Expand menu menu_open Minimize Go to startpage Home History history History expand_more
{{ item.displayTitle }}
navigate_next
No history yet!
Progress & Statistics equalizer Progress expand_more
Student
navigate_next
Teacher
navigate_next
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
No results
{{ searchError }}
search
menu
{{ courseTrack.displayTitle }}
{{ statistics.percent }}% Sign in to view progress
{{ printedBook.courseTrack.name }} {{ printedBook.name }}
search Use offline Tools apps
Digital tools Graphing calculator Geometry 3D Graphing calculator Geogebra Classic Mathleaks Calculator Codewindow
Course & Book Compare textbook Studymode Stop studymode Print course
Tutorials Video tutorials Formulary

Video tutorials

How Mathleaks works

Mathleaks Courses

How Mathleaks works

play_circle_outline
Study with a textbook

Mathleaks Courses

How to connect a textbook

play_circle_outline

Mathleaks Courses

Find textbook solutions in the app

play_circle_outline
Tools for students & teachers

Mathleaks Courses

Share statistics with a teacher

play_circle_outline

Mathleaks Courses

How to create and administrate classes

play_circle_outline

Mathleaks Courses

How to print out course materials

play_circle_outline

Formulary

Formulary for text courses looks_one

Course 1

looks_two

Course 2

looks_3

Course 3

looks_4

Course 4

looks_5

Course 5

Login account_circle menu_open

Parallelogram Opposite Angles Theorem

Proof

Parallelogram Opposite Angles Theorem

In a parallelogram, the opposite angles are congruent.

Consider the parallelogram PQRS,PQRS, the following angles are then congruent: PQRRSP and SPQQRS. \angle PQR \cong \angle RSP\text{ and }\angle SPQ \cong \angle QRS.

This can be proven with the ASA Congruence Theorem.

In the parallelogram PQRS,PQRS, the diagonal QS\overline{QS} is drawn.

According to the definition of a parallelogram, the opposite sides are parallel, PQSRPQ\parallel SR and QRPS.QR\parallel PS.

Then, by the Alternate Interior Angles Theorem, the angles PQSQSR\angle PQS\cong\angle QSR and PSQRQS.\angle PSQ\cong\angle RQS.

Since the triangles have two congruent angles and share one side, QS,\overline{QS}, the ASA Congruence Theorem applies. It states that two triangles are congruent if two angles and their including side are congruent. PQSRSQ \triangle PQS \cong \triangle RSQ As the triangles are congruent, all their corresponding parts are congruent. Therefore, the angles SPQ\angle SPQ and QRS\angle QRS are congruent as well.

Since the angles PQR\angle PQR and QRS\angle QRS are the sum of the same two congruent angles, they are congruent.

The proof can be summarized in a flowchart proof.