Chapter 10: Chapter 10

Problem 136

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

Short Answer

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.

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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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Chapter 10: Chapter 10

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

Short Answer

Step by step solution

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Step 1: Identify the corresponding angles

Step 2: Identify the alternate interior angles

Step 3: Use corresponding angle properties

Step 4: Use alternate interior angle properties

Step 5: Sum the angles

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

Step 7: Conclusion

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

Chapter 25

Chapter 26

Chapter 27

Chapter 28

Chapter 29

Chapter 30

Chapter 31

Chapter 32

Chapter 33

Chapter 34

Chapter 35

Chapter 36

Chapter 37

Chapter 38

Chapter 39

Chapter 40

Chapter 41

Chapter 42

Chapter 43

Chapter 44

Chapter 48

Chapter 49

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Chapter 51

Chapter 52

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