Solved examples: proving lines parallel with corresponding angles.

📐 Quick Study Guide

• 🔑 Corresponding Angles: When a transversal intersects two lines, corresponding angles are in the same relative position at each intersection. If the lines are parallel, corresponding angles are congruent (equal).
• 📏 Converse of Corresponding Angles Theorem: If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
• ➕ Using Algebra: Often, you'll need to set up an equation where two corresponding angles are equal and solve for a variable to prove parallelism.
• 💡 Key Idea: To prove lines are parallel using corresponding angles, show that the corresponding angles are congruent.

✍️ Practice Quiz

1. If angle 1 and angle 5 are corresponding angles and $m\angle 1 = (2x + 10)^\circ$ and $m\angle 5 = (3x - 5)^\circ$, what value of $x$ proves the lines are parallel?
   1. $5$
   2. $10$
   3. $15$
   4. $20$
2. Lines $a$ and $b$ are cut by a transversal. $m\angle 2 = (5x + 12)^\circ$ and $m\angle 6 = (7x - 8)^\circ$. Angles 2 and 6 are corresponding. What value of $x$ would make lines $a$ and $b$ parallel?
   1. $2$
   2. $4$
   3. $6$
   4. $10$
3. Given lines $m$ and $n$ cut by transversal $t$. If $m\angle 3 = (4x + 5)^\circ$ and $m\angle 7 = (6x - 15)^\circ$ are corresponding angles, what must $x$ be for lines $m$ and $n$ to be parallel?
   1. $5$
   2. $10$
   3. $15$
   4. $20$
4. Suppose $m\angle 4 = (9x - 20)^\circ$ and $m\angle 8 = (7x + 10)^\circ$ are corresponding angles formed by a transversal intersecting lines $p$ and $q$. Find $x$ such that $p \parallel q$.
   1. $5$
   2. $10$
   3. $15$
   4. $25$
5. If $m\angle A = (6x + 4)^\circ$ and $m\angle B = (8x - 12)^\circ$ are corresponding angles, what value of $x$ makes the lines containing these angles parallel?
   1. $4$
   2. $6$
   3. $8$
   4. $10$
6. Two lines are intersected by a transversal. One of the corresponding angles measures $(3y + 7)^\circ$, and the other measures $(5y - 3)^\circ$. For the lines to be parallel, what is the value of $y$?
   1. $2$
   2. $3$
   3. $5$
   4. $8$
7. Lines $r$ and $s$ are cut by a transversal such that $m\angle 1 = (4a - 15)^\circ$ and $m\angle 5 = (2a + 25)^\circ$ are corresponding angles. What value of $a$ makes lines $r$ and $s$ parallel?
   1. $10$
   2. $15$
   3. $20$
   4. $25$