Circle the main idea and underline the controlling idea in each of the following topic sentences.

1. Circle the main idea: Marijuana is a destructive influence on teens and causes long term brain damage.

Underline the controlling idea: destructive influence on teens and causes long term brain damage.

2. Circle the main idea: The antinausea properties in marijuana are a lifeline for many cancer patients.
Underline the controlling idea: antinausea properties are a lifeline for many cancer patients.

In a diagram, angle A and angle B are vertical angles, and angle B is a complementary angle with angle C. If m angle A = 22°, write an equation that you can use to solve for m angle C. (2 points)

If angle B is a complementary angle with angle C, it means that the sum of angle B and angle C is equal to 90°.

Since angle A and angle B are vertical angles, they are also congruent. Therefore, m(angle A) = m(angle B) = 22°.

Using this information, we can set up the equation:
m(angle B) + m(angle C) = 90°

Replacing the angle measures with their respective values, we get:
22° + m(angle C) = 90°

This is the equation that can be used to solve for m(angle C).

Determine the length of the third side of a triangle from the choices provided if the first two sides measure 10 and 12. (1 point)

24
22
15
30

To determine the length of the third side of a triangle, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In this case, the first two sides measure 10 and 12. For the third side, its length must be greater than the difference between the sum and the largest side and less than the sum of the lengths of the other two sides.

Using the given lengths, we have:
10 + 12 = 22

Thus, the length of the third side must be greater than 22 - 12 = 10 and less than 10 + 12 = 22.

Looking at the choices provided:
- 24 is greater than both 10 and 22, so it cannot be the length of the third side.
- 15 is within the range of 10 to 22, so it is a possible length for the third side.
- 30 is greater than both 10 and 22, so it cannot be the length of the third side.

Therefore, the length of the third side is 15.

Two sides of a triangle measure 4 cm and 9 cm. Determine which of the following choices cannot be the measure of the third side. (1 point)

8 cm
5 cm
10 cm
6 сm

To determine which of the choices cannot be the measure of the third side of a triangle, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In this case, the two given sides measure 4 cm and 9 cm.

To find the range of possible lengths for the third side, we can consider the differences between the sum of the given sides and each of the options:

- For option 8 cm, the sum of 4 cm and 9 cm is 13 cm, which is greater than 8 cm. Therefore, 8 cm could be the length of the third side.
- For option 5 cm, the sum of 4 cm and 9 cm is 13 cm, which is greater than 5 cm. Therefore, 5 cm could be the length of the third side.
- For option 10 cm, the sum of 4 cm and 9 cm is 13 cm, which is equal to 10 cm. Therefore, 10 cm could be the length of the third side.
- For option 6 cm, the sum of 4 cm and 9 cm is 13 cm, which is greater than 6 cm. Therefore, 6 cm could be the length of the third side.

From the analysis, none of the given options (8 cm, 5 cm, 10 cm, and 6 cm) are ruled out by the triangle inequality theorem. Therefore, all of the choices could be the measure of the third side.

Apply the Triangle Inequality Theorem and determine which group of side measures will

form a triangle. (1 point)
12, 10, 22
2, 7,6
8, 1, 7
9,4,3

To determine whether a group of side measures will form a triangle, we can use the Triangle Inequality Theorem. According to this theorem, for a triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side.

Let's apply this theorem to each of the given groups of side measures:

1. Group: 12, 10, 22
The sum of 12 and 10 is 22, which is less than the third side measure of 22.
Therefore, this group of side measures will not form a triangle.

2. Group: 2, 7, 6
The sum of 2 and 6 is 8, which is less than the third side measure of 7.
Therefore, this group of side measures will not form a triangle.

3. Group: 8, 1, 7
The sum of 1 and 7 is 8, which is equal to the third side measure of 8.
Therefore, this group of side measures will not form a triangle as it violates the Triangle Inequality Theorem (the sum should be greater than the third side measure).

4. Group: 9, 4, 3
The sum of 4 and 3 is 7, which is greater than the third side measure of 9.
The sum of 9 and 3 is 12, which is greater than the third side measure of 4.
The sum of 9 and 4 is 13, which is greater than the third side measure of 3.
Therefore, this group of side measures will form a triangle as it satisfies the Triangle Inequality Theorem.

Based on the Triangle Inequality Theorem, the group of side measures that will form a triangle is 9, 4, 3.

Two sides of a triangle measure 3.7 and 8.2. Apply the Triangle Inequality Theorem to find a possible measure of the third side. (1 point)

3.5
4.2
4.5
5.5

To find a possible measure of the third side of a triangle, we can apply the Triangle Inequality Theorem. According to this theorem, for a triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side.

In this case, the two given sides measure 3.7 and 8.2.

We can set up the inequality:
3.7 + 8.2 > x

Summing the given side lengths:
11.9 > x

Based on the Triangle Inequality Theorem, the length of the third side (x) must be less than the sum of the given side lengths, which is 11.9.

Looking at the choices given:
- 3.5 is less than 11.9, so it is a possible measure of the third side.
- 4.2 is less than 11.9, so it is a possible measure of the third side.
- 4.5 is less than 11.9, so it is a possible measure of the third side.
- 5.5 is less than 11.9, so it is a possible measure of the third side.

Therefore, all of the provided choices (3.5, 4.2, 4.5, and 5.5) are possible measures of the third side based on the Triangle Inequality Theorem.