/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 136 If \(\ell_{1} \| \ell_{2}\), pro... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Step by step solution

01

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.
02

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.
03

Use corresponding angle properties

By the corresponding angle property, angle 1 is equal to angle 3, i.e., \(\angle 1 = \angle 3\).
04

Use alternate interior angle properties

By the alternate interior angle property, angle 2 is equal to angle 4, i.e., \(\angle 2 = \angle 4\).
05

Sum the angles

Since angle 3 and angle 4 form a linear pair on line \(\ell_{2}\), their sum is 180 degrees. Therefore, \(\angle 3 + \angle 4 = 180^\circ\).
06

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: \(\angle 1 + \angle 2 = 180^\circ\)
07

Conclusion

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

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transversal
In geometry, when two parallel lines are intersected by a third line, this line is known as a transversal. The transversal creates several interesting angles with the parallel lines, which leads to unique geometric properties. To understand this concept:
  • The transversal intersects two or more lines at distinct points.
  • In the context of parallel lines, the angles formed have particular relationships.
  • These relationships are key to many geometric proofs and theorems.
A transversal crossing two parallel lines helps in identifying corresponding angles, alternate interior angles, and supplementary angles. This setup establishes a foundation for exploring how angles relate to each other, which is crucial in solving geometric problems.
Corresponding Angles
Corresponding angles are formed when a transversal crosses two parallel lines, resulting in pairs of angles that occupy the same relative position at each intersection. These angles have a special property:
  • Corresponding angles are equal in measure.
To identify corresponding angles:
  • Look for an angle on the first parallel line and locate that angle's counterpart in the same position on the second parallel line.
  • These angles are mirrored across the transversal.
This property is instrumental when proving angle congruence or calculating unknown angles in geometric figures.
Alternate Interior Angles
Alternate interior angles are another type of angle pair created by a transversal crossing parallel lines. These angles:
  • Lie between the two parallel lines.
  • Are positioned on opposite sides of the transversal.
  • Are equal in measure, according to the alternate interior angle theorem.
When solving problems involving these angles, consider:
  • Their location inside the parallel lines helps in identifying them easily.
  • Using their equality can simplify calculations in geometry problems, particularly in proving congruence or finding angles that are supplementary.
Alternate interior angles assist in identifying angle relations critical for geometric proofs.
Supplementary Angles
Supplementary angles are pairs of angles that add up to 180 degrees. This concept is crucial in understanding how different angles can relate to each other within various shapes or formed by intersecting lines. With parallel lines and a transversal, supplementary angles often appear:
  • On a straight line, where their measures add to 180 degrees.
  • In linear pairs, especially when formed by the intersection of the transversal with parallel lines.
For instance, knowing angles 1 and 2 are supplementary implies:
  • If one angle measure is known, the other's can be found simple by subtracting from 180.
  • This relationship is commonly used to demonstrate angle congruence or solve for unknown values in geometry.
Understanding how angles are supplementary is essential for mastering the broader topic of angle relationships in geometry.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Given: \(\underline{A D}\) and \(\underline{B C}\) intersect at \(E\). \(\underline{A B} \| C D\). \(C E=D E\) Prove: \(A E=B E\).

Prove the angle sum theorem: the sum of the measures of the angles of a triangle is \(180^{\circ}\).

In the accompanying figure, \(\underline{A D}=\underline{D C}\) and \(\underline{B D}=\underline{D E}\). Prove that \(\underline{A B} \| \underline{E C}\).

Given: \(\triangle \mathrm{ABC}\) is isosceles with base \(\mathrm{AB}\). \(\angle \mathrm{A} \cong \angle 1\) Prove: \(A B \| E D\).

Given: \(\underline{A D}\|\underline{B E}, \underline{B D}\| \underline{C E}\) and \(B\) is midpoint of \(\underline{A C}\). Prove: \(B E=A D\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.