Question

example
diagram
(3x + 10)°
(2x + 30)°
angle relationship
consecutive exterior
solve for x
3x + 10 + 2x + 30 = 180
5x + 40 = 180
5x = 140
x = 28

a
diagram (crossed lines, angles x+25 and 2x+5)
angle relationship
vertical angles
solve for x
x + 25 = 2x + 5 (handwritten)

b
diagram (a straight line with a ray, angles 7x - 46 and 3x + 6)

c
diagram (two parallel lines cut by a transversal, angles 5x - 34 and 3x + 16)

Explanation:

Let's solve for \( x \) in each diagram (A, B, C) one by one.

Diagram A: Vertical Angles

Vertical angles are equal, so we set the two angle expressions equal to each other.

Step 1: Set up the equation

The angles are \( x + 25 \) and \( 2x + 5 \). Since they are vertical angles:
\[
x + 25 = 2x + 5
\]

Step 2: Solve for \( x \)

Subtract \( x \) from both sides:
\[
25 = x + 5
\]
Subtract 5 from both sides:
\[
x = 25 - 5 = 20
\]

Diagram B: Supplementary Angles (Linear Pair)

A linear pair of angles is supplementary, so their sum is \( 180^\circ \). The angles are \( 7x - 46 \) and \( 3x + 6 \).

Step 1: Set up the equation

\[
(7x - 46) + (3x + 6) = 180
\]

Step 2: Simplify and solve for \( x \)

Combine like terms:
\[
10x - 40 = 180
\]
Add 40 to both sides:
\[
10x = 220
\]
Divide by 10:
\[
x = \frac{220}{10} = 22
\]

Diagram C: Alternate Interior Angles (or Equal Angles, since lines are parallel)

Assuming the lines are parallel, the alternate interior angles are equal. The angles are \( 5x - 34 \) and \( 3x + 16 \).

Step 1: Set up the equation

\[
5x - 34 = 3x + 16
\]

Step 2: Solve for \( x \)

Subtract \( 3x \) from both sides:
\[
2x - 34 = 16
\]
Add 34 to both sides:
\[
2x = 50
\]
Divide by 2:
\[
x = \frac{50}{2} = 25
\]

Final Answers:

• Diagram A: \( \boldsymbol{x = 20} \)
• Diagram B: \( \boldsymbol{x = 22} \)
• Diagram C: \( \boldsymbol{x = 25} \)

Answer:

Let's solve for \( x \) in each diagram (A, B, C) one by one.

Diagram A: Vertical Angles

Vertical angles are equal, so we set the two angle expressions equal to each other.

Step 1: Set up the equation

The angles are \( x + 25 \) and \( 2x + 5 \). Since they are vertical angles:
\[
x + 25 = 2x + 5
\]

Step 2: Solve for \( x \)

Subtract \( x \) from both sides:
\[
25 = x + 5
\]
Subtract 5 from both sides:
\[
x = 25 - 5 = 20
\]

Diagram B: Supplementary Angles (Linear Pair)

A linear pair of angles is supplementary, so their sum is \( 180^\circ \). The angles are \( 7x - 46 \) and \( 3x + 6 \).

Step 1: Set up the equation

\[
(7x - 46) + (3x + 6) = 180
\]

Step 2: Simplify and solve for \( x \)

Combine like terms:
\[
10x - 40 = 180
\]
Add 40 to both sides:
\[
10x = 220
\]
Divide by 10:
\[
x = \frac{220}{10} = 22
\]

Diagram C: Alternate Interior Angles (or Equal Angles, since lines are parallel)

Assuming the lines are parallel, the alternate interior angles are equal. The angles are \( 5x - 34 \) and \( 3x + 16 \).

Step 1: Set up the equation

\[
5x - 34 = 3x + 16
\]

Step 2: Solve for \( x \)

Subtract \( 3x \) from both sides:
\[
2x - 34 = 16
\]
Add 34 to both sides:
\[
2x = 50
\]
Divide by 2:
\[
x = \frac{50}{2} = 25
\]

Final Answers:

• Diagram A: \( \boldsymbol{x = 20} \)
• Diagram B: \( \boldsymbol{x = 22} \)
• Diagram C: \( \boldsymbol{x = 25} \)