https://file.aiquickdraw.com/mathbot/file/17241890403270gypift6.jpg

For what value of x is f ॥g? Which theorem justifies your answer?

To solve the problem, we need to determine the value of \( x \) that makes lines \( f \) and \( g \) parallel. The corresponding angles are given as \( (10x + 5)^\circ \) and \( (12x - 9)^\circ \).

### Step 1: Identify the Theorem
When two lines are parallel, corresponding angles or alternate interior angles are equal. The given angles appear to be alternate interior angles since they are on opposite sides of the transversal and between the two lines. 

### Step 2: Set up the Equation
Since the lines are parallel, we can set the two angles equal to each other:
\[
10x + 5 = 12x - 9
\]

### Step 3: Solve for \( x \)
Now, solve the equation:
\[
10x + 5 = 12x - 9
\]
Subtract \( 10x \) from both sides:
\[
5 = 2x - 9
\]
Add 9 to both sides:
\[
14 = 2x
\]
Divide by 2:
\[
x = 7
\]

### Step 4: Justification
The lines \( f \) and \( g \) are parallel when \( x = 7 \). The theorem that justifies this is the **Alternate Interior Angles Theorem**, which states that if a transversal intersects two parallel lines, the pairs of alternate interior angles are equal.

Would you like any further details or have any questions?

Here are some related questions:

1. What are corresponding angles, and how do they relate to parallel lines?
2. How would the solution change if the angles were on the same side of the transversal?
3. Can this problem be solved using the concept of supplementary angles?
4. What is the converse of the Alternate Interior Angles Theorem?
5. How can we apply the theorem to three or more parallel lines?

**Tip:** Always label angles and lines clearly when working with parallel lines and transversals to avoid confusion in identifying angle relationships.

Learn how to find the value of x that makes lines f and g parallel using the Alternate Interior Angles Theorem. Step-by-step solution provided for high school geometry students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation