Opposite Angles of a Parallelogram are Equal

Here we will discuss about the opposite angles of a parallelogram are equal.

In a parallelogram, each pair of opposite angles are equal.

Given: PQRS is a parallelogram in which PQ ∥ SR and QR ∥ PS

To prove: ∠P = ∠R and ∠Q = ∠S

Construction: Join PR and QS.

Opposite Angles of a Parallelogram are Equal

Proof:

            Statement:

In ∆PQR and ∆RSP;

1. ∠QPR = ∠PRS

2. ∠QRP = ∠SPR

3. ∠QPR + ∠SPR = ∠PRS + ∠QRP

⟹ ∠P = ∠R

4. Similarly, from ∆PQS and ∆RSQ, ∠Q = ∠S.     (Proved)

            Reason


1. PQ ∥ SR and PR is a transversal.

2. QR ∥ PS and PR is a transversal.

3. Adding statements 1 and 2.


Converse proposition of the above theorem

A quadrilateral is a parallelogram if each pair of opposite angles are equal.

Given: PQRS is a quadrilateral in which ∠P = ∠R and ∠Q = ∠S

Pair of Opposite Angles are Equal

To prove: PQRS is a parallelogram

Proof: ∠P + ∠Q + ∠R + ∠S = 360°, because the sum of the four angles of a quadrilateral is 360°.

Therefore, ∠2P + ∠2Q = 360°, (since ∠P = ∠R, ∠Q = ∠S)

Therefore, ∠P + ∠Q = 180° and so, ∠P + ∠S = 180°, (since ∠Q = ∠S)

∠P + ∠Q = 180°

⟹ PS ∥ QR (since sum of the co. interior angles is 180°)

∠P + ∠S = 180°

⟹ PQ ∥ SR (since sum of the co. interior angles is 180°)

Therefore, in the quadrilateral PQRS, PQ ∥ SR and PS ∥ QR. So, PQRS is a parallelogram.




9th Grade Math

From Opposite Angles of a Parallelogram are Equal to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 15, 25 02:46 AM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More

  2. 5th Grade Quadrilaterals | Square | Rectangle | Parallelogram |Rhombus

    Jul 15, 25 02:01 AM

    Square
    Quadrilaterals are known as four sided polygon.What is a quadrilateral? A closed figure made of our line segments is called a quadrilateral. For example:

    Read More

  3. Formation of Numbers | Smallest and Greatest Number| Number Formation

    Jul 14, 25 01:53 AM

    In formation of numbers we will learn the numbers having different numbers of digits. We know that: (i) Greatest number of one digit = 9,

    Read More

  4. 5th Grade Geometry Practice Test | Angle | Triangle | Circle |Free Ans

    Jul 14, 25 01:53 AM

    Name the Angles
    In 5th grade geometry practice test you will get different types of practice questions on lines, types of angle, triangles, properties of triangles, classification of triangles, construction of triang…

    Read More

  5. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 11, 25 02:14 PM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More