbered angle. 4. Ll 5. L2 6. L3 7. L4 8.'L5 9. L6 10. L7 11. L8 194 Chapter 7 Parallel Lines c d / "£ Exercises 1. Name each of the following. a. four pairs of corresponding angles b. three angles congruent to L 1 c. four angles supplementary to 4-1 2. Complete. a. If L.1 =. ,then /II n. b. If mL3 + m • = 180°, then /II n. c. If L.2 then /II n. =. , Find the measure of each numbered angle. .., ~ a ~ L2 6. L4 8. L.6 3.L.1 5. L3 7. L.5 b Find the measure of each numbered angle. 9. 10. E 4l 12. 11. Jm 2 I 'V ~ .._ 'D 13. What must be the measure of each angle in order that a II b? a. L.1 b. L.2 c. L3 Name the segments or lines that must be parallel. 14. ~c ~d 15. ~a ~b / 16. E A B C 7.3 Parallel Lines and Corresponding Angles D 195 r--..._ Name the lines, if any, that must be parallel if each statement is true. 17. L1 := L4 18. L4 := L6 19. mL2 + mL1 = 180° 20. L8 := L9 21. L4 := L5 22. mL4 + mL8 = 180° 23. Draw a figure like the one shown. a. Construct the line through P that is perpendicular to l. b. Construct the line through P that is perpendicular to the line constructed in part (a). c. How is the line constructed in part (b) related to line l? APPLICATIONS 24. Architecture In City Hall, Corridor One and Corridor Two are both perpendicular to Corridor Three. Why can you say that Corridor One and Corridor Two are parallel to each other? 25. Algebra Two parallel lines are cut by a transversal so that two corresponding angles have measures (3x - 8) 0 and (2x + 10) 0 • Find the value of x and the measures of the angles. 26. Computer The following LOGO program is for a turtle that likes to travel backwards. Rewrite the program so it allows the user to decide what the angles of the parallelogram will be. To parallelogram repeat 2 [bk 50 rt 40 rt 180 bk 20 rt 140 rt 180] end Test Yourself Name each of the following. 1. two pairs of alternate interior angles 2. two pairs of same-side interior angles 3. four pairs of corresponding angles 4. all angles supplementary to L 1 5. all angles congruent to L 1 6. If mL2 = 60°, find the measures of L5, L6, and L7. What must be the measure of each angle so that a II b? 7. L2 8. L3 9. L4 196 Chapter 7 Parallel Lines p. 7.4 Constructing Parallel Lines Objective: To construct the line parallel to a given line through a point not on the line. You can use the properties of angles formed by a transversal to construct a line parallel to a given line. - - - - - - - - · CONSTRUCTION 11 - - - - - - - - · Construct the line through P that is parallel to line l. p• 1. Draw a line l and a point P not on l. Draw a line through P intersecting l. 2. _At vertex P, construct L.2 congruent to L.1. Draw line m. Then m II l. m 1. In Construction 11 , L.2 was constructed congruent to L.1. Why does this guarantee that m II l ? 2. The figures show an alternative construction of line m parallel to the given line l through the given point P. a. Describe Step 1. b. Describe Step 2. m c. Why is line m parallel to line l? d. Why can this construction be considered a special case of Construction 11? 7.4 Constructing Parallel Lines 197 Class Exercise 1. Draw a line m and a point A not on m. Construct the line through A that is parallel to m. Exercises Using only the given information, can you conclude that a II b? 1. ~ 2. • 3. a b b 4. 6. 5. a b Draw a figure like the one shown. Construct the line through A that is parallel to Bc. 7. ~ 8. C B 9. A •A c A B • c 10. Draw a line l. Construct two lines, one on each side of l, that are parallel to l. Draw a figure like the one shown. Construct the figure described. 11. rhombus JKLM with KL on line a J 12. parallelogram ABCD A • • a C B K 13. square EFGH with FG on line l E• 14. trapezoid ABCD such that AD II BC and AD= 2·BC A• B 198 Chapter 7 Parallel Lines C x 15. Draw a figure like the one shown and use it to construct square WXYZ. y .4 P I• I I I 16. Use the construction suggested by Postulate 10 on page 189 to construct the line through P that is parallel to l. 1 -; APPLICATIONS 17. Space Geometry Given a point P not on line l, how many lines through P do not intersect l? 18. Paper Folding Draw a line l and a point P not on l. Use paper folding to determine the line through P that is parallel to l. Thinking in Geometry The following steps show an alternative construction of a line through P parallel to l. = Step 1: From P, draw an arc intersecting l at any point A. p p • ~ A<::::::::_ ~ Step 3: Draw PC. Step 2: Construct AB PA on l. With the same compass setting and with the compass point on P and then B, draw arcs intersecting at some point C. I • 1. Why is h. PAC ::: h. BCA? . 4. What kind of figure is PABC? ... )B AC:Z::: • ' P / / / / // • A<--- / -----2 I )B AC:Z::: ~c / ---- • ~ 1---- --~" / "1c'\ ... p . I ~ 2. Why is L1 ::: L2? 3. Why is Pc Ill? .. ~ • )II; ... / )/ // I ~ • w 5. Draw a figure like the one shown. Without copying an angle, construct parallelogram WXYZ. • X y 7.4 Constructing Parallel Lines 199 Thinking About Proof Converses On page 90 you learned that a conditional is a statement that can be written in if-then form. When you interchange the hypothesis and conclusion of a conditional statement, you get the converse of the original statement. Example: Write the converse of each conditional. a. If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent. b. A rectangle is a parallelogram. Solution: a. If two lines are cut by a transversal so that each pair of alternate interior angles are congruent, then the lines are parallel. b. Rewrite the conditional in if-then form: If a figure is a rectangle, then it is a parallelogram. Converse: If a figure is a parallelogram, then it is a rectangle. Part (b) of the Example demonstrates that the converse of a true conditional does not have to be true. When a conditional and its converse are true, you can combine them and write the combined statement in if-and-only-ifforrn: Two lines cut by a transversal are parallel if and only if each pair of alternate interior angles are congruent. Exercises Write the converse of each conditional. 1. If two angles of a triangle are congruent, then the sides opposite those angles are congruent. 2. An equilateral triangle is also equiangular. 3. The three medians of a triangle are concurrent. 4. Vertical angles are congruent. 5. Which converses in Exercises 1-4 are true? 6. Combine Theorems 5.9 and 5.10 on page 138 in if-and-only-if form. 7. What is the converse of the converse of a conditional? 200 Chapter 7 Parallel Lines 7.5 Interior Angle Measures of a Triangle Objective: To use the sum of the interior angle measures of a triangle. You know about the relationships among angles formed by parallel lines and a transversal. Important relationships also exist among the angles of a triangle. EXPLORING r/; . . ,. . . ,. _ LJJ~~ 1. Use the Geometric Supposer: Triangles disk to draw an acute triangle, !:J.ABC. Through point C, draw DE so that it is parallel to side AB. 2. Sketch the figure. Use the Measure option to measure all the angles formed. Record the measures on your sketch. What observations and conjectures can you make about the angles of !:J.ABC? Repeat using a right triangle, an isosceles triangle, and an equilateral triangle. 3. Explain why any of your conjectures must be true. EXPLORING Part A 1. Draw and cut out a scalene triangle. Fold the triangle as shown below so that all three vertices meet on one side of the triangle. /"\ / // /\ \ // \. / /fti\ // /_:~_ \ \ / / \ \ // /~ \ \ ~ 2. What is the sum of the measures of the three angles of the triangle? PartB c 1. What is mL5 + mL3 + mL4? l 2. Why is L5 =: Ll? Why is L4 =: L2? 3. What is mLl + mL3 + mL2? A .,L..__:__---i--~ 7.5 Interior Angle Measures of a Triangle 201 The angles of a triangle are sometimes called the interior angles of the triangle. The EXPLORING activities demonstrate the following theorem. THEOREM 7.5 (Triangle Ang le-Sum Theorem): The sum of the measures of the interior angles of any triangle is 180°. EXPLORING Part A 1. What kind of triangle is XYZ? X ~z 2. What is mLX + mLY + mLZ? mLY? mLX + mLZ? y 3. Describe the relationship of LX andLZ. PartB 1. Draw an equilateral /:::,. RST and name the parts that are congruent. 2. What is the measure of each interior angle of /:::,. RST? The Triangle Angle-Sum Theorem has the following corollaries. COROLLARY 1 : The acute angles of any right triangle are complementary. COROLLARY 2: The measure of each interior angle of an equilateral triangle is 60°. Example: Find the measure of each numbered angle. a. ~ 60° ~2 Solution: a. mL1 + 75° + 60° mL1 + 135° mL1 mL1 mL2 mL2 202 Chapter 7 Parallel Lines b. M ~ • = 180° = 180° = 180° = 45° = 180° = 135° b. - 135° - 45° mL3 mL4 + 70° + 70° mL4 + 140° mL4 mL4 = 70° = 180° = 180° = 180° = 40° - 140° Thinking Critically 1. Is Part A of the second Exploring activity on page 201 an example of inductive reasoning or deductive reasoning? Which type of reasoning is Part B? 2. Explain why the following statements are true. a. A triangle must have at least one angle that measures at least 60°. b. Every triangle must have at least two acute angles. Class Exercises n Find the measure of each numbered angle. 1. 50° 4. 6 70° 3. 2. [i (1 1 5. 6. Exercises True or false? Give a reason or example to support each answer. 1. A triangle can have two 89° angles. 2. The acute angles of any right triangle are supplementary. 3. An equiangular triangle is a special kind of acute triangle. 4. A triangle can have angles of measure 74°, 43°, and 62°. 5. A right triangle can be equilateral. 6. An isosceles triangle can be obtuse. 7. A triangle can have no obtuse angles. 8. A triangle can have two right angles. 9. A right triangle can have an obtuse angle. 10. A triangle can have exactly one acute angle. 7.5 Interior Angle Measures of a Triangle 203 Find the measure of each numbered angle. 11. il 12. 13. 15.6 16. 17. 18. 19. 20. 21. 22. 14. ~ ..,1450 S 5 ( ) " 5 23. a. Ll ~ c 18"135" b. LDAB c. LDCB d. L2 I A B 24. a. Ll b. LPTR c. L2 d. L3 Tv ~ • 25. One base angle of an isosceles triangle has a measure of 35°. a. Find the measure of the other base angle. b. Find the measure of the vertex angle. 26. The vertex angle of an isosceles triangle has a measure of 150°. Find the measure of each base angle. 27. One acute angle of a right triangle has a measure of 28°. Find the measure of the other acute angle. 28. fJ. PQR is isosceles. One of the base angles, LP, has a measure - of 25°, and PQ QR. = a. FindmLR. 204 b. FindmLQ. Chapter 7 Parallel Lines c. Classify fJ. PQR by its angles. -- 'R 29. a. Find mLl. b. Find mL2. c. Classify ~ABC by its sides. Give a reason for your answer. A.....___ _~~~ 30. One angle of an obtuse isosceles triangle has a measure of 40°. Find the measures of the other two angles. 31. To be Proven: If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent (Theorem 6.1). Given: LA ::: LD; LB := LE Prove: LC ::: LF a. Since LA ::: LD and LB ::: LE, b. c. d. e. F A ~BD~E . mLA = • andmLB = • . mLA + mLB = • What are mLC and mLF? Why? Substitute your answer from part (b) into one of your equations in part (c). Is mLC = mLF? Why is LC ::: LF? APPLICATIONS 32. Algebra Two angles of a triangle are congruent. The third angle has measure equal to the sum of the measures of the other two. Find the measures of all three angles. 33. Algebra The measure of one acute angle of a right triangle is five times the measure of the other acute angle. Find the measure of each acute angle. 34. Algebra and Mental Math The measure of the largest angle of a triangle is five times the measure of the smallest angle. The measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles. Computer Use the Your Own option on the Geometric Supposer: Triangles disk. Try to make a triangle such that mLBA C = 120°, AB = 5, and mL CBA = 60°. What happens? Explain. 7.5 Interior Angle Measures of a Triangle 205 7.6 Exterior Angle Measures of a Triangle Objective: To use the relationship between an exterior angle of a triangle and its remote interior angles. An exterior angle of a polygon is formed by one side of the polygon and the extension of the adjacent side. For f:.ABC , L 1 is an exterior angle. Every triangle has six exterior angles. The exterior angles of f:. DEF are Ll , L2, L3, L4, L5, and L6. At each vertex, the two exterior angles are congruent. Why? Ll =: L2 L3 =: L4 L5 =: L6 Each exterior angle of a triangle has one adjacent interior angle and two remote interior angles. Example: For each exterior angle shown, name one adjacent interior angle and two remote interior angles. Solution: Exterior angle ~---~ B A ~~ .. -~~-.. , D t6 2 ', E ~ ~~ ~--~ Adjacent interior angles Remote interior angles L7 L8 L8 , L9 Ll L4 L7,L9 EXPLORING • 1. For each triangle, one exterior angle is shown. Find the measure of each n~ umbered angle . ...,._4v 3 370 37° .._ __2 A ~-· 2. Compare the measure of each exterior angle shown with the measure of each of its remote interior angles. Are they the same? If not, which is greater? 3. Compare the measure of each exterior angle shown with the sum of the measures of its remote interior angles. Are they the same? 206 Chapter 7 Parallel Lines The EXPLORING activity demonstrates the following. THEOREM 7.6: In a triangle, the measure of each exterior angle is equal to the sum of the measures of its two remote interior angles. COROLLARY: In a triangle, the measure of each exterior angle is greater than the measure of either of its remote interior angles. Thinking Critically 1. DoLl and L2 have the same adjacent interior angle and remote interior angles? 2. Why is Ll ::: L2? c / \ 1A~B \ 3. What is the relationship between an exterior angle of a triangle and its adjacent interior angle? ' 4. Classify the exterior angles for an acute triangle, a right triangle, and an obtuse triangle. Class Exercises 1. Name the exterior angle adjacent to L2. p~ 2. Name the remote interior angles for L4. 3. Name an angle that is supplementary to L2. 4. mLl + mL3 = m • 5. mL4 = m • Q + m• Draw a triangle similar to, but larger than, each one shown. Then draw and label an exterior angle at vertex A. 6. FVA 1.v<Jz A 8. R '-----~A Find the measure of each numbered angle. 9. 50'l.d7 10. ~ ~ 11 . ....:--~ ~ 7.6 Exterior Angle Measures of a Triangle 207 Exercises 2 Name the adjacent interior angle and the two remote interior angles for each exterior angle. 1. L 1 2. L2 3. L3 ~ I R~T-• ..lllfk-'".,. S 3 Name each of the following. 4. three interior angles 5. six exterior angles 6. three angles that are neither exterior nor interior angles 7. the adjacent interior angle and two remote interior angles for L4 8. two exterior angles for which L3 and L5 are the two remote interior angles 9. six pairs of vertical angles 12 ~~----------~ 10 Draw a triangle similar to, but larger than, the one shown. Then draw and label an exterior angle at vertex A. ~T 10. S 11. A 12. A B~C X A~G Find the measure of each numbered angle. 13. ~ 14. ~~ . 16. A 15. ~ 4\ 17. D ' ~~~ 19. 20. ....__...... ~ / ~ 208 ... 5 3\ 1 2' .... ~ Chapter 7 Parallel Lines A ~--£_1 ~ )1 8 18. ._h 150° -~ '~ 11 True or false? 21. All exterior angles of an acute triangle are obtuse angles. 22. All exterior angles of an equiangular triangle are congruent. 23. All exterior angles of an obtuse triangle are acute angles. X 24. All exterior angles of a right triangle are right angles. Find the measure of each numbered angle. 25. 26. 48° 27.G 28.6 55° 97° 2 1 - 1__ ..,.. 2 116° --· 29. In 6 ABC, mLA = 45° and mLB = 80°. Find the measure of LC and of an exterior angle at C. 30. In 6 PQR, mLP = 35° and the measure of an exterior angle at Q is 105°. Find the measures of LR and LQ. - - 31. Explain why AB and AC could not both be perpendicular to BD. • A ~ ..-..... D BC APPLICATION 32. Calculator The measure of an exterior angle of a triangle is twenty-three times the measure of its adjacent interior angle. Find the measures of both angles. Calculator The measures of two angles of a triangle are 28° and 94 o. Andy and Christine used different methods to find the measure of the third angle with their calculators: Andy's Method 28 180 1±1 E1 94 El 122 Christine's Method 122 El 180 E1 28 E1 94 El 58 58 Which method do you think was better? Give a reason for your answer. 7.6 Exterior Angle Measures of a Triangle 209 7.7 Angle Measures of a PolygQn Objective: To find the sum of angle measures of any polygon. You can use the sum of the interior angle measures of a triangle to determine angle measures in other polygons. EXPLORING 1. Draw a quadrilateral like the one shown. Use a protractor to measure LQ, LQUA, LA, and LADQ. Use a calculator to find the sum. 2. Use the Triangle Angle-Sum Theorem to find the sum of the measures of the interior angles of DQUA. Compare your results with Step 1. The EXPLORING activity demonstrates the following theorem. THEOREM 7. 7: The sum of the measures of the interior angles of any quadrilateral is 360°. To find the sum of the measures of the interior angles of any polygon, draw all possible diagonals from one vertex. Then find the sum of the measures of the interior angles of the triangles formed. Triangles I Sum of Interior Formed Angle Measures Polygon Sides Pentagon 5 ~ ~ 3 3 X 180° = 540° Hexagon 6 8 4 4 Figure - '' X 180° = 720° - These results and the EXPLORING activity suggest the following theorem. THEOREM 7.8 (Interior Angle Sum Theorem): The sum of the measures of the interior angles of a polygon with n sides is (n - 2)180°. 210 Chapter 7 Parallel Lines hA QU/ D Example 1: Find mLl. ~ Solution: mL1 + 98° + 115° + 90° + 121 o = (5-2) 180° mL1 + 424° = 540° mL1 = 540° - 424° = 116° EXPLORING 1. What is mL 1 + mL2? What is the sum of the measures of each exterior angle and adjacent interior angle? 2. What is the sum of the measures of all the numbered angles shown? 3. What is the sum of the measures of the interior angles? 4. Subtract the sum of the measures of the interior angles (Step 3) from the sum in Step 2 to find the sum of the measures of the exterior angles, one at each vertex. .lllllt:' ........ D 5. Draw a triangle or quadrilateral. Draw one exterior angle at each vertex. Number these exterior angles and the interior angles. Repeat Steps 1-4 for your figure . This EXPLORING activity demonstrates the following theorem. THEOREM 7.9 (Exterior Angle Sum Theorem): The sum of the measures of the exterior angles, one at each vertex, of any polygon is 360°. Example 2: The measure of each exterior angle of a regular polygon is 18°. Find the number of sides. Solution: Let n = the number of sides. 18° • n = 360° n = 20 There are 20 sides. 7.7 Angle Measures of a Polygon 211 1. Why are the exterior angles of a regular polygon congruent? 2. For what kind of polygon does the sum of the interior angles equal the sum of the exterior angles, one exterior angle at each vertex? ~ Class Exercises Find the sum of the measures of the interior angles of a polygon with the given number of sides. 1. 7 sides 2. 9 sides 3. 10 sides 4. The measure of each interior angle of a regular polygon is 135°. a. Find the measure of each exterior angle. b. Find the number of sides. Find the measure of each numbered angle. 5. ~ 7. 6. ~-~ Exercises Copy and complete the table. Sum of the Interior Angle Measures Number of Sides of Polygon Sum of the Exterior Angle Measures i 1. 2. 3. 4. 5. • •• 4 360° •• •• 4 • 180 = 720° 6 • 7 • 180 = 1,260° 16 24 7. 0 :r ! ! I .i Find the measure of each numbered angle. 6. 8. /\ .,._ .., 50° 4 90° 212 105°1 Chapter 7 2 10oo ~..... Parallel Lines Jl- 16 ~ 9. regular pentagon 3\ 2 ~ \ 110" .... €J 2 _,... Copy and complete the table . . . - Number of Sides of Regular Polygon 10. 5 11. 9 M easure of Each Inter ior Angle Measure of Each Exterior Angle • • •• •• • 12. 13. 14. D Gl n 90 144° 160° I The sum of the measures of the interior angles is given for some polygons. Find the number of sides of each polygon. 15. 540° 16. 900° 17. 1,440° 18. 1,800° 19. 2,340° 20. 3,240° Draw and label a figure for each polygon described, if possible. If it is not possible, explain why. 21. triangle: exterior angles-obtuse angles 22. triangle: exterior angles-2 right angles, 4 obtuse angles 23. quadrilateral: exterior angles---4 obtuse angles, 4 acute angles 24. quadrilateral: interior angles- 4 acute angles 25. quadrilateral: interior angles-2 obtuse angles, 1 right angle, 1 acute angle 26. pentagon: interior angles- 5 acute angles APPLICATION 27. Sports Research the shape of the home plate used in baseball. Sketch home plate and give the measures of its interior angles. Everyday Geometry Bees construct honeycombs in the shape of regular hexagons. Do library research to find out what advantages this structure offers the bees. 7.7 Angle Measures of a Polygon 213 7.8 Problem Solving Application: Interpreting Contour Maps Objective: To solve problems involving contour maps. If a plane intersects a portion of Earth's surface, the intersection is a closed curve. In the figure below, parallel planes Land M intersect the mountain at elevations of 7,000 ft and 8,000 ft respectively. The resulting contour lines show the shape of the mountain at the two elevations. A map depicting Earth's surface at different cross sections is a contour map. The closer the lines are to each other, the steeper the terrain. Hikers, mountain climbers, and highway engineers use contour maps to plan routes between places. => contour map Example: Find the elevations of points A, B, and Con the contour map. contour interval: 200ft Solution: Point A lies on the 4,400-ft contour line, so its elevation is exactly 4,400 ft. Point B lies between 4,200 ft and 4,400 ft. Point C lies inside the 4,600-ft curve, so its elevation is greater than 4,600 ft but less than 4,800 ft. 214 Chapter 7 Parallel Lines Class Exercises Use the contour map shown on this page to answer these questions. 1. What is the difference in elevation between adjacent contour lines? 2. What is the elevation of Bascom Lodge? 3. Describe the difference in elevation between Wilbur Clearing Lean-to and Bascom Lodge. If you cannot write an exact elevation, give a range. Exercises Use the contour map shown on this page to answer these questions. 1. State the elevations of the Money Brook Lean-to and the intersection of the Hopper and Overlook Trails. If you cannot write an exact elevation, give a range. 2. Suppose you hiked along the trail from the Mt. Williams campsite to Bascom Lodge. What is the change in elevation? 3. What is the elevation of the highest point on Mt. Prospect Trail? 4. Explain why roads and hiking trails do not usually proceed in a straight line, but curve and loop as you travel between places. - - Trail 1::t Lean-to/Lodge Elevation is given in feet. LEGEND .A. Campsite Road Draw sketches of contour maps showing the following types of terrain. 5. a steep mountainside 6. a gentle sloping hillside 7.8 Problem Solving Application: Interpreting Contour Maps 215 Chapter 7 Review Vocabulary You should be able to write a brief description, draw a picture, or give an example to illustrate the meaning of each of the following terms. Vocabulary converse (p. 200) corollary (p. 194) transversal (p. 185) Angles determined by a transversal alternate interior angles (p. 185) corresponding angles (p. 185) exterior angles (p. 185) interior angles (p.185) same-side exterior angles (p. 186) same-side interior angles (p. 186) Angles of a triangle adjacent interior angle (p. 206) exterior angle (p. 206) interior angles (p. 202) remote interior angles (p. 206) Summary The following list indicates the major skills, facts, and results you should have mastered in this chapter. 7. 1 Identify the interior, exterior, alternate interior, corresponding, and same-side interior and exterior angles formed when two lines are cut by a transversal. (pp. 185-187) 7.2 Use facts about the interior angles formed when two parallel lines are cut by a transversal and use relationships between interior angles to determine when two lines are parallel. (pp. 188-192) 7.3 Use facts about parallel lines cut by a transversal and pairs of corresponding angles. (pp. 193-196) 7.4 Construct the line parallel to a given line through a point not on the line. (pp. 197-199) 7. 5 Use the fact that the sum of the interior angle measures of a triangle is 180°. (pp. 201-205) 7.6 Use the relationship between an exterior angle of a triangle and its remote interior angles. (pp. 206-209) 7.7 Find the sum of angle measures of any polygon. (pp. 210-213) 7.8 Solve problems involving contour maps. (pp. 214-215) 216 Chapter 7 Review Exercises Describe the angle or pair of angles. Use interior, exterior, alternate interior, corresponding, same-side interior, or same-side exterior. 2. L5 5. L4 and L5 8. L2 and L6 1. L7 4. L2 and L8 7. L1 and L5 3. L4 6. L6 and L4 9. L1 and L7 j ~ 3 L/1.. ABCD is a parallelogram. Classify each statement as true or false. A 10. L2 ::: L5 11. mL3 + mL4 = 180° 12. mL? + mL8 = 180° 14. L7 ::: L4 13. mL3 + mL6 = 180° 15. mLABC + mL4 = 180° 16. L1 ::: L7 17. LADC ::: L8 B 1 2 6 5 3 7 • 4 8 D C Name the lines, if any, that must be parallel if each statement is true. b 18. L1 ::: L6 19. L1 ::: L2 c 20. L1 ::: L5 21. L2 ::: L4 22. mL3 + mL5 = 180° 23. L4::: L6 d 24. Draw an acute triangle ABC. Construct a line parallel to BC through A and a line parallel to AB through C. Find the measure of each numbered angle. · 25. - ~ 85° 1 35° 26. ~1 ~ ~ 27. ~---~\____A 28. Find the sum of the measures of the interior angles of a 30-sided polygon. 29. The sum of the measures of the interior angles of a polygon is 2,160°. Find the number of sides. PROBLEM SOLVING contour 30. Estimate the difference in elevation between interval: points A and B. ' - - - - - - - - - - - - ' 50.ft Chapter 7 Review 217 ··'? Chapi~~~1Z2Je~t ' ,,·~;· i ,_._·::/fV:.-)7-;,c~~:(~JJti:,;;,:;::;.'~i ' __ :·.r:~, :,:·," :•.. '- -:•:,_;:, -:,:/, .:~- -:.: - ~"' 1. Name the transversal. 2. Name three exterior angles and three interior angles. 3. N arne one pair of same-side interior angles and one pair of same-side exterior angles. 4. N arne one pair of alternate interior angles. 5. N arne two pairs of corresponding angles. Find the measure of each angle. Assume that a II b, c II d, and e llf. 6. Ll 7. L3 8. L6 1c Name the lines, if any, that must be parallel if each statement is true. 9. L5 := L6 10. Ll := L4 11. L3 := L7 12. mL8 = 95° 13. mL2 = 45° 14. mL3 = 45° '! a 15. Draw a line k and a point P not on k. Choose any point Q on k and construct rhombus PQRS with QR on k. Complete. 16. In h. ABC, if mLA = 54° and mLB = 71°, then mLC = • . 17. The sum of the measures of the interior angles of a polygon is 6,840°. The polygon has • sides. 18. The sum of the measures of the interior angles of a 36-sided polygon is • . The sum of the measures of the exterior angles is • . Find the measure of each numbered angle. 19. ~ 78° 56° 20. . ---· 1 21. ~ ~ PROBLEM SOLVING 22. State the elevations of points A, B, C, and D. If you cannot write an exact elevation, give a range. 218 Chapter 7 Test contour interval: 100ft