Carlos constructed the angle bisector of ∠JKI by constructing ray KM . Carlos then constructed the angle bisector of ∠MKI by constructing ray KN . If the measurement of ∠JKI is 56 degrees, what is the measurement of ∠NKI ?(1 point)

Question

Responses

The measurement of ∠NKI is 7 degrees.
The measurement of angle upper N upper K upper I is 7 degrees.

The measurement of ∠NKI is 28 degrees.
The measurement of angle upper N upper K upper I is 28 degrees.

The measurement of ∠NKI is 112 degrees.
The measurement of angle upper N upper K upper I is 112 degrees.

The measurement of ∠NKI is 14 degrees.

Answer 1

To find the measurement of angle \( \angle NKI \), we start with the given measurement of \( \angle JKI \), which is 56 degrees.

1. Constructing the angle bisector \( KM \): Carlos bisects \( \angle JKI \) into two equal parts. Therefore, each angle \( \angle JKM \) and \( \angle MKI \) will be half of \( \angle JKI \):

   \[ \angle JKM = \angle MKI = \frac{56}{2} = 28 \text{ degrees} \]

2. Constructing the angle bisector \( KN \): Next, Carlos bisects \( \angle MKI \) (which is 28 degrees) into two equal parts. Hence:

   \[ \angle NKJ = \angle NKI = \frac{28}{2} = 14 \text{ degrees} \]

Now, we are specifically asked for \( \angle NKI \):

\[ \angle NKI = 14 \text{ degrees} \]

Thus, the correct response is:

The measurement of \( \angle NKI \) is 14 degrees.

Answer 2

Which set of side lengths does not form a triangle? All lengths are given in inches.(1 point)
Responses

18, 10, 8
18, 10, 8

11, 19, 9
11, 19, 9

9, 17, 11
9, 17, 11

18, 11, 8

Answer 3

To determine which set of side lengths does not form a triangle, we can use the triangle inequality theorem. This theorem states that for any three side lengths \(a\), \(b\), and \(c\) (where \(a\), \(b\), and \(c\) are the lengths of the sides of a triangle), the following must hold true:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

We will apply this theorem to each set of side lengths provided.

1. Set: 18, 10, 8

   • \(18 + 10 > 8\) → \(28 > 8\) (True)
   • \(18 + 8 > 10\) → \(26 > 10\) (True)
   • \(10 + 8 > 18\) → \(18 > 18\) (False)

   This set does not form a triangle.

2. Set: 11, 19, 9

   • \(11 + 19 > 9\) → \(30 > 9\) (True)
   • \(11 + 9 > 19\) → \(20 > 19\) (True)
   • \(19 + 9 > 11\) → \(28 > 11\) (True)

   This set forms a triangle.

3. Set: 9, 17, 11

   • \(9 + 17 > 11\) → \(26 > 11\) (True)
   • \(9 + 11 > 17\) → \(20 > 17\) (True)
   • \(17 + 11 > 9\) → \(28 > 9\) (True)

   This set forms a triangle.

4. Set: 18, 11, 8

   • \(18 + 11 > 8\) → \(29 > 8\) (True)
   • \(18 + 8 > 11\) → \(26 > 11\) (True)
   • \(11 + 8 > 18\) → \(19 > 18\) (True)

   This set forms a triangle.

Conclusion

The set of side lengths that does not form a triangle is:

18, 10, 8.

Answer 4

In 1–2 sentences, define parallel and perpendicular lines and explain the difference between the two.(2 points)

Answer 5

Parallel lines are lines in a plane that never intersect and are always the same distance apart, meaning they have the same slope. Perpendicular lines, on the other hand, are lines that intersect at a right angle (90 degrees), indicating they have slopes that are negative reciprocals of each other. The key difference is that parallel lines never meet, while perpendicular lines intersect at a specific angle.

Answer 6

In 1–2 sentences, distinguish between definitions and undefined terms in geometry. (2 points)

Answer 7

In geometry, definitions are precise explanations of concepts that are made using other previously established terms, providing clarity and context, such as defining a triangle as a three-sided polygon. Undefined terms, on the other hand, are foundational concepts that are not formally defined within the geometry framework but are understood intuitively, such as points, lines, and planes, serving as the building blocks for more complex definitions and theorems.

Answer 8

Joanna has four straws of different lengths. The table shows the lengths of the straws.

Straw A B C D
Length in inches 7 9 3 10Joanna makes as many triangles as she can using combinations of three of these straws. Determine how many different triangles she can make. What combinations of straws does she use? Show your work.(4 points)

Answer 9

To determine how many triangles Joanna can make with the straws of lengths \(A = 7\), \(B = 9\), \(C = 3\), and \(D = 10\), we need to use the triangle inequality theorem. According to this theorem, for any three lengths \(a\), \(b\), and \(c\) to form a triangle, the following conditions must be satisfied:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

We will check the combinations of three straws that can be formed from the given lengths:

All Combinations of Three Straws:

1. \(A, B, C\) → lengths 7, 9, 3
2. \(A, B, D\) → lengths 7, 9, 10
3. \(A, C, D\) → lengths 7, 3, 10
4. \(B, C, D\) → lengths 9, 3, 10

Now, let’s verify each combination to see if they satisfy the triangle inequality conditions:

1. Combination: \(A, B, C\) (7, 9, 3)

• Check:
  • \(7 + 9 > 3\) → \(16 > 3\) (True)
  • \(7 + 3 > 9\) → \(10 > 9\) (True)
  • \(9 + 3 > 7\) → \(12 > 7\) (True)

Result: This combination can form a triangle.

2. Combination: \(A, B, D\) (7, 9, 10)

• Check:
  • \(7 + 9 > 10\) → \(16 > 10\) (True)
  • \(7 + 10 > 9\) → \(17 > 9\) (True)
  • \(9 + 10 > 7\) → \(19 > 7\) (True)

Result: This combination can form a triangle.

3. Combination: \(A, C, D\) (7, 3, 10)

• Check:
  • \(7 + 3 > 10\) → \(10 > 10\) (False)
  • \(7 + 10 > 3\) → \(17 > 3\) (True)
  • \(3 + 10 > 7\) → \(13 > 7\) (True)

Result: This combination cannot form a triangle.

4. Combination: \(B, C, D\) (9, 3, 10)

• Check:
  • \(9 + 3 > 10\) → \(12 > 10\) (True)
  • \(9 + 10 > 3\) → \(19 > 3\) (True)
  • \(3 + 10 > 9\) → \(13 > 9\) (True)

Result: This combination can form a triangle.

Summary of Combinations

Based on the checks above, Joanna can form triangles from the following combinations:

1. \(A, B, C\) (7, 9, 3)
2. \(A, B, D\) (7, 9, 10)
3. \(B, C, D\) (9, 3, 10)

Final Count

Joanna can make 3 different triangles using the combinations:

1. 7, 9, 3
2. 7, 9, 10
3. 9, 3, 10