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Problem 138

# If line $$\underline{\mathrm{AB}}$$ is parallel to line $$\underline{\mathrm{CD}}$$ and line $$\underline{\mathrm{EF}}$$ is parallel to line $$\underline{\mathrm{GH}}$$, prove that $$\mathrm{m}<1=\mathrm{m} \angle 2$$.

Expert verified
To prove that $$\mathrm{m}<1=\mathrm{m} \angle 2$$, we use the Alternate Interior Angles theorem which states that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. First, we identify that angles 1 and 3 are alternate interior angles between lines $$\underline{\mathrm{AB}}$$ and $$\underline{\mathrm{CD}}$$, so $$\mathrm{m}<1=\mathrm{m} \angle 3$$. Then, we identify that angles 3 and 2 are alternate interior angles between lines $$\underline{\mathrm{EF}}$$ and $$\underline{\mathrm{GH}}$$, so $$\mathrm{m}<3=\mathrm{m} \angle 2$$. Since $$\mathrm{m}<1=\mathrm{m} \angle 3$$ and $$\mathrm{m}<3=\mathrm{m} \angle 2$$, we can conclude that $$\mathrm{m}<1=\mathrm{m} \angle 2$$, proving the statement.
See the step by step solution

## Step 1: Identify transversals

We will identify lines that are transversals to both pairs of parallel lines. These transversals will be key in finding the relationship between angle 1 and angle 2.

## Step 2: Apply the Alternate Interior Angles Theorem

The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. We will apply this theorem to find a relationship between angle 1 and angle 2.

## Step 3: Angle relationships from the first pair of parallel lines

Assuming angles 1 and 3 are on the same side of the first transversal and between lines $$\underline{\mathrm{AB}}$$ and $$\underline{\mathrm{CD}}$$, we can say that angles 1 and 3 are alternate interior angles. Since lines $$\underline{\mathrm{AB}}$$ and $$\underline{\mathrm{CD}}$$ are parallel, we know that $$\mathrm{m}<1=\mathrm{m} \angle 3$$.

## Step 4: Angle relationships from the second pair of parallel lines

Next, we look at the second pair of parallel lines, $$\underline{\mathrm{EF}}$$ and $$\underline{\mathrm{GH}}$$. Assuming angles 3 and 2 are on the same side of the second transversal and between these lines, they are also alternate interior angles. Since lines $$\underline{\mathrm{EF}}$$ and $$\underline{\mathrm{GH}}$$ are parallel, we know that $$\mathrm{m}<3=\mathrm{m} \angle 2$$.

## Step 5: Prove that the measure of angle 1 is equal to the measure of angle 2

From step 3, we know that $$\mathrm{m}<1=\mathrm{m} \angle 3$$, and from step 4, we know that $$\mathrm{m}<3=\mathrm{m} \angle 2$$. Therefore, we can conclude that $$\mathrm{m}<1=\mathrm{m} \angle 2$$, which is the statement we were trying to prove.

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