There are two types of pairings of angles: adjacent angles and vertical angles. The major idea of geometry presented to us in post-primary classes is angles. Angles are a fundamental idea that is explored in all academic disciplines. At school, we study a variety of angles and their properties.
The angle generated by the ray between its initial and final positions is the measure of rotation of a ray when rotated about its terminus. Two rays that are joined end-to-end make an angle.
A pair of angles is sometimes used in geometry. Complementary angles, adjacent angles, linear pairs of angles, opposite angles, and so on are all examples of pairs of angles. We’ll go over the definitions of adjacent angles and vertical angles in-depth in this article.
When two angles have the same vertex and side, they are said to be neighbouring angles. Vertex of the angle is a point where the rays meet to form the sides of the angle. When two adjacent angles share the same vertex and side, they might be complementary or supplementary angles.
- If the sum of neighbour angles equals 90°, then it can be called as the complimentary.
- If the total of adjacent angles equals 180°, then they are said to be supplementary.
Example of Adjacent Angles
Consider a wall clock. The minute hand and second hand of the clock make one angle, denoted by ∠AOC, and the hour hand and second hand form another, denoted by ∠COB. Both of these pairs of angles, ∠AOC and ∠COB, are known as adjacent angles since they are near to each other. The uncommon arms are on either side of the common arms in ∠AOC and ∠COB. They share a common vertex and a common arm. Adjacent angles are those that are adjacent to one another.
Properties of Adjacent Angles
The following are some of the essential properties of adjacent angles:
Two angles are adjacent if they are in the same plane.
- They all have the same vertex.
- They have the same arm.
- It doesn’t have a common interior-point, hence adjacent angles don’t overlap.
Vertical angles, also known as vertically opposite angles, are generated when two lines intersect. The sum of two vertically opposite angles is always the same. A vertical angle and its adjacent angle are also called supplementary angles since they sum up to 180 degrees. When two lines connect and form an angle, such as X=45°, the opposite angle is also 45°. And the angle next to angle X will be 180 – 45 = 135 degrees.
A pair of non-adjacent angles formed by the intersection of two straight lines is known as a vertical angle. In simple terms, vertical angles are placed in the corners of the “X” created by two straight lines, across from one another. Because they are perpendicular to each other, they are also known as vertically opposite angles.
Theorem Related to Vertical Angles
Angles that are vertically opposite angles theorem Two opposed vertical angles created when two lines intersect one another are always equal (congruent) to each other, according to the theorem.
Proof:- considering two intersecting lines and at the intersecting point four angles are formed. Let us say them ∠1, ∠2,∠3,∠4 where (∠1, ∠2),( ∠2,∠3),(∠3,∠4),(∠1, ∠4) are the adjacent angles such that sum of all these adjacent angles are 180° as they are linear pair of angles.
∠1 + ∠2 = 180° (both are the linear pair of angles) ——— (1)
∠1 +∠4 = 180° (both are the linear pair of angles) ——— (2)
From equations (1) and (2), ∠1 + ∠2 = 180° = ∠1 +∠4.
From the property of transitive, if a = b and b = c then a = c. Therefore, we can write the same statement as ∠1 + ∠2 = ∠1 +∠4. ——–(3) By removing ∠1 from both sides of the equation (3), we get ∠2 = ∠4. Similarly. we will use the same process to find out the ∠1 = ∠3. Therefore, we come to the point that the vertically opposite angles are equal.
The term “neighbouring angles” refers to two angles that have the same vertex and side. This is because the ray endpoints, which generate the angle’s side, constitute the vertex of an angle. When adjacent angles share the same vertex and side, they are referred to as complementary or supplementary angles. The vertical angle theorem asserts that a pair of crossing lines’ vertically opposite angles are equivalent. Angles in the vertical plane are always equal. Vertical angles can be both complementary and supplementary. Vertical angles are never parallel.
PROOF OF AREA OF SIMILAR TRIANGLE:-
Statement:- The square of the ratio of any pair of their respective sides is equal to the ratio of the areas of two identical triangles.
Given: Consider the following two triangles: ∆ABC and ∆DEF. And ∆ABC ~ ∆DEF.
To prove:- Area of ABC/Area of DEF = (AB)²/(DE)² = (BC)²/(EF)²= (AC)²/(DF)² .
Construction:- To the sides BC and EF, draw the perpendicular AP and DQ, respectively.
Proof:- Since ∠B = ∠E ……..(ΔABC ~ ΔDEF)
and, ∠APB = ∠DQE…..[AP and DQ are perpendicular on sides BC and EF, respectively, and both angles are 90 degrees]
We can see that ABP and DEQ are equiangular using the AA property of triangle similarity.
As a result, ΔABP ~ ΔDEQ .As a result, AP/DQ = AB/DE. This also suggests that:BC/EF = AP/DQ ——- (1)
Thus, Area(ΔABC)/Area(ΔDEF) = [(1/2) × BC × AP]/[(1/2) × EF × DQ]
= (BC/EF) × (AP/DQ)
= (BC/EF) × (BC/EF) ….[from (1)]
⇒ Area(ΔABC)/Area(ΔDEF) = (BC/EF)²
In the similar manner, we may demonstrate that,
Area of ΔABC/Area of ΔDEF = (AB)²/(DE)² = (BC)²/(EF)² = (AC)²/(DF)²
POINTS TO KEPT IN MIND:-
- The square of the ratio of any pair of their respective sides is equal to the ratio of the areas of two identical triangles.
- Area of ABC/Area of DEF = (AB)2/(DE)2 = (BC)2/(EF)2 = (AC)2/(EF)2
- For identical triangles, all corresponding angle pairs are equal, and all corresponding sides are proportional.
- The ratio of the squares of the matching medians determines the area of two identical triangles.
- The ratio of the squares of corresponding elevations is the area of two similar triangles.
- The ratio of the squares of the respective angle bisector segments determines the areas of two identical triangles.
- When the areas of two identical triangles are equal, the triangles are congruent, meaning they are equal and similar.
The square of the ratio of any pair of comparable triangles’ corresponding sides is equal to the ratio of the area of two similar triangles. When two triangles are comparable, their respective angle pairs are equal and their corresponding sides are proportional.
Geometric figures with the same shape but various sizes are known as similar figures. Two triangles are considered to be similar if their respective angles are equal and their corresponding sides are proportionate.