When two angles have a common vertex and a common arm whose interiors do not overlap are called?

Pairs of angles are discussed here in this lesson.

1. Complementary Angles:

Two angles whose sum is 90° (that is, one right angle) are called complementary angles and one is called the complement of the other.

Here, ∠AOB = 40° and ∠BOC = 50°

Therefore, ∠AOB + ∠BOC = 90°Here, ∠AOB and ∠BOC are called complementary angles.∠AOB is complement of ∠BOC and ∠BOC is complement of ∠AOB.

For Example:

(i) Angles of measure 60° and 30° are complementary angles because 60° + 30° = 90°

Thus, the complementary angle of 60° is the angle measure 30°. The complementary angle angle of 30° is the angle of measure 60°.

(ii) Complement of 30° is → 90° - 30° = 60°(iii) Complement of 45° is → 90° - 45° = 45°(iv) Complement of 55° is → 90° - 55° = 35°(v) Complement of 75° is → 90° - 75° = 15°

Working rule: To find the complementary angle of a given angle subtract the measure of an angle from 90°.

So, the complementary angle = 90° - the given angle.

2. Supplementary Angles:

Two angles whose sum is 180° (that is, one straight angle) are called supplementary angles and one is called the supplement of the other. Here, ∠PQR = 50° and ∠RQS = 130°

∠PQR + ∠RQS = 180° Hence, ∠PQR and ∠RQS are called supplementary angles and ∠PQR is supplement of ∠RQS and ∠RQS is supplement of ∠PQR.

For Example:

(i) Angles of measure 100° and 80° are supplementary angles because 100° + 80° = 180°.

Thus the supplementary angle of 80° is the angle of measure 100°.

(ii) Supplement of - 55° is 180° - 55° = 125°

(iii) Supplement of 95° is 180° - 95° = 85°

(iv) Supplement of 135° is 180° - 135° = 45°

(v) Supplement of 150° is 180° - 150° = 30°

Working rule: To find the supplementary angle of a given angle, subtract the measure of angle from 180°.

So, the supplementary angle = 180° - the given angle.

3. Adjacent Angles:

Two non – overlapping angles are said to be adjacent angles if they have:

(a) a common vertex

(b) a common arm

In the above given figure, ∠AOB and ∠BOC are non – overlapping, have OB as the common arm and O as the common vertex. The other arms OC and OA of the angles ∠BOC and ∠AOB are an opposite sides, of the common arm OB.

Hence, the arm ∠AOB and ∠BOC form a pair of adjacent angles.

4. Vertically Opposite Angles:

Two angles formed by two intersecting lines having no common arm are called vertically opposite angles.

In the above given figure, two lines \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\) intersect each other at a point O.

They form four angles ∠AOC, ∠COB, ∠BOD and ∠AOD in which ∠AOC and ∠BOD are vertically opposite angles. ∠COB and ∠AOD are vertically opposite angle.

∠AOC and ∠COB, ∠COB and ∠BOD, ∠BOD and ∠DOA, ∠DOA and ∠AOC are pairs of adjacent angles.

Similarly we can say that, ∠1 and ∠2 form a pair of vertically opposite angles while ∠3 and ∠4 form another pair of vertically opposite angles.

When two lines intersect, then vertically opposite angles are always equal.

∠1 = ∠2

∠3 = ∠4

5. Linear Pair:

Two adjacent angles are said to form a linear pair if their sum is 180°.

These are the pairs of angles in geometry.

● Angle.

Interior and Exterior of an Angle.

Measuring an Angle by a Protractor.

Types of Angles.

Pairs of Angles.

Bisecting an angle.

Construction of Angles by using Compass.

Worksheet on Angles.

Geometry Practice Test on angles.

5th Grade Geometry Page

5th Grade Math Problems

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Mathematics Standard 7 Term I>Geometry>Try these>Q 3

1. Adjacent Angles:

(i) Two angles which have a common vertex and a common arm, whose interiors do not overlap are called adjacent angles.

(ii) Here, ∠1 and ∠2 are adjacent angles.

2. Linear Pair Angles:

(i) The adjacent angles that are supplementary are called linear pair of angles.

(ii) Here, ∠1 and ∠2 are pair of linear angles.

3. Sum of Angles:

(i) The sum of all angles formed at a point on a straight line is 180°.

(ii) The sum of the angles at a point is 360°.

4. Pair of Angles formed by Two Intersecting Lines:

(i) When two lines intersect each other, two pairs of nob-adjacent angle formed are called vertically opposite angles.

(ii) Here, ∠1 and ∠3 are called vertically opposite angles and ∠2 and ∠4 are called vertically opposite angles.

5. Transversal:

(i) A transversal is a line that intersects two or more lines at distinct points.

(ii) When two parallel lines are cut by a transversal,

(a) each pair of corresponding angles are equal.

(b) each pair of alternate interior angles are equal.

(d) interior angles on the same side of the transversal are supplementary.

(e) exterior angles on the same side of the transversal are supplementary.

6. Bisector:

(i) A perpendicular line which divides a line segment into two equal parts is a perpendicular bisector of the given line segment.

(ii) If a line or line segment divides an angle into two equal angles, then the line or line segment is called angle bisector of the given angle.

7. Properties of Angles of a Triangle:

(i) The sum of three angles in a triangle is 180°.

(ii) The exterior angle of a triangle is equal to the sum of the two interior opposite angles.

(iii) The exterior angles of a triangle add up to 360° .

8. Congruent figures:

(i) Two lines segments are congruent if they have the same length.

(ii) Two angles are congruent if the measures of the angles are equal.

(iii) If the corresponding sides and corresponding angles of two plane figures are equal then they are called congruent figures.

9. Congruency of Triangles:

(i) If three sides of one triangle are equal to the corresponding sides of the other triangle then the two triangles are congruent. This is known as Side–Side–Side criterion.

(ii) If two sides and the included angle of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the two triangles are congruent to each other. This is known as Side–Angle–Side criterion.

(iii) If two angles and the included side of one triangle are congruent to the corresponding two angles and the included side of another triangle, then the triangles are congruent This is known Angle-Side-Angle criterion.

(iv) In right angled triangle, the side which is opposite to right angle is the largest side called Hypotenuse.

(v) If the hypotenuse and one side of a right angled triangle is equal to the hypotenuse and one side of another right angled triangle then the two right angled triangles are congruent. This is known as Right Angle-Hypotenuse Side criterion.

10. Symmetry through transformations:

(i) A transformation is a specific set of rules that change the preimage onto the image.

(ii) A translation is a transformation that moves all points of a figure in the same distance in the same direction.

(iii) In horizontal, the right-side movement is denoted by→and the left side movement is denoted by←.

(iv) In vertical, the upside movement is denoted by ↑ and the downward movement is denoted by ↓.

(v) A reflection is a transformation that “flips” or “reflects” a figure about a line. The line that a figure is flipped over is called a line of reflection.

(vi) A rotation is a transformation that turns every point of the pre-image through a specified angle and direction about a point. The fixed point is called the centre of rotation. The angle is called the angle of rotation.

(vii) A rotation is also called a turn.

(viii) The default direction of a rotation is the anti-clockwise direction.

(ix) Rotation of 360° is called a full turn, rotation of 180° is called a half turn, rotation of 90° is called a quarter turn.

11. Circle and its Related Terms:

(i) The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle. The fixed point is called the centre of the circle and the fixed distance is called the radius of the circle.

(ii) If two points on a circle are joined by a line segment, then the line segment is called a chord of the circle.

(iii) The chord, which passes through the centre of the circle, is called a diameter of the circle.

12. Concentric Circles:

(i) Circles drawn in a plane with a common centre and different radii are called concentric circles.

(ii) The area between the two concentric circles is known as circular ring.

(iii) Width of the circular ring w=r2-r1.

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