Question

proving lines are parallel with algebra
directions: find x so that ( a parallel b ). state the converse used.
1.
( (10x - 23)^circ ) ( 137^circ )
converse:
2.
( 74^circ ) ( (9x + 11)^circ )
converse:
3.
( 96^circ ) ( (6x - 30)^circ )
converse:
4.
( 33^circ ) ( (7x + 5)^circ )
converse:
5.
( (8x - 17)^circ ) ( (5x + 19)^circ )
converse:
6.
( (12x - 29)^circ ) ( (4x + 1)^circ )
converse:
7.
( (6x + 5)^circ ) ( (8x - 37)^circ )
converse:
8.
( (4x - 18)^circ ) ( 90^circ )
converse:
10.
( (13x - 2)^circ ) ( (4x - 5)^circ )
converse:

Explanation:

Let's solve problem 1 step by step:

We need to find \( x \) such that lines \( a \) and \( b \) are parallel. The angles given are alternate interior angles. The converse of the alternate interior angles theorem states that if alternate interior angles are congruent, then the lines are parallel. So we set the two angle expressions equal to each other.

Step 1: Set up the equation

The angle \( (10x - 23)^\circ \) and \( 137^\circ \) are alternate interior angles. For \( a \parallel b \), these angles must be equal. So we write the equation:
\[
10x - 23 = 137
\]

Step 2: Solve for \( x \)

Add 23 to both sides of the equation:
\[
10x - 23 + 23 = 137 + 23
\]
\[
10x = 160
\]
Divide both sides by 10:
\[
x = \frac{160}{10}
\]
\[
x = 16
\]

Step 3: State the converse

The converse used is the Alternate Interior Angles Converse, which states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

Answer:

\( x = 16 \), Converse: Alternate Interior Angles Converse