Question

3 the figure shows two lines cut by a transversal. what value of x shows that the lines are parallel? show your work.
4 which statements about the figure are true? select all that apply. a ∠2≅∠3 b ∠10 and ∠11 are supplementary angles. c ∠2 are ∠5 are same - side interior angles. d ∠8 and ∠9 are supplementary angles. e ∠9 and ∠12 are same - side exterior angles. f m∠3 + m∠6 = 180°
5 the figure shows two lines cut by a transversal. what value of x shows that the lines parallel? show your work.

Explanation:

Step1: Identify angle - relationship for parallel lines in problem 3

For parallel lines cut by a transversal, corresponding angles are equal. Here, the two given angles \(5x\) and \(10x - 30\) are corresponding angles. So, we set up the equation \(5x=10x - 30\).
\[5x=10x - 30\]

Step2: Solve the equation for \(x\) in problem 3

Subtract \(5x\) from both sides: \(0 = 10x-5x - 30\), which simplifies to \(0 = 5x - 30\). Then add 30 to both sides: \(30 = 5x\). Divide both sides by 5: \(x = 6\).
\[

$$\begin{align*} 0&=5x - 30\\ 30&=5x\\ x&=6 \end{align*}$$

\]

Step3: Identify angle - relationship for parallel lines in problem 5

The two given angles \((2x + 10)\) and \((x - 10)\) are same - side interior angles. For parallel lines, same - side interior angles are supplementary, so \((2x + 10)+(x - 10)=180\).
\[2x + 10+x - 10=180\]

Step4: Solve the equation for \(x\) in problem 5

Combine like terms: \(3x=180\). Divide both sides by 3: \(x = 60\).
\[x=\frac{180}{3}=60\]

Step5: Analyze statements in problem 4

A:

\(\angle2\) and \(\angle3\) are vertical angles, and vertical angles are always congruent, so \(\angle2\cong\angle3\) is True.

B:

\(\angle10\) and \(\angle11\) are not supplementary. They are vertical angles. So, this statement is False.

C:

\(\angle2\) and \(\angle5\) are not same - side interior angles. So, this statement is False.

D:

\(\angle8\) and \(\angle9\) are adjacent angles that form a straight - line, so \(\angle8+\angle9 = 180^{\circ}\), and they are supplementary. This statement is True.

E:

\(\angle9\) and \(\angle12\) are not same - side exterior angles. So, this statement is False.

F:

\(\angle3\) and \(\angle6\) are same - side interior angles for parallel lines. If the lines are parallel, \(m\angle3 + m\angle6=180^{\circ}\). But we don't know if the lines are parallel from the given information. However, if we assume the general property for parallel lines, this statement is True.

Answer:

• Problem 3: \(x = 6\)
• Problem 4: A. \(\angle2\cong\angle3\), D. \(\angle8\) and \(\angle9\) are supplementary angles, F. \(m\angle3 + m\angle6 = 180^{\circ}\) (assuming parallel lines)
• Problem 5: \(x = 60\)