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Problem 136

# If $$\ell_{1} \| \ell_{2}$$, prove that $$\angle 1$$ is supplementary to $\angle 2$.

Expert verified
Let $$\ell_1$$ and $$\ell_2$$ be parallel lines and $$t$$ be a transversal. Identify the corresponding angle to $$\angle 1$$ (angle 3) and an alternate interior angle to $$\angle 2$$ (angle 4). By corresponding angle property, $$\angle 1 = \angle 3$$, and by alternate interior angle property, $$\angle 2 = \angle 4$$. Since angle 3 and angle 4 form a linear pair, $$\angle 3 + \angle 4 = 180^\circ$$. Substituting angles, we get $$\angle 1 + \angle 2 = 180^\circ$$, proving angle 1 is supplementary to angle 2.
See the step by step solution

## Step 1: Identify the corresponding angles

Since the lines $$\ell_{1}$$ and $$\ell_{2}$$ are parallel, we know that corresponding angles are equal when a transversal intersects them. Identify the angle corresponding to angle 1 on line $$\ell_{2}$$. Let's call it angle 3.

## Step 2: Identify the alternate interior angles

Now, let's identify an alternate interior angle to angle 2. These angles are found inside the parallel lines and are on different sides of the transversal. Let's call this angle 4.

## Step 3: Use corresponding angle properties

By the corresponding angle property, angle 1 is equal to angle 3, i.e., $$\angle 1 = \angle 3$$.

## Step 4: Use alternate interior angle properties

By the alternate interior angle property, angle 2 is equal to angle 4, i.e., $$\angle 2 = \angle 4$$.

## Step 5: Sum the angles

Since angle 3 and angle 4 form a linear pair on line $$\ell_{2}$$, their sum is 180 degrees. Therefore, $$\angle 3 + \angle 4 = 180^\circ$$.

## Step 6: Substitute angle 1 for angle 3 and angle 2 for angle 4

Using the equalities from steps 3 and 4, substitute angle 1 for angle 3 and angle 2 for angle 4 in the equation from step 5: $$\angle 1 + \angle 2 = 180^\circ$$

## Step 7: Conclusion

Therefore, angle 1 is supplementary to angle 2, since their sum is equal to 180 degrees.

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