Chapter 10: Chapter 10

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 \(\mathrm{m}<1=\mathrm{m} \angle 2\).

Short Answer

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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.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

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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Chapter 10: Chapter 10

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 \(\mathrm{m}<1=\mathrm{m} \angle 2\).

Short Answer

Step by step solution

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Step 1: Identify transversals

Step 2: Apply the Alternate Interior Angles Theorem

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

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

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

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