Geometry Proofs ( Similarity of Triangles)

We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.

We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.

Affiliations with Schools & Educational institutions are also welcome.

Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit

We will be happy to post videos as per your requirements also. Do write to us.

In this section we will discuss Geometry proofs on similar triangles.
Basic Proportionality Theorem( Thales theorem):
If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.

1) In triangle ABC, DE || BC and intersects AB in D and AC in E.
Given : DE || BC
To Prove thant : AD / DB = AE / EC
Construction : Join BC,CD and draw EF ┴ BA and DG ┴CA.



Proof :
Statements
Reasons
1) EF ┴ BA 1) Construction
2) EF is the height of ∆ADE and ∆DBE 2) Definition of perpendicular
3)Area(∆ADE) = (AD .EF)/2 3)Area = (Base .height)/2
4)Area(∆DBE) =(DB.EF)/2 4) Area = (Base .height)/2
5)(Area(∆ADE))/(Area(∆DBE)) = AD/DB 5) Divide (4) by (5)
6) (Area(∆ADE))/(Area(∆DEC)) = AE/EC 6) Same as above
7) ∆DBE ~∆DEC 7) Both the ∆s are on the same base and
between the same || lines.
8) Area(∆DBE)=area(∆DEC) 8) If the two triangles are similar their
areas are equal
9) AD/DB =AE/EC 9) From (5) and (6) and (7)

2) In triangle ABC, DE || AC where D is a point on AB.F and E are any two points on BC, such that DF || AE.
Given :DE || AC and DF || AE .
Prove that : BF / FE = BE / EC
Proof :
Statements
Reasons
1) DE || AC 1) Given
2) BD/AD = BE/AC 2) Basic Proportionality Theorem
3) DF || AE 3)Given
4) BD/AD = BF/FE 4) Basic Proportionality Theorem in ∆ ABE
5) BE/AC = BF/FE 5) From (2) and (4) (Transitivity)

___________________________________________________________
Note : The converse of this theorem is also true.

If a line divides any two sides of triangle in the same ratio, then the line intersect must be parallel to the third side.
• Geometry proofs
• GeometryProof-1
• GeometryProof 2
• Proofs on Area of similar triangles
• Pythagorean theorem

Home

Russia-Ukraine crisis update - 3rd Mar 2022

The UN General assembly voted at an emergency session to demand an immediate halt to Moscow's attack on Ukraine and withdrawal of Russian troops.