The Triangle Angle Sum Theorem states that the sum of the angles inside a triangle must equal 180°. Using this theorem, find the measure of angle B.
An image of a triangle is shown. The triangle has A, B, and C marked on the corners. A is 100°, and C is 55°.
The Triangle Angle Sum Theorem states that the sum of the angles inside a triangle must equal 180°. Using this theorem, find the measure of angle C.
To find the measure of angle C, we will use the Triangle Angle Sum Theorem, which states that the sum of the angles in a triangle is 180°.
Given:
Angle A = 100°
Angle B = 25°
Let's substitute the given angle measures into the theorem:
Angle A + Angle B + Angle C = 180°
100° + 25° + Angle C = 180°
125° + Angle C = 180°
Angle C = 180° - 125°
Angle C = 55°
Therefore, the measure of angle C is 55°.
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.
30° 30° and 120°
15° 60° and 105°
30° 60° and 90°
15° 75° and 90°
Let's calculate the angle measures for each set of angles using the given equations:
1. For the angles 30°, 30°, and 120°:
Let the angle measures be 2x, 4x, and 6x.
2x + 4x + 6x = 180°
12x = 180°
x = 15°
Now, substitute the value of x back into the angle measures:
2x = 2(15) = 30°
4x = 4(15) = 60°
6x = 6(15) = 90°
Therefore, the angles in this triangle are 30°, 60°, and 90°.
2. For the angles 15°, 60°, and 105°:
Let the angle measures be 2x, 4x, and 6x.
2x + 4x + 6x = 180°
12x = 180°
x = 15°
Now, substitute the value of x back into the angle measures:
2x = 2(15) = 30°
4x = 4(15) = 60°
6x = 6(15) = 90°
Therefore, the angles in this triangle are 30°, 60°, and 90°.
3. For the angles 30°, 60°, and 90°:
This set of angles already satisfies the conditions of a triangle as their sum is 180°.
4. For the angles 15°, 75°, and 90°:
Let the angle measures be 2x, 4x, and 6x.
2x + 4x + 6x = 180°
12x = 180°
x = 15°
Now, substitute the value of x back into the angle measures:
2x = 2(15) = 30°
4x = 4(15) = 60°
6x = 6(15) = 90°
Therefore, the angles in this triangle are 30°, 60°, and 90°.
By applying the Triangle Angle Sum Theorem to each case, we have determined the measures of the angles in these triangles.
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.
3 mm
5 mm
1 mm
2 mm
The Triangle Inequality Theorem states that 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, two sides of the triangle are 10 mm and 7 mm, and we need to determine the possible length of the third side.
Let x represent the length of the third side.
According to the Triangle Inequality Theorem:
- The sum of any two sides must be greater than the third side.
- So, we have:
10 + 7 > x
17 > x
Because x must be greater than 17 to satisfy the Triangle Inequality Theorem, the only option from the given choices that is greater than 17 mm is 5 mm.
Therefore, the length of the third side must be 5 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?
10 in.
2 in.
1 in.
To determine which length could be the third side of the triangle using the Triangle Inequality Theorem, we need to check if the sum of the two given sides is greater than the length of the third side.
Given two sides of lengths 6 in. and 8 in., let's denote the length of the third side as x.
According to the Triangle Inequality Theorem:
- The sum of any two sides must be greater than the length of the third side.
- So, we have:
6 + 8 > x
14 > x
Among the choices given:
- 10 in. satisfies the inequality as 14 > 10.
- 2 in. does not satisfy the inequality as 14 < 2.
- 1 in. does not satisfy the inequality as 14 < 1.
Therefore, the length of 10 in. could be the length of the third side of the triangle, based on the Triangle Inequality Theorem.
Using the Triangle Inequality Theorem (see picture above), determine which three sides form a triangle.
8, 8. 20
10, 20, 30
10, 15, 20
4, 8, 12
According to the Triangle Inequality Theorem, for a triangle to be formed, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's check each set of three sides given:
1. 8, 8, 20
The sum of the two equal sides, 8 and 8, is 16. Is 16 greater than 20? No, this set of sides cannot form a triangle.
2. 10, 20, 30
The sum of the two smaller sides, 10 and 20, is 30. Is 30 greater than the largest side, 30? No, this set of sides cannot form a triangle.
3. 10, 15, 20
The sum of the two smaller sides, 10 and 15, is 25. Is 25 greater than the largest side, 20? Yes, this set of sides can form a triangle.
4. 4, 8, 12
The sum of the two smaller sides, 4 and 8, is 12. Is 12 greater than the largest side, 12? No, this set of sides cannot form a triangle.
Therefore, the set of sides 10, 15, 20 can form a triangle based on the Triangle Inequality Theorem.