Question

directions: solve for x in each of the triangles below. the triangle sum theorem states that the interior angles of a triangle always add up to 180°. your first step should be to set up an equation where the sum of the angles adds up to 180. solve the equation for x, then plug that value back in to the expressions to find the measure of the missing angles.
1)
triangle with vertices a, b, c. angle at a: x, angle at b: 3x, angle at c: x.
∠a = ____
x = __
∠b = ____
∠c = ____
2)
triangle with vertices a, b, c. angle at a: x+52, angle at b: x+8, angle at c: x.
∠a = ____
x = __
∠b = ____
∠c = ____
3)
triangle with vertices a, b, c. angle at a: 5x, angle at b: 4x, angle at c: 3x.
∠a = ____
x = __
∠b = ____
∠c = ____
4)
triangle with vertices a, b, c. angle at a: x+60, angle at b: 4x, angle at c: x.
∠a = ____
x = __
∠b = ____
∠c = ____
5)
triangle with vertices a, b, c. angle at a: 4x, angle at b: 2x, angle at c: 3x.
∠a = ____
x = __
∠b = ____
∠c = ____
6)
triangle with vertices a, b, c. angle at a: x, angle at b: 3x, angle at c: 6x+10.
∠a = ____
x = __
∠b = ____
∠c = ____
7)
triangle with vertices a, b, c. angle at a: 2x+30, angle at b: 2x+25, angle at c: x.
∠a = ____
x = __
∠b = ____
∠c = ____
8)
triangle with vertices a, b, c. angle at a: 2x, angle at b: 3x-30, angle at c: 2x.
∠a = ____
x = __
∠b = ____
∠c = ____
9)
triangle with vertices a, b, c. angle at a: x, angle at b: 2x, angle at c: 2x+75.
∠a = ____
x = __
∠b = ____
∠c = ____
10)
triangle with vertices a, b, c. angle at a: x+40, angle at b: 2x+20, angle at c: 2x-20.
∠a = ____
x = __
∠b = ____
∠c = ____
11)
triangle with vertices a, b, c. angle at a: 5x, angle at b: 4x+24, angle at c: 3x.
∠a = ____
x = __
∠b = ____
∠c = ____
12)
triangle with vertices a, b, c. angle at a: 2x, angle at b: 2x, angle at c: x.
∠a = ____
x = __
∠b = ____
∠c = ____

Explanation:

Step1: Solve for \( x \) in Triangle 1

The angles are \( x \) (∠A), \( 3x \) (∠B), and \( x \) (∠C). By the Triangle Sum Theorem:
\( x + 3x + x = 180 \)
\( 5x = 180 \)
\( x = \frac{180}{5} = 36 \)

Step2: Find Angles for Triangle 1

- ∠A: \( x = 36^\circ \)
- ∠B: \( 3x = 3 \times 36 = 108^\circ \)
- ∠C: \( x = 36^\circ \)

Step1: Solve for \( x \) in Triangle 2

Angles: \( x + 52 \) (∠A), \( x + 8 \) (∠B), \( x \) (∠C).
\( (x + 52) + (x + 8) + x = 180 \)
\( 3x + 60 = 180 \)
\( 3x = 120 \)
\( x = 40 \)

Step2: Find Angles for Triangle 2

- ∠A: \( x + 52 = 40 + 52 = 92^\circ \)
- ∠B: \( x + 8 = 40 + 8 = 48^\circ \)
- ∠C: \( x = 40^\circ \)

Step1: Solve for \( x \) in Triangle 3

Angles: \( 5x \) (∠A), \( 4x \) (∠B), \( 3x \) (∠C).
\( 5x + 4x + 3x = 180 \)
\( 12x = 180 \)
\( x = \frac{180}{12} = 15 \)

Step2: Find Angles for Triangle 3

- ∠A: \( 5x = 5 \times 15 = 75^\circ \)
- ∠B: \( 4x = 4 \times 15 = 60^\circ \)
- ∠C: \( 3x = 3 \times 15 = 45^\circ \)

Step1: Solve for \( x \) in Triangle 4

Angles: \( x + 60 \) (∠A), \( 4x \) (∠B), \( x \) (∠C).
\( (x + 60) + 4x + x = 180 \)
\( 6x + 60 = 180 \)
\( 6x = 120 \)
\( x = 20 \)

Step2: Find Angles for Triangle 4

- ∠A: \( x + 60 = 20 + 60 = 80^\circ \)
- ∠B: \( 4x = 4 \times 20 = 80^\circ \)
- ∠C: \( x = 20^\circ \)

Step1: Solve for \( x \) in Triangle 5

Angles: \( 2x \) (∠A), \( 4x \) (∠B), \( 3x \) (∠C).
\( 2x + 4x + 3x = 180 \)
\( 9x = 180 \)
\( x = 20 \)

Step2: Find Angles for Triangle 5

- ∠A: \( 2x = 2 \times 20 = 40^\circ \)
- ∠B: \( 4x = 4 \times 20 = 80^\circ \)
- ∠C: \( 3x = 3 \times 20 = 60^\circ \)

Step1: Solve for \( x \) in Triangle 6

Angles: \( x \) (∠A), \( 3x \) (∠B), \( 6x + 10 \) (∠C).
\( x + 3x + (6x + 10) = 180 \)
\( 10x + 10 = 180 \)
\( 10x = 170 \)
\( x = 17 \)

Step2: Find Angles for Triangle 6

- ∠A: \( x = 17^\circ \)
- ∠B: \( 3x = 3 \times 17 = 51^\circ \)
- ∠C: \( 6x + 10 = 6 \times 17 + 10 = 112^\circ \)

Step1: Solve for \( x \) in Triangle 7

Angles: \( 2x + 30 \) (∠A), \( 2x + 25 \) (∠B), \( x \) (∠C).
\( (2x + 30) + (2x + 25) + x = 180 \)
\( 5x + 55 = 180 \)
\( 5x = 125 \)
\( x = 25 \)

Step2: Find Angles for Triangle 7

- ∠A: \( 2x + 30 = 2 \times 25 + 30 = 80^\circ \)
- ∠B: \( 2x + 25 = 2 \times 25 + 25 = 75^\circ \)
- ∠C: \( x = 25^\circ \)

Step1: Solve for \( x \) in Triangle 8

Angles: \( 2x \) (∠A), \( 3x - 30 \) (∠B), \( 2x \) (∠C).
\( 2x + (3x - 30) + 2x = 180 \)
\( 7x - 30 = 180 \)
\( 7x = 210 \)
\( x = 30 \)

Step2: Find Angles for Triangle 8

- ∠A: \( 2x = 2 \times 30 = 60^\circ \)
- ∠B: \( 3x - 30 = 3 \times 30 - 30 = 60^\circ \)
- ∠C: \( 2x = 60^\circ \)

Step1: Solve for \( x \) in Triangle 9

Angles: \( x \) (∠A), \( 2x \) (∠B), \( 2x + 75 \) (∠C).
\( x + 2x + (2x + 75) = 180 \)
\( 5x + 75 = 180 \)
\( 5x = 105 \)
\( x = 21 \)

Step2: Find Angles for Triangle 9

- ∠A: \( x = 21^\circ \)
- ∠B: \( 2x = 2 \times 21 = 42^\circ \)
- ∠C: \( 2x + 75 = 2 \times 21 + 75 = 117^\circ \)

Step1: Solve for \( x \) in Triangle 10

Angles: \( x + 40 \) (∠A), \( 2x + 20 \) (∠B), \( 2x - 20 \) (∠C).
\( (x + 40) + (2x + 20) + (2x - 20) = 180 \)
\( 5x + 40 = 180 \)
\( 5x = 140 \)
\( x = 28 \)

Step2: Find Angles for Triangle 10

- ∠A: \( x + 40 = 28 + 40 = 68^\circ \)
- ∠B: \( 2x + 20 = 2 \times 28 + 20 = 76^\circ \)
- ∠C: \( 2x - 20 = 2 \times 28 - 20 = 36^\circ \)

Step1: Solve for \( x \) in Triangle 11

Angles: \( 5x \) (∠A), \( 4x + 24 \) (∠B), \( 3x \) (∠C).
\( 5x + (4x + 24) + 3x = 180 \)
\( 12x + 24 = 180 \)
\( 12x = 156 \)
\( x = 13 \)

Step2: Find Angles for Triangle 11

- ∠A: \( 5x = 5 \times 13 = 65^\circ \)
- ∠B: \( 4x + 24 = 4 \times 13 + 24 = 76^\circ \)
- ∠C: \( 3x = 3 \times 13 = 39^\circ \)

Step1: Solve for \( x \) in Triangle 12

Angles: \( 2x \) (∠A), \( 2x \) (∠B), \( x \) (∠C).
\( 2x + 2x + x = 180 \)
\( 5x = 180 \)
\( x = 36 \)

Step2: Find Angles for Triangle 12

- ∠A: \( 2x = 2 \times 36 = 72^\circ \)
- ∠B: \( 2x = 72^\circ \)
- ∠C: \( x = 36^\circ \)

Answer:

Triangle 1:

\( x = 36 \), \( \angle A = 36^\circ \), \( \angle B = 108^\circ \), \( \angle C = 36^\circ \)

Triangle 2:

\( x = 40 \), \( \angle A = 92^\circ \), \( \angle B = 48^\circ \), \( \angle C = 40^\circ \)

Triangle 3:

\( x = 15 \), \( \angle A = 75^\circ \), \( \angle B = 60^\circ \), \( \angle C = 45^\circ \)

Triangle 4:

\( x = 20 \), \( \angle A = 80^\circ \), \( \angle B = 80^\circ \), \( \angle C = 20^\circ \)

Triangle 5:

\( x = 20 \), \( \angle A = 40^\circ \), \( \angle B = 80^\circ \), \( \angle C = 60^\circ \)

Triangle 6:

\( x = 17 \), \( \angle A = 17^\circ \), \( \angle B = 51^\circ \), \( \angle C = 112^\circ \)

Triangle 7:

\( x = 25 \), \( \angle A = 80^\circ \), \( \angle B = 75^\circ \), \( \angle C = 25^\circ \)

Triangle 8:

\( x = 30 \), \( \angle A = 60^\circ \), \( \angle B = 60^\circ \), \( \angle C = 60^\circ \)

Triangle 9:

\( x = 21 \), \( \angle A = 21^\circ \), \( \angle B = 42^\circ \), \( \angle C = 117^\circ \)

Triangle 10:

\( x = 28 \), \( \angle A = 68^\circ \), \( \angle B = 76^\circ \), \( \angle C = 36^\circ \)

Triangle 11:

\( x = 13 \), \( \angle A = 65^\circ \), \( \angle B = 76^\circ \), \( \angle C = 39^\circ \)

Triangle 12:

\( x = 36 \), \( \angle A = 72^\circ \), \( \angle B = 72^\circ \), \( \angle C = 36^\circ \)