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# Lines and Similarity of Triangles Tips and Tricks

In this article we will discuss about basic of Geometry that is** Line and Angle similarity** concepts. This article is very important as it will clear your doubts about angle and triangle similarity.

**Line:**

- A line is made up of infinite points and it has no end point.
- A line segment is a part of line which has two end point.
- A ray has only one point and it will goes up to infinity in one direction.

**Collinear points:**

Three or more than three points are said to be collinear if a line contains all of them.

Here A, B and C are collinear point as all of them lie on the same point.

**Concurrent Line:**

Three or more than three lines are said to be collinear if there is a point which lie on all of them..

AB, CD & EF are concurrent line.

**Parallel lines:**

- If two lines have no point of intersection if they produced up to infinity then the lines are parallel to each other.
- Intersecting lines:
- If two lines have a point in common they are said to be intersecting lines.

Angle:

an angle is formed when two or more than two lines or line segment meets.

**Here are some important angel:**

**Acute angle :**

An angle which is smaller than 90° is called an acute angle.

**Right angle :**

An angle of 90° is called an right angle.

**Obtuse angel:**

Angles greater than 90° are called obtuse angle.

**Straight angle:**

Angle of 180° is known as straight angle as it is formed on a straight line.

**Reflex angle:**

Angle greater than 180° but smaller than 360° are known as reflex angle.

180°<r<360°

**Complementary angle:**

When the sum of two angles measure is 90° then the angles are said to be complementary angle.

**Supplementary angle:**

when the sum of two angle measure is 180° then the angles are said to be supplementary angle.

here sum of angle x and y is 180° as AB is a straight line. So angle x and y are supplementary.

**Adjacent angle:**

Two angle are said to be adjacent angle if ;

- They have a common vertex.
- They have a common arm.
- The non-common arm are on either side of the common arm.

**Linear pair:**

A linear pair is pair of adjacent angles whose non-common sides are opposite to each other. That is supplementary angle are always linear pair.

**Vertically opposite angles.**

when two lines intersect they form two pair of vertically opposite angles.

Vertically opposite angle always equal to each other.

Here ∠AOD=∠BOC

∠AOC=∠BOD

**Angles made by a transverse line:**

If two parallel lines are intersected by a transverse line then:

- Each pair of corresponding angles are equal i.e.( ∠1= ∠5, ∠2= ∠6, ∠4= ∠7 and ∠3= ∠8)
- Each pair of alternate interior angles are equal i.e.( ∠3= ∠5 and ∠4= ∠6)
- Interior angles on the same side of the transverse are supplementary.

### Congruent and Similar triangle:

**Congruent triangle:**

- Two triangle are said to be congruent if they are
**equal**in all respects i.e. - Each of the
**three sides**of one triangle must be equal to the three respective sides of the other. - Each of three
**angles**of one triangle must be equal to the other three respective angles of the other.

AB=PQ, AC=PR & BC=QR

∠A=∠P, ∠B=∠Q & ∠C=∠R

**Condition of Congruent Triangle:**

**S-S-S( Side- Side-Side)**

If AB=PQ, AC=PR & BC=QR then triangle ABC and triangle PQR are congruent to each other.

**S-A-S(Side-Angle-Side)**

If two sides and the angle between the two sides are equal then both the triangle are congruent to each other.

if in the above figure AB=PQ , BC= QR and the angle ABC = angle PQR

**A-S-A(Angle-Side-Angle)**

If ∠B=∠Q, ∠C=∠R and the sides BC=QR then above two triangle are congruent.

**A-A-S(Angle-Angle-Side)**

if ∠B=∠Q, ∠C=∠R and AC=PR then the above two triangle are congruent to each other.

**R-H-S(Right-Hypotenuse-Side)**

if ∠B=∠Q, BC=QR and AC=PR then the above triangle are congruent to each other.

**Similarity of Two triangle:**

Two triangle are similar are said to be similar if the corresponding angles are congruent and the corresponding sides are in proportions.

**Condition for similarity of Triangles:**

- A-A( Angle-Angle)
- S-A-S(Side-Angle -Side)
- S-S-S(side-Side-Side)
- Congruent triangle are always similar to each other.

**Some Properties of Similar Triangle**

**(1)**If two triangle are similar to each other then:

Ratio of sides= ratio of heights

= Ratio of medians

=Ratio of angle bisector

= Ratio of inradius

= Ratio of circumradius

**(2)**Ratio of area of two similar triangle = ratio of squares of corresponding sides

i.e. if triangle ABC is similar to traingle PQR then

**(3) **If a line drawn parallel to one side of the triangle to intersect the other two sides in distinct points , the other sides are divided in the same ratio.

and if D and E are the mid points of AB and AC and DE parallel to BC then

DE=BC/2

**(4) ** Two triangle between the two parallel lines are parallel to each other.

** **