Question

the measures of two sides are given. between what two numbers must the third side fall.

1. 9 and 15
2. write an inequality to represent your answer:
3. 11 and 20
4. write an inequality to represent your answer:
5. 23 and 14
6. write an inequality to represent your answer:
7. 5 and 8
8. write an inequality to represent your answer:
9. 15 and 18
10. write an inequality to represent your answer:
11. 22 and 34
12. write an inequality to represent your answer:
13. 47 and 71
14. write an inequality to represent your answer:
15. 21 and 47
16. write an inequality to represent your answer:

name the largest and the smallest angle.
19)
20)
21)
list the angles of △abc from the smallest to the largest.

1. ab = 17, bc = 21, ac = 18
2. ab = 15, ac = 16, bc = 17

Explanation:

Step1: Recall triangle - side length rule

The length of the third side \(x\) of a triangle with two side - lengths \(a\) and \(b\) (\(a\geq b\)) satisfies the inequality \(|a - b|\lt x\lt a + b\).

Step2: Solve for problem 11

Given \(a = 15\) and \(b = 9\), then \(|15 - 9|=6\) and \(15 + 9 = 24\). The inequality is \(6\lt x\lt24\).

Step3: Solve for problem 12

Given \(a = 20\) and \(b = 11\), then \(|20 - 11| = 9\) and \(20+11 = 31\). The inequality is \(9\lt x\lt31\).

Step4: Solve for problem 13

Given \(a = 23\) and \(b = 14\), then \(|23 - 14|=9\) and \(23 + 14 = 37\). The inequality is \(9\lt x\lt37\).

Step5: Solve for problem 14

Given \(a = 8\) and \(b = 5\), then \(|8 - 5| = 3\) and \(8+5 = 13\). The inequality is \(3\lt x\lt13\).

Step6: Solve for problem 15

Given \(a = 18\) and \(b = 15\), then \(|18 - 15|=3\) and \(18 + 15 = 33\). The inequality is \(3\lt x\lt33\).

Step7: Solve for problem 16

Given \(a = 34\) and \(b = 22\), then \(|34 - 22| = 12\) and \(34+22 = 56\). The inequality is \(12\lt x\lt56\).

Step8: Solve for problem 17

Given \(a = 71\) and \(b = 47\), then \(|71 - 47|=24\) and \(71 + 47 = 118\). The inequality is \(24\lt x\lt118\).

Step9: Solve for problem 18

Given \(a = 47\) and \(b = 21\), then \(|47 - 21| = 26\) and \(47+21 = 68\). The inequality is \(26\lt x\lt68\).

Step10: Recall angle - side relationship in a triangle

In a triangle, the smallest angle is opposite the shortest side and the largest angle is opposite the longest side.

Step11: Solve for problem 19

In \(\triangle ABC\) with side - lengths \(AB = 7\), \(BC = 6\), and \(AC = 5\), the smallest angle is \(\angle A\) (opposite \(BC\)) and the largest angle is \(\angle C\) (opposite \(AB\)).

Step12: Solve for problem 20

In \(\triangle DEF\) with side - lengths \(DE = 4\), \(EF = 3\), and \(DF = 5\), the smallest angle is \(\angle F\) (opposite \(DE\)) and the largest angle is \(\angle D\) (opposite \(EF\)).

Step13: Solve for problem 21

In \(\triangle JHG\) with side - lengths \(JH = 5\), \(HG = 5.1\), and \(JG = 5.03\), the smallest angle is \(\angle G\) (opposite \(JH\)) and the largest angle is \(\angle H\) (opposite \(JG\)).

Step14: Solve for problem 22

In \(\triangle ABC\) with \(AB = 17\), \(BC = 21\), and \(AC = 18\), the order of angles from smallest to largest is \(\angle C,\angle A,\angle B\) (since the sides in ascending order are \(AB,AC,BC\)).

Step15: Solve for problem 23

In \(\triangle ABC\) with \(AB = 15\), \(AC = 16\), and \(BC = 17\), the order of angles from smallest to largest is \(\angle C,\angle B,\angle A\) (since the sides in ascending order are \(AB,AC,BC\)).

Answer:

1. \(6\lt x\lt24\)
2. \(9\lt x\lt31\)
3. \(9\lt x\lt37\)
4. \(3\lt x\lt13\)
5. \(3\lt x\lt33\)
6. \(12\lt x\lt56\)
7. \(24\lt x\lt118\)
8. \(26\lt x\lt68\)
9. Smallest: \(\angle A\), Largest: \(\angle C\)
10. Smallest: \(\angle F\), Largest: \(\angle D\)
11. Smallest: \(\angle G\), Largest: \(\angle H\)
12. \(\angle C,\angle A,\angle B\)
13. \(\angle C,\angle B,\angle A\)