Pairs of Angles

1. Pairs of Angles LESSON 7-1 Problem of the Day Students at West Ridge Middle School are going on a trip to the museum. Nine have never gone before. Twelve have gone once, half as many as that have gone twice, and none have gone more than that. How many students are going on the trip? 27 7-1

2. Pairs of Angles LESSON 7-1 Check Skills You’ll Need (For help, go to Lesson 1-6.) 1.Vocabulary Review What is the inverse operation of addition? Solve each equation. 2. a + 14 = 32 3. b – 5 = 26 4. 10 + c = –31 5. –48 = d – 19 Check Skills You’ll Need 7-1

3. Pairs of Angles LESSON 7-1 Check Skills You’ll Need Solutions 1. subtraction 2. 18 3. 31 4. –41 5. –29 7-1

4. Simplify. The sum of the measures of supplementary angles is 180º. x° + m IGJ = 180° x° + 145° = 180° Substitute 145º for mDEF. x° + 145° – 145° = 180° – 145° Subtract 145º from each side. The measure of the supplement of mIGJ is 35º. x° = 35° Pairs of Angles LESSON 7-1 Additional Examples Find the measure of the supplement of  IGJ. Quick Check 7-1

5. The adjacent angles are HGK and KGJ; KGJ and JGI; JGI and IGH; IGH and HGK. HGI and KGJ. The vertical angles are JGI and HGK; Since vertical angles are congruent, mHGK = mJGI = 145°. Pairs of Angles LESSON 7-1 Additional Examples Name a pair of adjacent angles and a pair of vertical angles in the figure. Find mHGK. Quick Check 7-1

6. DKE and FKE are supplementary. m DKE + 90° = 180° m DKE = 90° Subtract 90º from each side. Pairs of Angles LESSON 7-1 Additional Examples In this figure, if mDKH = 73°, find the measures of GKJ and JKF. 7-1

7. KHE and DKH are complementary. m KHE + 73° = 90° m KHE = 17° Subtract 73º from each side. GKJ and KHE are vertical angles. m GKJ = mKHE = 17° JKF and DKH are vertical angles. m JKF = mDKH = 73° So, the measure of GKJ is 17° is and the measure of JKF is 73°. Pairs of Angles LESSON 7-1 Additional Examples (continued) Quick Check 7-1

8. AXD and BXC; AXB and DXC AXD and BXC Pairs of Angles LESSON 7-1 Lesson Quiz Use the diagram to answer Questions 1 – 3. 1. List all pairs of vertical angles. 2. List any angles adjacent to CXD. 3. If mAXB = 110°, find mDXC. 4. An angle measures 57°. What is the measure of its supplement? 110° 123° 7-1

9. 1 6 5 8 13 24 4 – = 3 ; true Angles and Parallel Lines LESSON 7-2 Problem of the Day Write the following sentence in mathematical symbols. Then indicate whether the sentence is true or false: Four and one sixth minus five eighths equals three and thirteen twenty fourths. 7-2

10. Angles and Parallel Lines LESSON 7-2 Check Skills You’ll Need (For help, go to Lesson 7-1.) 1.Vocabulary Review Which of the following pairs of angles are supplementary? 50° and 40°, 100° and 90°, 120° and 60°, 75° and 125° Find the measure of the supplement of each angle. 2. 48° 3. 119° 4. 67° 5. 131° Check Skills You’ll Need 7-2

11. Angles and Parallel Lines LESSON 7-2 Check Skills You’ll Need Solutions 1. 120° and 60° 2. 132° 3. 61° 4. 113° 5. 49° 7-2

12. 1 and 3, 2 and 4, 5 and 7, 6 and 8 are pairs of corresponding angles. 2 and 7, 3 and 6, are pairs of alternate interior angles. Angles and Parallel Lines LESSON 7-2 Additional Examples Identify each pair of corresponding angles and each pair of alternate interior angles. Quick Check 7-2

13. Alternate interior angles are congruent. m 6 = m 3 = 56° m 6 = 56° Angles and Parallel Lines LESSON 7-2 Additional Examples If p is parallel to q, and m 3 = 56º, find m 6. Quick Check 7-2

14. p || q because 5 and 7 are congruent alternate interior angles. s || t because 6 and 7 are congruent corresponding angles. Angles and Parallel Lines LESSON 7-2 Additional Examples In the diagram below, m 5 = m 6 = and m 7 = 80º. Explain why p and q are parallel and why s and t are parallel. Quick Check 7-2

15. Angles and Parallel Lines LESSON 7-2 Lesson Quiz Use the diagram to answer the questions. 1. Classify 4 and 7 as 2. Classify 2 and 8 as alternate interior angles, alternate interior angles, corresponding angles, corresponding angles, or neither. or neither. 3. If a || b and m 8 = 80°, 4. Suppose m 5 = 100° and find m 4. m 3 = 100°. What can you conclude about line a and line b? neither alternate interior angles 80° a || b 7-2

16. 4 5 n 27 3 5 n = 21 Congruent Polygons LESSON 7-3 Problem of the Day Solve the proportion: = . 7-3

17. Congruent Polygons LESSON 7-3 Check Skills You’ll Need (For help, go to Lesson 4-4.) 1.Vocabulary Review Congruent angles have ? measures. Are the polygons similar? Explain. 2. Check Skills You’ll Need 7-3

18. = / Congruent Polygons LESSON 7-3 Check Skills You’ll Need Solutions 1. equal 2. ; ; not similar; corresponding sides are not in proportion. 5 15 4 10 LN ZY LM XZ 7-3

19. Congruent Angles A M B N C O D P Congruent Sides AB MN BC NO CD OP DA PM Since A corresponds to M, B corresponds to N, C corresponds to O, and D corresponds to P, a congruence statement is ABCDMNOP. Congruent Polygons LESSON 7-3 Additional Examples In the diagram below, list the congruent parts of the two figures. Then write a congruence statement. Quick Check 7-3

20. Q EAngle SQVTSide Q TAngle QPEYSide QRTUSide Q YAngle QPR EYT by ASA. SQR VTU bySAS. Congruent Polygons LESSON 7-3 Additional Examples Show that each pair of triangles is congruent. a. b. Quick Check 7-3

21. J H Both are right angles. JI HI Both measure 48 ft. KIJ GIH They are vertical angles. So ∆GHI ∆ JI by ASA Congruent Polygons LESSON 7-3 Additional Examples A surveyor drew the diagram below to find the distance from J to I across the canyon. Show that GHIKJI. Then find JK. Corresponding parts of congruent triangles are congruent. JK corresponds to HG, so JK is 36 ft. Quick Check 7-3

22. Congruent Polygons LESSON 7-3 Lesson Quiz Use ABC and XYZ to answer the questions. 1.Suppose AC=XZ, AB =XY, and BC =YZ. Write a congruence statement for the figures. 2. Suppose ABC and XYZ are congruent. If AB = 5 cm, BC = 8 cm, and AC = 10 cm, find XZ. 3. Suppose BY, AX, and ABXY. Why is ABCXYZ? ∆ABC∆XYZ 10 cm ASA 7-3

23. ∆ABC ∆XYZ by SAS; m C = 35° Congruent Polygons LESSON 7-3 Lesson Quiz 4. Let AB =XY = 9 inches; BC =YZ = 24 inches; and m B = 85°, m Z = 35°, and m Y = 85°. Prove that the triangles are congruent and find m C. 7-3

24. Classifying Triangles and Quadrilaterals LESSON 7-4 Problem of the Day Ruth was born on February 29, 1984. If she insists on celebrating her birthday only on February 29, when will she celebrate her 12th birthday? 2032 7-4

25. Classifying Triangles and Quadrilaterals LESSON 7-4 Check Skills You’ll Need (For help, go to the Skills Handbook page 640.) 1.Vocabulary Review How many degrees does a right angle have? Classify each angle as acute, right, obtuse, or straight. 2.3. Check Skills You’ll Need 7-4

26. Classifying Triangles and Quadrilaterals LESSON 7-4 Check Skills You’ll Need Solutions 1. 90° 2. acute 3. obtuse 7-4

27. Classifying Triangles and Quadrilaterals LESSON 7-4 Additional Examples Classify LMN by its sides and angles. The triangle has two sides that are congruent and three acute angles. It is an isosceles acute triangle. Quick Check 7-4

28. Classifying Triangles and Quadrilaterals LESSON 7-4 Additional Examples What is the best name for figure WXYZ? Explain your choice. WXYZ has both pairs of opposite sides parallel, but adjacent sides are not equal, so it is a parallelogram. Quick Check 7-4

29. Classifying Triangles and Quadrilaterals LESSON 7-4 Lesson Quiz 1. Classify the triangle according to its angles and sides. 3. Determine the best name for the quadrilateral. obtuse isosceles triangles 2. A triangle’s sides are all congruent and its angles all measure 60°. Classify the triangle. equilateral, acute rhombus 7-4

30. Classifying Triangles and Quadrilaterals LESSON 7-4 Lesson Quiz 4.What is the best name for a figure that has four sides congruent, corresponding sides parallel, and all four angles congruent? square 7-4

31. Angles and Polygons LESSON 7-5 Problem of the Day A pound of turkey has 144 g of protein and will serve 4 people. If 4 people are served equal amounts, how many grams of protein will each receive? 36 g 7-5

32. Angles and Polygons LESSON 7-5 Check Skills You’ll Need (For help, go to the Lesson 1-1.) 1.Vocabulary Review How do you evaluate an algebraic expression? Evaluate each expression for a = 8. 2. 3(a + 1) 3. 5a + 8 a 4. (a –2)6 Check Skills You’ll Need 7-5

33. Angles and Polygons LESSON 7-5 Check Skills You’ll Need Solutions 1. You replace each variable in the expression with a number and then simplify. 2. 27 3. 6 4. 36 7-5

34. (n – 2) 180º = (8 – 2) 180º Substitute 8 for n. = 1,080º = (6) 180º Simplify. Subtract. Angles and Polygons LESSON 7-5 Additional Examples Find the sum of the measures of the interior angles of an octagon. An octagon has 8 sides. The sum of the interior angles of an octagon is 1,080. Quick Check 7-5

35. (n – 2) 180° = (6 – 2) 180° Substitute 6 for n. Simplify. = 720° Angles and Polygons LESSON 7-5 Additional Examples Find the missing angle measure in the hexagon. Step 1 Find the sum of the measures of the interior angles of a hexagon. 7-5

36. 720° = 120° + 115° + 136° + 80° + 147° + x° Write an equation. Add. 720° = 598° + x° Subtract 598º from each side. 122° = x° Angles and Polygons LESSON 7-5 Additional Examples (continued) Step 2 Write an equation. Let x = the missing angle measure. The missing angle measure is 122º. Quick Check 7-5

37. (n – 2) 180° = (9 – 2) 180° Substitute 9 for n since a nonagon has 9 sides. Simplify. = 1,260° Divide the sum by the number of angles in a nonagon. 1,260° ÷ 9 = 140° Angles and Polygons LESSON 7-5 Additional Examples A design on a tile is in the shape of a regular nonagon. Find the measure of each angle. Each angle of a regular nonagon has a measure of 140°. Quick Check 7-5

38. Angles and Polygons LESSON 7-5 Lesson Quiz 1. Find the sum of the measures of the interior angles of a polygon having 17 sides. 2. Five angles of a hexagon measure 128°, 190°, 112°, 154°, and 90°. Find the measure of the missing angle. 3. Find the measure of each angle of a regular polygon having 24 sides. 2,700° 46° 165° 4. A regular figure has an interior angle measure of 135°. How many sides does it have? 8 7-5

39. Areas of Polygons LESSON 7-6 Problem of the Day Lucy predicted that her final average for math class would be at least a 93. Her test grades were 88, 90, 92, 97. What is the lowest test score she can make on the last test to make this true? 98 7-6

40. Areas of Polygons LESSON 7-6 Check Skills You’ll Need (For help, go to Lesson 2-6.) 1.Vocabulary Review What is a formula? Find the area of each figure. 2.3. Check Skills You’ll Need 7-6

41. Areas of Polygons LESSON 7-6 Check Skills You’ll Need Solutions 1. A formula is a rule that shows the relationship between two or more quantities. 2.A = • w3. A = s2 = 10 • 8 = 72 A = 80 cm2A = 49 ft2 7-6

42. 1 2 A = bh Use the area of a triangle formula. 1 2 = • 36 •21 Substitute 36 for b and 21 for h. = 378 Multiply. Areas of Polygons LESSON 7-6 Additional Examples Find the area of the triangular part of the doghouse. The area of the triangular part of the doghouse is 378 in.2. Quick Check 7-6

43. 1 2 A = h(b1+b2) Use the formula. 1 2 Substitute 4.4 for h, 6.7 for b1, and 9.3 for b2. = (4.4) (6.7 + 9.3) Simplify. = 35.2 Areas of Polygons LESSON 7-6 Additional Examples Find the area of the trapezoid. Quick Check The area of the trapezoid is 35.2 in.2. 7-6

44. Areas of Polygons LESSON 7-6 Lesson Quiz 1. Find the area. 14 cm2 2. An architect is designing a restaurant with a triangular entrance. The base of the triangle is 10 ft wide. The entrance is 14 ft tall. Find the area of the triangle. 70 ft2 7-6

45. Areas of Polygons LESSON 7-6 Lesson Quiz 3. Find the area. 56 ft2 4.If both bases of the figure in Exercise 3 are doubled, what is the new area of the trapezoid? 112 ft2 7-6

46. Circumference and Area of a Circle LESSON 7-7 Problem of the Day Opal, Charles, Jean, and Scott had an earthworm-catching contest. Jean caught one fourth as many worms as Opal and twice as many as Charles. Opal caught 3 times as many worms as Scott. How many worms did each of the other three contestants catch if Scott caught 8 worms? Opal, 24; Jean, 6; Charles, 3 7-7

47. Circumference and Area of a Circle LESSON 7-7 Check Skills You’ll Need (For help, go to Lesson 7-6.) 1.Vocabulary Review Explain the difference between perimeter and area. Find the area of the figure below. 2. Check Skills You’ll Need 7-7

48. 1 2 1 2 Circumference and Area of a Circle LESSON 7-7 Check Skills You’ll Need • Solutions • 1. Perimeter is the distance around a figure. Area is the number of square • units a figure encloses. • 2. A = bh • = • 13 • 7 A = 45.5 ft² 7-7

49. Use the formula for circumference. C = d = (125) Substitute 125 for d. 125 Use a calculator. 392.6990817 Circumferences and Areas of Circles LESSON 7-7 Additional Examples The diameter of a tractor tire is 125 cm. Find the circumference and area. Round to the nearest tenth. The circumference is about 392.7 cm. 7-7

50. A = r 2 = (62.5) 2 The radius is 125 ÷ 2, or 62.5. Substitute 62.5 for r. 62.5 Use a calculator. 12271.8463 Circumferences and Areas of Circles LESSON 7-7 Additional Examples (continued) Use the formula for the area of a circle. The area is about 12,271.8 cm2. Quick Check 7-7