Triangle Angle Sum

Show your work for each problem.

1.
What is the sum of all interior angles in any triangle?
1. a) $270°$
2. b) $360°$
3. c) $180°$
4. d) $90°$
Answer: $180°$
The Triangle Angle Sum Theorem states that the sum of all three interior angles in any triangle is always $180°$ . This is true for all triangles regardless of their shape.
2.
A triangle has angles of $50°$ and $70°$ . What is the third angle?
Answer: 60
$180° - 50° - 70° = 60°$ . The three angles must sum to $180°$ , 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) $90°$
2. b) $120°$
3. c) $45°$
4. d) $60°$
Answer: $60°$
In an equilateral triangle, all angles are equal. Since they must sum to $180°$ : each angle = $180° \div 3 = 60°$ .
4.
A triangle has angles of $80°$ , $55°$ , and an unknown angle $x$ . Find $x$ .
Answer: 45
1. What do all three angles sum to? 180
2. What is the sum of $80° + 55°$ ? 135
3. What is $180° - 135°$ ? 45
5.
In a right triangle, one acute angle is $25°$ . What is the other acute angle?
Answer: 65
In a right triangle: $90° + 25° + x = 180°$ , so $x = 180° - 90° - 25° = 65°$ . Alternatively, since the two acute angles sum to $90°$ : $90° - 25° = 65°$ .
6.
An isosceles triangle has two equal angles of $70°$ 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 $180° - 140°$ ? 40
7.
A triangle has angles of $35°$ , $95°$ , and $x$ . What type of angle is $x$ ?
1. a)Acute ( $30°$ )
2. b)Acute ( $50°$ )
3. c)Right ( $90°$ )
4. d)Obtuse ( $130°$ )
Answer: Acute ( $50°$ )
$x = 180° - 35° - 95° = 50°$ . Since $50° < 90°$ , it's an acute angle.
8.
The angles of a triangle are in the ratio $2:3:4$ . What is the largest angle, in degrees?
Answer: 80
Call the angles $2x$ , $3x$ , $4x$ . Then $2x + 3x + 4x = 180°$ , so $9x = 180°$ and $x = 20°$ . The angles are $40°$ , $60°$ and $80°$ , so the largest is $4x = 80°$ .
9.
In a triangle, one angle is twice another, and the third angle is $30°$ . Find all three angles.
Answer: 50, 100, 30
1. What do the three angles add up to? 180
2. Call the smaller unknown angle $x$ , so the other one is $2x$ . Then $x + 2x + 30° = 180°$ . What is $3x$ ? 150
3. What is $x$ ? 50
4. What is the other unknown angle, $2x$ ? 100
10.
Can a triangle have angles of $60°$ , $70°$ , and $60°$ ?
1. a)Yes, they sum to $180°$
2. b)No, a triangle cannot have two equal angles
3. c)No, they sum to $170°$
4. d)No, they sum to $190°$
Answer: No, they sum to $190°$
$60° + 70° + 60° = 190°$ . Since this exceeds $180°$ , these angles cannot form a valid triangle.
11.
In a triangle, one angle is $20°$ more than another. The third angle is $80°$ . Find the smallest angle.
Answer: 40
Let the smallest angle be $x$ . Then: $x + (x + 20°) + 80° = 180°$ , so $2x + 100° = 180°$ , giving $2x = 80°$ and $x = 40°$ .
12.
In a triangle, the angles are $(2x + 10)°$ , $(3x - 20)°$ , and $(x + 40)°$ . Find the value of $x$ and all three angles.
Answer: x=25, angles: 60, 55, 65
1. Write the sum equation: $(2x + 10) + (3x - 20) + (x + 40) = ?$ 180
2. Combine like terms: How many $x$ 's total? 6
3. Combine constants: $10 - 20 + 40 = ?$ 30
4. Solve $6x + 30 = 180$ . What is $x$ ? 25
13.
Why can't a triangle have two obtuse angles?
1. a)Every triangle needs at least one right angle
2. b)Obtuse angles are not allowed in triangles
3. c)Two obtuse angles would sum to more than $180°$
4. d)Triangles can only have acute angles
Answer: Two obtuse angles would sum to more than $180°$
Each obtuse angle is greater than $90°$ . If two angles were both obtuse, their sum alone would exceed $180°$ (e.g., $91° + 91° = 182°$ ), leaving no room for a third positive angle.
14.
An isosceles triangle has a vertex angle of $46°$ . What is each base angle?
Answer: 67
Let each base angle be $x$ . Then $46° + x + x = 180°$ , so $46° + 2x = 180°$ , meaning $2x = 134°$ and $x = 67°$ .
15.
In a right triangle, one acute angle is $10°$ less than three times the other. Find both acute angles.
Answer: 25, 65
1. In a right triangle, what do the two acute angles add up to? 90
2. Call one acute angle $x$ ; the other is $3x - 10$ . Then $x + (3x - 10) = 90$ . What is $4x$ ? 100
3. What is $x$ ? 25
4. What is the other acute angle, $3x - 10$ ? 65
16.
An exterior angle of a triangle measures $130°$ . The two non-adjacent interior angles are in the ratio $2:3$ . 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 three interior angles sum to $180°$ . What do the two non-adjacent angles add up to? 130
3. If the ratio is $2:3$ , let them be $2x$ and $3x$ . Solve $2x + 3x = 130$ . What is $x$ ? 26
4. What is the smaller non-adjacent angle ( $2x$ )? 52