Question

directions: please use a separate paper to show work and do the problems!!! use the triangle sum theorem: the sum of the interior angles of a triangle is 180°. for each problem: 1. write an equation 2. solve for x 3. find the measure of each angle show all work.

Explanation:

Since the specific triangle angle measures (in terms of \( x \)) are not provided, we can't solve a particular problem. However, here's a general example of how to use the Triangle Sum Theorem:

Example:

Suppose a triangle has angles \( (2x + 5)^\circ \), \( (3x - 10)^\circ \), and \( (x + 15)^\circ \).

Step 1: Write the equation

Using the Triangle Sum Theorem (\( \text{Sum of interior angles} = 180^\circ \)):
\( (2x + 5) + (3x - 10) + (x + 15) = 180 \)

Step 2: Solve for \( x \)

Combine like terms:
\( 2x + 3x + x + 5 - 10 + 15 = 180 \)
\( 6x + 10 = 180 \)

Subtract 10 from both sides:
\( 6x = 180 - 10 \)
\( 6x = 170 \) (Wait, correction: \( 5 - 10 + 15 = 10 \), so \( 6x + 10 = 180 \) → \( 6x = 170 \) is wrong. Let's recalculate: \( 5 - 10 + 15 = 10 \)? No: \( 5 - 10 = -5 \), \( -5 + 15 = 10 \). Wait, no, \( 2x + 3x + x = 6x \); \( 5 - 10 + 15 = 10 \). So \( 6x + 10 = 180 \). Subtract 10: \( 6x = 170 \)? No, \( 180 - 10 = 170 \)? Wait, no, \( 180 - 10 = 170 \)? Wait, no, \( 6x + 10 = 180 \) → \( 6x = 180 - 10 = 170 \)? Wait, that can't be. Wait, maybe my example angles are bad. Let's use a better example: angles \( (x)^\circ \), \( (x + 20)^\circ \), \( (x + 40)^\circ \).

Step 1 (corrected example):

Equation: \( x + (x + 20) + (x + 40) = 180 \)

Step 2 (corrected example):

Combine like terms:
\( 3x + 60 = 180 \)

Subtract 60:
\( 3x = 120 \)

Divide by 3:
\( x = 40 \)

Step 3: Find each angle

• First angle: \( x = 40^\circ \)
• Second angle: \( x + 20 = 40 + 20 = 60^\circ \)
• Third angle: \( x + 40 = 40 + 40 = 80^\circ \)

To solve your specific problem, follow these steps:

1. Identify the three angle expressions (in terms of \( x \)).
2. Set their sum equal to \( 180^\circ \) (Triangle Sum Theorem).
3. Solve for \( x \) (combine like terms, isolate \( x \)).
4. Substitute \( x \) back into each angle expression to find the measure of each angle.

If you provide the specific angle expressions (e.g., \( (ax + b)^\circ \), \( (cx + d)^\circ \), \( (ex + f)^\circ \)), I can help you solve it step-by-step.

Answer:

Step 1: Write the equation

Using the Triangle Sum Theorem (\( \text{Sum of interior angles} = 180^\circ \)):
\( (2x + 5) + (3x - 10) + (x + 15) = 180 \)

Step 2: Solve for \( x \)

Combine like terms:
\( 2x + 3x + x + 5 - 10 + 15 = 180 \)
\( 6x + 10 = 180 \)

Subtract 10 from both sides:
\( 6x = 180 - 10 \)
\( 6x = 170 \) (Wait, correction: \( 5 - 10 + 15 = 10 \), so \( 6x + 10 = 180 \) → \( 6x = 170 \) is wrong. Let's recalculate: \( 5 - 10 + 15 = 10 \)? No: \( 5 - 10 = -5 \), \( -5 + 15 = 10 \). Wait, no, \( 2x + 3x + x = 6x \); \( 5 - 10 + 15 = 10 \). So \( 6x + 10 = 180 \). Subtract 10: \( 6x = 170 \)? No, \( 180 - 10 = 170 \)? Wait, no, \( 180 - 10 = 170 \)? Wait, no, \( 6x + 10 = 180 \) → \( 6x = 180 - 10 = 170 \)? Wait, that can't be. Wait, maybe my example angles are bad. Let's use a better example: angles \( (x)^\circ \), \( (x + 20)^\circ \), \( (x + 40)^\circ \).

Step 1 (corrected example):

Equation: \( x + (x + 20) + (x + 40) = 180 \)

Step 2 (corrected example):

Combine like terms:
\( 3x + 60 = 180 \)

Subtract 60:
\( 3x = 120 \)

Divide by 3:
\( x = 40 \)

Step 3: Find each angle

• First angle: \( x = 40^\circ \)
• Second angle: \( x + 20 = 40 + 20 = 60^\circ \)
• Third angle: \( x + 40 = 40 + 40 = 80^\circ \)

To solve your specific problem, follow these steps:

1. Identify the three angle expressions (in terms of \( x \)).
2. Set their sum equal to \( 180^\circ \) (Triangle Sum Theorem).
3. Solve for \( x \) (combine like terms, isolate \( x \)).
4. Substitute \( x \) back into each angle expression to find the measure of each angle.

If you provide the specific angle expressions (e.g., \( (ax + b)^\circ \), \( (cx + d)^\circ \), \( (ex + f)^\circ \)), I can help you solve it step-by-step.