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1. Point - A Point is that which has no component. It is represented by a dot.
2. Line - When we join two distinct points then we get a line. A line has no endpoints it can be extended infinitely.
3. Line Segment - It is the part of the line which has two endpoints.
4. Ray - Ray is also a part of the line which has only one endpoint and has no end on the other side.
5. Collinear and Non-collinear points – Points lie on the same line are known as collinear points and the points that don't lie on the same line are known as Non-Collinear Points.
When two rays begin from the same endpoint then they form an Angle. The two rays are the arms of the angle and the endpoint is the vertex of the angle.
Complementary Angles are those which have the sum of two angles as 90°.
Supplementary Angles are those which have the sum of two angles as 180°.
There are two ways to draw two lines-
1. The lines which cross each other from a particular point is called Intersecting Lines.
2. The lines which never cross each other at any point are called Non-intersecting Lines. These lines are called Parallel Lines and the common length between two lines is the distance between parallel lines.
1. If a ray stands on a line, then the sum of two adjacent angles formed by that ray is 180°.
This shows that the common arm of the two angles is the ray which is standing on a line and the two adjacent angles are the linear pair of the angles. As the sum of two angles is 180° so these are supplementary angles too.
2. If the sum of two adjacent angles is 180°, then the arms which are not common of the angles form a line.
This is the reverse of the first axiom which says that the opposite is also true.
When two lines intersect each other, then the vertically opposite angles so formed will be equal.
AC and BD are intersecting each other so ∠AOD = ∠BOC and ∠AOB = DOC.
If a line passes through two distinct lines and intersects them at distant points then this line is called Transversal Line.
Here line “l” is transversal of line m and n.
Exterior Angles - ∠1, ∠2, ∠7 and ∠8
Interior Angles - ∠3, ∠4, ∠5 and ∠6
Pairs of angles formed when a transversal intersects two lines-
1. Corresponding Angles :
∠ 1 and ∠ 5
∠ 2 and ∠ 6
∠ 4 and ∠ 8
∠ 3 and ∠ 7
2. Alternate Interior Angles :
∠ 4 and ∠ 6
∠ 3 and ∠ 5
3. Alternate Exterior Angles:
∠ 1 and ∠ 7
∠ 2 and ∠ 8
4. Interior Angles on the same side of the transversal:
∠ 4 and ∠ 5
∠ 3 and ∠ 6
1. If a transversal intersects two parallel lines, then
Each pair of corresponding angles will be equal.
Each pair of alternate interior angles will be equal.
Each pair of interior angles on the same side of the transversal will be supplementary.
2. If a transversal intersects two lines in such a way that
Corresponding angles are equal then these two lines will be parallel to each other.
Alternate interior angles are equal then the two lines will be parallel.
Interior angles on the same side of the transversal are supplementary then the two lines will be parallel.
Find ∠DGH.
Here, AB ∥ CD and EH is transversal.
∠EFB + ∠BFG = 180° (Linear pair)
∠BFG = 180°- 133°
∠BFG = 47°
∠BFG =∠DGH (Corresponding Angles)
∠DGH = 47°
If two lines are parallel with a common line then these two lines will also be parallel to each other.
As in the above figure if AB ∥ CD and EF ∥ CD then AB ∥ EF.
1. The sum of the angles of a triangle is 180º.
∠A + ∠B + ∠C = 180°
2. If we produce any side of a triangle, then the exterior angle formed is equal to the sum of the two interior opposite angles.
∠BCD = ∠BAC + ∠ABC
Find x and y.
Solution
Here, ∠A + ∠B + ∠C = 180° (Angle sum property)
30°+ 42° + x = 180°
x = 180°- (30° + 42°)
x = 108°
And y is the exterior angle and the two opposite angles are ∠A and ∠B.
So,
∠BCD = ∠A + ∠B (Exterior angle is equal to the sum of the two interior opposite angles).
y = 30°+ 42°
y = 72°
We can also find it by linear pair axiom as BC is a ray on the line AD, so
x + y = 180° (linear pair)
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