Parallel and Intersecting Lines - Corresponding Angles

In the given figure, line $l$ is parallel to line $m$ ($l \parallel m$) and line $t$ is a transversal. If one of the corresponding angles is $(2x + 10)^\circ$ and the other is $70^\circ$, find the value of $x$.

Since $l \parallel m$, the corresponding angles must be equal. $(2x + 10)^\circ = 70^\circ$ $2x = 70 - 10$ $2x = 60$ $x = \frac{60}{2}$ $x = 30$

According to the corresponding angles axiom, when two parallel lines are intersected by a transversal, the corresponding angles are equal. We set the expressions equal to each other and solve for $x$.

Two lines $p$ and $q$ are cut by a transversal $r$. The measures of a pair of corresponding angles are $115^\circ$ and $(125 - y)^\circ$. What value of $y$ would make the lines $p$ and $q$ parallel?

For lines $p$ and $q$ to be parallel, the corresponding angles must be equal. $115^\circ = 125^\circ - y$ $y = 125^\circ - 115^\circ$ $y = 10^\circ$

By the converse of the corresponding angles axiom, if the corresponding angles are equal, the lines are parallel. Therefore, we equate the two angles to find the value of $y$ that satisfies this condition.

If $\angle 1$ and $\angle 2$ are corresponding angles formed by a transversal of two parallel lines, and $\angle 1 = 85^\circ$, find the supplement of $\angle 2$.

Step 1: Find $\angle 2$. Since the lines are parallel, corresponding angles are equal: $\angle 2 = \angle 1 = 85^\circ$ Step 2: Find the supplement of $\angle 2$. Supplement of $\angle 2 = 180^\circ - \angle 2$ $\begin{array}{r} 180 \\ - 85 \\ \hline 95 \end{array}$ So, the supplement is $95^\circ$.

First, we use the property of parallel lines to determine that the corresponding angle $\angle 2$ is also $85^\circ$. Then, we calculate the supplement by subtracting the angle from $180^\circ$.

$p \parallel q \text{ (Given)}$ $\angle a = \angle b \text{ (Corresponding angles postulation)}$ $3x - 20^\circ = 2x + 10^\circ$ $3x - 2x = 10^\circ + 20^\circ$ $x = 30^\circ$

Since the lines $p$ and $q$ are parallel, the corresponding angles formed by the transversal $s$ are equal. We set the algebraic expressions for $\angle a$ and $\angle b$ equal to each other and solve for $x$.

$\text{Given: Pair of corresponding angles are } 122^\circ \text{ and } 122^\circ.$ $\text{Since } 122^\circ = 122^\circ, \text{ the corresponding angles are equal.}$ $\text{By the converse of the corresponding angles property, } L \parallel M.$

If a transversal intersects two lines such that the corresponding angles are equal, then the lines are parallel. Here, both angles measure $122^\circ$, confirming the lines are parallel.