Complete the condition statements that must be met in order for three side lengths—a, b, and c—to create a triangle.

a < b + c and a > |b - c|

Use the image to answer the question.

Complete the inequality so that it represents the whole-number values that side a could be to create a triangle.
An illustration of a triangle shows an equation along the base as b equals 6 and the hypotenuse as c equals 7. The third side on the triangle is labeled as a.

__< a < __

2 < a < 13

Apply the Triangle Inequality Theorem to determine which three side lengths form a triangle.

8, 4, 12

20, 10, 30

10, 20, 15

8, 8, 20

To determine if the three side lengths form a triangle, we need to apply the Triangle Inequality Theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

1. For the first set, (8, 4, 12):
8 + 4 = 12 < 12 (not a triangle)

2. For the second set, (20, 10, 30):
20 + 10 = 30, but 30 > 30 (not a triangle)

3. For the third set, (10, 20, 15):
10 + 15 = 25 > 20, 10 + 20 = 30 > 15, 15 + 20 = 35 > 10 (forms a triangle)

4. For the fourth set, (8, 8, 20):
8 + 8 = 16 < 20, 8 + 20 = 28 > 8, 8 + 20 = 28 > 8 (not a triangle)

Therefore, only the set (10, 20, 15) forms a triangle according to the Triangle Inequality Theorem.

Apply the Triangle Inequality Theorem to determine the possible whole number measures of the third side of a triangle if the first two sides measure 6 and 2. List them in ascending order.

The measure of the third side could be
__, __, or __.

If the first two sides measure 6 and 2, according to the Triangle Inequality Theorem, the possible whole number measures of the third side of a triangle in ascending order would be:

3, 4, or 5

So, the measure of the third side could be 3, 4, or 5.

Use the image to answer the question.

{A number line with arrows at both the ends ranges from negative 5 to 5 in unit increments. The number line shows closed points on the following numbers along with letters labeled above the points: A for negative 4, B for negative 1, C for 3, and D for 4.}

A triangle has no congruent sides. The lengths of two sides of the triangle are AB and BD . Use the number line above to determine the lengths of AB and BD . Then, use the Triangle Inequality Theorem to determine which two segments have lengths that could be used as the third side of the triangle. Select a response containing 2 possible answers.

A. BC and AD

B. CD and AC

C. BC and CD

D. AC and BC

On the number line provided, the lengths of AB and BD can be determined as follows:

AB = |A - B| = |-4 - (-1)| = |-4 + 1| = 3
BD = |B - D| = |-1 - 4| = |-1 + 4| = 3

So, the lengths of AB and BD are both 3.

According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, we need to check which pairs of segments satisfy this condition.

For the triangle with sides AB, BD, and CD:
AB + BD = 3 + 3 = 6
CD = 3
6 > 3

For the triangle with sides AB, BD, and BC:
AB + BD = 3 + 3 = 6
BC = 5
6 > 5

Therefore, the lengths of CD and BC could be used as the third side in the triangle. The possible answers are:
A. BC and AD
C. BC and CD