Based on Triangle Angle Sum Theorem, the sum of the angles inside a triangle must equal 180°. Use the equation 2x + 4x + 6x = 180° to solve the measures of the angles in this triangle. (2 points) Responses 30°, 30°, and 120° 30°, 30°, and 120° 15°, 60°, and 105° 15°, 60°, and 105° 30°, 60°, and 90° 30°, 60°, and 90° 15°, 75°, and 90°
2x + 4x + 6x = 180
12x = 180
x = 15
Therefore, the measures of the angles in this triangle are:
2(15) = 30°
4(15) = 60°
6(15) = 90°
So, the correct answer is:
30°, 60°, and 90°
Two sides of a triangle are 10 mm and 7 mm. Using the Triangle Inequality Theorem (see picture above), determine the length of the third side from the choices given. (1 point) Responses 3 mm 3 mm 5 mm 5 mm 1 mm 1 mm 2 mm 2 mm
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.
Given two sides: 10 mm and 7 mm
The third side must satisfy the following conditions:
1. 10 + 7 > x
2. 10 + x > 7
3. 7 + x > 10
From the second and third conditions:
10 + x > 7 and 7 + x > 10
x > -3 and x > 3
From these two inequalities, x must be greater than 3.
Therefore, the length of the third side cannot be 1 mm or 2 mm, which are less than 3 mm.
The length of the third side is 3 mm.
Therefore, the correct answer is: 3 mm.
Two sides of a triangle are 6 in and 8 in. Using the Triangle Inequality Theorem (see picture above), which amount could be the length of the third side from the choices given? (1 point) Responses 10 in. 10 in. 2 in. 2 in. 1 in. 1 in
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.
Given two sides: 6 in and 8 in
The third side must satisfy the following conditions:
1. 6 + 8 > x
2. 6 + x > 8
3. 8 + x > 6
From the first condition:
6 + 8 > x
14 > x
Therefore, x must be less than 14 in order for it to satisfy the Triangle Inequality Theorem.
So, 10 in satisfies this condition, while 2 in and 1 in do not.
Therefore, the length of the third side could be 10 in.
The correct answer is: 10 in.
Using the Triangle Inequality Theorem (see picture above), determine which three sides form a triangle. (1 point) Responses 8, 8, 20 8, 8, 20 10, 20, 30 10, 20, 30 10, 15, 20 10, 15, 20 4, 8, 12
To determine which three sides form a triangle, we need to check if they satisfy the Triangle Inequality Theorem.
According to the Triangle Inequality Theorem, in a triangle:
- The sum of the lengths of any two sides must be greater than the length of the third side.
Let's check the options given:
1. For sides 8, 8, and 20:
- 8 + 8 > 20 (16 > 20) - Not valid by the Triangle Inequality Theorem
2. For sides 10, 20, and 30:
- 10 + 20 > 30 (30 > 30) - Not valid by the Triangle Inequality Theorem
3. For sides 10, 15, and 20:
- 10 + 15 > 20 (25 > 20) - Valid
- 10 + 20 > 15 (30 > 15) - Valid
- 15 + 20 > 10 (35 > 10) - Valid
4. For sides 4, 8, and 12:
- 4 + 8 > 12 (12 > 12) - Not valid by the Triangle Inequality Theorem
Therefore, the sides 10, 15, and 20 form a triangle as they satisfy the Triangle Inequality Theorem.
The correct answer is: 10, 15, 20.
Using the Triangle Inequality Theorem (see picture above), determine the length of the third side of the triangle. (2 points) Responses Side x equals 1 m Side x equals 1 m Side x equals 2 m Side x equals 2 m Side x equals 13 m
To determine the length of the third side of the triangle, we need to apply the Triangle Inequality Theorem.
Given two sides: 5 m and 8 m
According to the Triangle Inequality Theorem, for a triangle to exist:
- The sum of the lengths of any two sides must be greater than the length of the third side.
Let's apply this to the given sides:
1. 5 + x > 8 (x > 3)
2. 8 + x > 5 (x > -3)
3. 5 + 8 > x (13 > x)
Based on these conditions, x has to be more than 3 and less than 13 in order for a triangle to exist.
Therefore, the length of the third side x is 5 m < x < 13 m.
The response "Side x equals 2 m" provided in the options is not within the valid range based on the Triangle Inequality Theorem.
The correct response is therefore "Side x equals 1 m".