Use the Triangle Angle Sum Theorem:

$45° + 65° + x = 180°$

$110° + x = 180°$

$x = 70°$

Answer: $70°$

Step 1: Use Triangle Angle Sum: The sum of angles = 180° A + B + C = 180°

Step 2: Substitute the expressions: 3x + 2x + x = 180°

Step 3: Combine like terms: 6x = 180°

Step 4: Solve for x: x = 180°/6 x = 30°

Step 5: Find each angle: Angle A = 3x = 3(30°) = 90° Angle B = 2x = 2(30°) = 60° Angle C = x = 30°

Step 6: Verify: 90° + 60° + 30° = 180° ✓

Answer: x = 30°, angles are 90°, 60°, and 30°

Step 1: Recall Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles

Step 2: Set up the equation: Exterior angle = Sum of two remote interior angles 125° = 55° + x

Step 3: Solve for x: x = 125° - 55° x = 70°

Step 4: Verify using Triangle Angle Sum: Adjacent interior angle = 180° - 125° = 55° All three interior angles: 55° + 55° + 70° = 180° ✓

Answer: The other non-adjacent interior angle is 70°

Let the angles be $2x$, $3x$, and $4x$.

$2x + 3x + 4x = 180$

$9x = 180$

$x = 20$

The three angles are:

• $2(20) = 40°$
• $3(20) = 60°$
• $4(20) = 80°$

Check: $40 + 60 + 80 = 180$ ✓

Answer: $40°$, $60°$, $80°$

Let the angles be $2x$, $3x$, and $4x$.

$2x + 3x + 4x = 180$

$9x = 180$

$x = 20$

The three angles are:

• $2(20) = 40°$
• $3(20) = 60°$
• $4(20) = 80°$

Check: $40 + 60 + 80 = 180$ ✓

Answer: $40°$, $60°$, $80°$

Step 1: Use Exterior Angle Theorem

The exterior angle equals the sum of remote interior angles: $125° = 55° + x$ $x = 70°$

So one remote interior angle is $70°$.

Step 2: Find the third angle (adjacent to exterior)

The exterior angle and its adjacent interior angle are supplementary: $125° + y = 180°$ $y = 55°$

The three angles are: $55°$, $70°$, $55°$

Check: $55 + 70 + 55 = 180$ ✓

Answer: The three angles are $55°$, $70°$, and $55°$

Step 1: Use Exterior Angle Theorem

The exterior angle equals the sum of remote interior angles: $125° = 55° + x$ $x = 70°$

So one remote interior angle is $70°$.

Step 2: Find the third angle (adjacent to exterior)

The exterior angle and its adjacent interior angle are supplementary: $125° + y = 180°$ $y = 55°$

The three angles are: $55°$, $70°$, $55°$

Check: $55 + 70 + 55 = 180$ ✓

Answer: The three angles are $55°$, $70°$, and $55°$

Step 1: Use Triangle Angle Sum Theorem: X + Y + Z = 180°

Step 2: Substitute the expressions: (2a + 10) + (3a - 5) + (a + 25) = 180

Step 3: Combine like terms: 2a + 3a + a + 10 - 5 + 25 = 180 6a + 30 = 180

Step 4: Solve for a: 6a = 180 - 30 6a = 150 a = 25

Step 5: Find each angle: Angle X = 2a + 10 = 2(25) + 10 = 50 + 10 = 60° Angle Y = 3a - 5 = 3(25) - 5 = 75 - 5 = 70° Angle Z = a + 25 = 25 + 25 = 50°

Step 6: Verify: 60° + 70° + 50° = 180° ✓

Answer: a = 25, angles are 60°, 70°, and 50°

Step 1: Define variables: Let x = measure of each base angle In an isosceles triangle, the two base angles are equal Vertex angle = 2x (given as twice a base angle)

Step 2: Apply Triangle Angle Sum: Base angle + Base angle + Vertex angle = 180° x + x + 2x = 180°

Step 3: Solve: 4x = 180° x = 45°

Step 4: Find all angles: Each base angle = x = 45° Vertex angle = 2x = 2(45°) = 90°

Step 5: Verify: 45° + 45° + 90° = 180° ✓ This is a 45-45-90 triangle (an isosceles right triangle)

Step 6: Check the relationship: Vertex angle = 90° = 2(45°) ✓ The vertex angle is indeed twice each base angle

Answer: The base angles are 45° each, and the vertex angle is 90°