/*! 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 135 Given: \(\angle 1\) and \(\angle... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 135

Given: \(\angle 1\) and \(\angle 2\) are supplementary. Prove: \(\ell_{1} \| \ell_{2}\)

Short Answer

Expert verified
Given that \(\angle 1\) and \(\angle 2\) are supplementary, their sum equals 180 degrees. As alternate interior angles are congruent when lines are parallel and \(\angle 1 \cong \angle 2\), we can use the alternate interior angle theorem to conclude that \(\ell_{1}\) is parallel to \(\ell_{2}\). Therefore, we have proven that \(\ell_{1} \| \ell_{2}\).

Step by step solution

01

Draw a diagram

First, let's sketch a diagram to help visualize the problem. Draw lines \(\ell_{1}\) and \(\ell_{2}\), with a transversal line \(\ell_{3}\) cutting through both parallel lines. Mark angle 1 on line \(\ell_{1}\) and angle 2 on line \(\ell_{2}\) so that \(\angle 1\) and \(\angle 2\) are alternate interior angles with respect to \(\ell_{3}\).
02

Write down what is given

We are given that \(\angle 1\) and \(\angle 2\) are supplementary, which means their sum equals 180 degrees. So, we can write the equation: \[\angle 1 + \angle 2 = 180^\circ\]
03

Use the alternate interior angle theorem

The alternate interior angle theorem states that if two lines are parallel and cut by a transversal line, then the alternate interior angles are congruent. Specifically, if \(\ell_{1} \| \ell_{2}\), then \(\angle 1 \cong \angle 2\).
04

Prove that the lines are parallel

We want to prove that \(\ell_{1} \| \ell_{2}\). Since we know that \(\angle 1 + \angle 2 = 180^\circ\), we can deduce that \(\angle 1\) and \(\angle 2\) are congruent to each other, since they are both supplementary to the same angle. We already established that \(\angle 1 \cong \angle 2\) if and only if \(\ell_{1} \| \ell_{2}\). Since \(\angle 1 \cong \angle 2\), by the alternate interior angle theorem, we can conclude that \(\ell_{1}\) is parallel to \(\ell_{2}\). Therefore, we have proven that $\ell_{1} \| \ell_{2}$.

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.

Parallel Lines
When we talk about parallel lines, we refer to two lines that are in the same plane but do not intersect, no matter how far they are extended. Think of the lanes on a straight road or the lines on notebook paper; these are examples of parallel lines in real life.

In geometry, the importance of understanding parallel lines lies in their properties and how they interact with other lines and angles. For instance, when another line crosses two parallel lines, this line is known as a transversal, and it creates several angles with unique relationships, some of which are the focus of our problem. The fact that parallel lines never meet is because they have the same gradient or slope, which is a measure of how steep a line is. This equality in a slope is the core principle we leverage to solve various geometric problems.
Alternate Interior Angles
Within the world of geometry, alternate interior angles are pairs of angles that are formed when a transversal crosses two (usually parallel) lines and lie on opposite sides of the transversal, but inside the two lines. These angle pairs are like mirror images or reflections of each other.

The Alternate Interior Angles Theorem provides a fascinating insight: if the lines are parallel, each pair of alternate interior angles are equal in measure. This theorem is often used to prove whether two lines are parallel, as was done in the step-by-step solution we are discussing. Recognizing these angles is a crucial skill when solving geometric proofs and problems.
Transversal Line
A transversal line is a straight line that intersects two or more other lines at distinct points. When the lines cut by the transversal are parallel, the angles formed exhibit special relationships: corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary (adding up to 180 degrees).

In the context of our geometric problem, the transversal is the line that intersects the two lines we're analyzing (ll_{1} and ll_{2}) to help us determine their relationship. By understanding the role of the transversal and how it interacts with other lines and angles, we unlock the ability to navigate through a plethora of geometric problems with confidence and clarity.

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

A collapsible ironing board is constructed so that the supports bisect each other. Show why the board will always be parallel to the floor.

Show that any two medians of a triangle intersect at a point in the interior of the triangle.

Prove that if two adjacent angles of a quadrilateral are right angles, the bisectors of the other two angles are perpendicular to each other.

Given: \(\underline{A B} \| D C\) and \(\underline{A B} \cong \underline{C D}\). Prove: \(\angle \mathrm{A} \cong \angle \mathrm{C}\).

Given: \(\underline{A C}\) and \(\underline{E B}\) bisect each other at \(D\). Prove: \(\underline{\mathrm{AE}} \| \underline{\mathrm{BC}}\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.