Triangle Angle Sum

Show your work for each problem.

1.
What is the sum of all interior angles in any triangle?
1. a)
2. b)
3. c)
4. d)
Answer:
The Triangle Angle Sum Theorem states that the sum of all three interior angles in any triangle is always . This is true for all triangles regardless of their shape.
2.
A triangle has angles of and . What is the third angle?
Answer: 60
. The three angles must sum to , so we subtract the known angles to find the missing one.
3.
In an equilateral triangle, all three angles are equal. What is the measure of each angle?
1. a)
2. b)
3. c)
4. d)
Answer:
In an equilateral triangle, all angles are equal. Since they must sum to : each angle = .
4.
A triangle has angles of , , and an unknown angle . Find .
Answer: 45
1. What do all three angles sum to? 180
2. What is the sum of ? 135
3. What is ? 45
5.
In a right triangle, one acute angle is . What is the other acute angle?
Answer: 65
In a right triangle: , so . Alternatively, since the two acute angles sum to : .
6.
An isosceles triangle has two equal angles of each. Find the third angle.
Answer: 40
1. What is the sum of the two equal angles? 140
2. What must all three angles sum to? 180
3. What is ? 40
7.
A triangle has angles of , , and . What type of angle is ?
1. a)Obtuse ()
2. b)Acute ()
3. c)Acute ()
4. d)Right ()
Answer: Acute ()
. Since , it's an acute angle.
8.
The angles of a triangle are in the ratio . What is the largest angle?
Answer: 90
Let the angles be , , . Then , so , giving . The largest angle is .
9.
In a triangle, one angle is twice another, and the third angle is . Find all three angles.
Answer: 40, ~46.67, ~93.33
1. Let one angle be . What is the 'twice' angle in terms of ? 2x
2. Write the equation: 180
3. Simplify: . What is ? 140
4. What is (rounded to nearest whole number)? 47
10.
Can a triangle have angles of , , and ?
1. a)No, they sum to
2. b)No, a triangle cannot have two equal angles
3. c)No, they sum to
4. d)Yes, they sum to
Answer: No, they sum to
. Since this exceeds , these angles cannot form a valid triangle.
11.
In a triangle, one angle is more than another. The third angle is . Find the smallest angle.
Answer: 40
Let the smallest angle be . Then: , so , giving and .
12.
In a triangle, the angles are , , and . Find the value of and all three angles.
Answer: x=25, angles: 60, 55, 65
1. Write the sum equation: 180
2. Combine like terms: How many 's total? 6
3. Combine constants: 30
4. Solve . What is ? 25
13.
Why can't a triangle have two obtuse angles?
1. a)Triangles can only have acute angles
2. b)Two obtuse angles would sum to more than
3. c)The sides would not connect
4. d)Obtuse angles are not allowed in triangles
Answer: Two obtuse angles would sum to more than
Each obtuse angle is greater than . If two angles were both obtuse, their sum alone would exceed (e.g., ), leaving no room for a third positive angle.
14.
An isosceles triangle has a vertex angle of . What is each base angle?
Answer: 67
Let each base angle be . Then , so , meaning and .
15.
A right triangle has one angle that is less than three times another. Find both acute angles.
Answer: 26.25, 63.75
1. In a right triangle, what do the two acute angles sum to? 90
2. Let one acute angle be . The other is . Write the equation: 90
3. Simplify: . What is ? 105
4. What is ? 26.25
16.
The exterior angle of a triangle is . The two non-adjacent interior angles are in the ratio . Find all three interior angles.
Answer: 50, 52, 78
1. The exterior angle and its adjacent interior angle are supplementary. What is the adjacent interior angle? 50
2. The two non-adjacent angles sum to the exterior angle. What do they sum to? 130
3. If the ratio is , let them be and . Solve . What is ? 26
4. What is the smaller non-adjacent angle ()? 52