Opposite Angles
When two lines intersect each other, four angles are formed. Among these angles, there are two pairs of non-adjacent angles. These are called opposite angles, vertical angles or vertically opposite angles. These angles are equal in measure. However, in geometry, the term 'opposite angles' is also used in quadrilaterals. These opposite angles are the angles that are diagonally opposite to each other. In other words, they are the angles that are connected through diagonals. Let us understand the opposite angles with respect to both the concepts.
| 1. | What are Opposite Angles? |
| 2. | Difference Between Opposite Angles and Adjacent Angles |
| 3. | Opposite Angles in a Parallelogram |
| 4. | Opposite Angles in a Cyclic Quadrilateral |
| 5. | FAQs on Opposite Angles |
What are Opposite Angles?
When any two straight lines intersect each other, there are different pairs of angles that are formed. The angles that are directly opposite to each other are known as opposite angles. They are also termed as vertical angles or vertically opposite angles and are equal to each other. Observe the following figure in which lines 'a' and 'b' intersect each other and form two pairs of opposite angles. The opposite angles are ∠1 = ∠3 and ∠2 = ∠4.
Difference Between Opposite Angles and Adjacent Angles
The intersection of any two lines results in adjacent and opposite angles in them. However, these two angles are different from each other and can be identified easily with the help of their properties. Observe the following figure and the table which shows the difference between opposite angles and adjacent angles.
| Adjacent Angles | Opposite Angles |
|---|---|
|
Adjacent angles share a common arm. For example, in the figure given above, ∠1 and ∠2 share a common arm AO. |
Opposite angles do not share a common arm. For example, ∠1 and ∠3 do not share a common arm. |
|
Adjacent angles may or may not be equal in measure. |
Opposite angles are always equal. |
| Two adjacent angles are always located next to each other. | Opposite angles are always located opposite to each other |
| In the given figure, the adjacent angles are: ∠1 and ∠2; ∠2 and ∠3; ∠ 3 and ∠4; ∠4 and ∠1 | In the given figure, the opposite angles are: ∠1 and ∠3; ∠2 and ∠4 |
We have understood the concept of opposite angles with respect to intersecting lines. Now, let us understand the other concept of opposite angles with reference to a parallelogram and a cyclic quadrilateral.
Opposite Angles in a Parallelogram
The opposite angles in a quadrilateral are those angles that are located diagonally opposite to each other. In other words, they are the angles that are connected through diagonals. For example, in the following parallelogram ABCD, ∠A and ∠C are called opposite angles. Similarly, ∠B and ∠D are opposite angles. One of the properties of a parallelogram states that the opposite angles are equal in measure.
Opposite Angles in a Cyclic Quadrilateral
A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle and it is also known as a quadrilateral inscribed in a circle. In other words, it is a quadrilateral that is inside a circle and all its vertices touch the circle. There are many theorems related to a cyclic quadrilateral and the one related to opposite angles states that," The opposite angles in a cyclic quadrilateral are supplementary, that is, the sum of the opposite angles is equal to 180°". Observe the following figure which shows that the opposite angles in a cyclic quadrilateral are supplementary.
Related Links
Check out the following pages related to opposite angles.
Opposite Angles Examples
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Example 1: Find the pairs of opposite angles in the given figure.
Solution:
In the given figure, there are two pairs of opposite angles.
∠a and ∠c, and ∠b and ∠d are opposite angles and they are equal to each other, that is, ∠a = ∠c, and ∠b = ∠d
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Example 2: Write true or false for the following statements related to opposite angles.
a.) When two straight lines intersect, the opposite angles are always complementary.
b.) Opposite angles always lie next to each other.
Solution:
Using the properties of opposite angles, we can answer the questions.
a.) False, opposite angles may not be always complementary, however, they are always equal.
b.) False, opposite angles always lie opposite to each other.
FAQs on Opposite Angles
What are Opposite Angles Called?
When any two straight lines intersect each other, then four angles are formed. The angles that are directly opposite to each other are known as opposite angles. They are also called vertical angles or vertically opposite angles.
What is the Difference Between Adjacent Angles and Opposite Angles?
Adjacent angles share a common arm between them and they are always located next to each other. Opposite angles are formed when two lines intersect each other and they are always located opposite to each other.
What are Vertically Opposite Angles?
Opposite angles are also called vertically opposite angles or vertical angles. So, when two straight lines intersect each other, the angles that lie opposite to each other at a vertex are called vertically opposite angles.
What are Opposite Angles in a Parallelogram?
The opposite angles in a parallelogram are those angles that are located diagonally opposite to each other. In a parallelogram, the opposite angles are always equal.
How Many pairs of Opposite Angles are there in a Quadrilateral?
There are 2 pairs of opposite angles in a quadrilateral. The angles connected by the two diagonals are the opposite angles.
Are the Opposite Angles of a Rhombus Congruent?
Yes, the opposite angles of a rhombus are congruent.
Are the Opposite Angles of a Kite Equal?
Only one pair of opposite angles in a kite are equal in measure. In a kite, there are two pairs of opposite angles. In these, there is one set of opposite angles that lies between the sides that are of different lengths. This pair of opposite angles is equal. Therefore, one pair of opposite angles in a kite are equal.