The following side lengths, in meters, were given to a carpenter to build a front porch with a triangular design. The carpenter needs to determine which set of lengths will make a triangle to be able to use it in his design. (Hint: Use the Triangle Inequality Theorem)

Question

Option 1: Side lengths: 3, 3, 6
Option 2: Side lengths: 6, 8, 10
Option 3: Side lengths: 4, 6, 13(1 point)
Responses

Option 1
Option 1

Option 2
Option 2

Option 3

Answer 1

To determine which set of lengths can form a triangle, we can use the Triangle Inequality Theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold true:

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

Let's check each option using these inequalities.

Option 1: Side lengths 3, 3, 6

• \(3 + 3 = 6\) (Not greater than 6)
• \(3 + 6 = 9\) (Greater than 3)
• \(3 + 6 = 9\) (Greater than 3)

Here, the first condition fails (\(3 + 3\) is not greater than \(6\)). Therefore, Option 1 cannot form a triangle.

Option 2: Side lengths 6, 8, 10

• \(6 + 8 = 14\) (Greater than 10)
• \(6 + 10 = 16\) (Greater than 8)
• \(8 + 10 = 18\) (Greater than 6)

All conditions are satisfied. Therefore, Option 2 can form a triangle.

Option 3: Side lengths 4, 6, 13

• \(4 + 6 = 10\) (Not greater than 13)
• \(4 + 13 = 17\) (Greater than 6)
• \(6 + 13 = 19\) (Greater than 4)

Here, the first condition fails (\(4 + 6\) is not greater than \(13\)). Therefore, Option 3 cannot form a triangle.

Conclusion

The only option that can form a triangle is Option 2 (side lengths: 6, 8, 10).

Answer 2

are you sure

Answer 3

Yes, let's recap the analysis again, just to ensure clarity and accuracy about the applicability of the Triangle Inequality Theorem to each set of lengths.

Option 1: Side lengths 3, 3, 6

Using the Triangle Inequality Theorem:

1. \(3 + 3 > 6\) → \(6 > 6\) (False)
2. \(3 + 6 > 3\) → \(9 > 3\) (True)
3. \(3 + 6 > 3\) → \(9 > 3\) (True)

Since the first condition is not satisfied, Option 1 cannot form a triangle.

Option 2: Side lengths 6, 8, 10

Using the Triangle Inequality Theorem:

1. \(6 + 8 > 10\) → \(14 > 10\) (True)
2. \(6 + 10 > 8\) → \(16 > 8\) (True)
3. \(8 + 10 > 6\) → \(18 > 6\) (True)

All conditions are satisfied, so Option 2 can form a triangle.

Option 3: Side lengths 4, 6, 13

Using the Triangle Inequality Theorem:

1. \(4 + 6 > 13\) → \(10 > 13\) (False)
2. \(4 + 13 > 6\) → \(17 > 6\) (True)
3. \(6 + 13 > 4\) → \(19 > 4\) (True)

Since the first condition is not satisfied, Option 3 cannot form a triangle.

Final Conclusion

After reviewing all the calculations:

• Option 1 does not satisfy the triangle inequality.
• Option 2 does satisfy the triangle inequality.
• Option 3 does not satisfy the triangle inequality.

The set of side lengths that can form a triangle is indeed Option 2 (6, 8, 10). Thank you for your patience, and if you have any further questions or need additional clarification, feel free to ask!

Answer 4

Determine if each of the following statements is always true, sometimes
true or never true.
A. The sum of the measures of two supplementary angles is 180°.
B. The sum of complimentary angles is 180°.
C. Two adjacent angles are complementary.

(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
A. The sum of the measures of two supplementary angles is 180°
B. The sum of complimentary angles is 180°.
C. Two adjacent angles are complementary.

Answer 5

Let's evaluate each statement individually:

A. The sum of the measures of two supplementary angles is 180°.

• This statement is always true. By definition, supplementary angles are two angles whose measures add up to 180°.

B. The sum of complementary angles is 180°.

• This statement is never true. By definition, complementary angles are two angles whose measures add up to 90°.

C. Two adjacent angles are complementary.

• This statement is sometimes true. Two adjacent angles can be complementary (if they sum to 90°), but they can also be supplementary or of other measures. Therefore, not all adjacent angles are necessarily complementary.

Summary of Responses:

• A: Always true
• B: Never true
• C: Sometimes true

You can insert these responses into the designated inputs:

• A: Always true
• B: Never true
• C: Sometimes true