# Prove Triangle Angles <= 30 Degrees | Let ABC & P Be a Triangle

• bogdan
In summary, At least one of the angles PAB, PBC or PCA measures less or equal to 30 degrees, as proven by the fact that any of the angles (Aº, Bº or Cº) greater than 60º would result in the smallest angle being less than 60º. Therefore, PAB, PBC or PCA will always measure less than or equal to 30 degrees, demonstrated by the formula Aº+Bº+Cº = 180 and the fact that either <PAC or <PAB will be less than or equal to 30. This question has appeared in the International Mathematical Olympiad (IMO) of 1991.

#### bogdan

Let ABC be a triangle and P a point inside it...
Prove that at least one of the angles PAB, PBC or PCA measures less or equal to 30 degrees...

Aº+Bº+Cº = 180
Aº=Bº=Cº=60º is the largest value for the smallest angle.
(60º)/2=30º

If any of the angles (Aº,Bº or Cº) are greater than 60º then the smallest angle is less than 60º. Thus PAB, PBC or PCA measures less or equal to 30 degrees.

Why 60/2 ?

Because 60 is the greatest possible value that the smallest angle can be.

60+60+60=180 smallest 60
59+61+60=180 smallest 59
30+60+90=180 smallest 30

Its impossible for the smallest value to be greater than 60

Take a look...
http://www.angelfire.com/pro/fbi/tri.bmp

Last edited by a moderator:
PCA is less that 30 in that pic.
No matter where the point P is there will always be an angle equal to (in the case of an equalateral) or less than (in all other cases) 30.

Yeah...but I need a demonstration...

What do u mean by demonstration.

proof...logical...mathematical...

Aº+Bº+Cº = 180
Aº=180-Bº-Cº where Aº is the smallest angle.
<PAC+<PAB = Aº
either <PAC or <PAB is less than or equal to 30
Is that mathematical enough

another IMO question

This question appears in IMO 1991

Of course...but it "looks" simple enough to be solved by anyone...

Originally posted by KL Kam
This question appears in IMO 1991

would this be online and if so could you post the link?

is there a website with past IMO?

Originally posted by marcus
would this be online and if so could you post the link?

is there a website with past IMO?

no need to post it, thanks anyway

## 1. What does it mean to prove triangle angles are less than or equal to 30 degrees?

When we say "prove triangle angles are less than or equal to 30 degrees," we are asking for a mathematical proof that shows all three angles of a triangle are either less than or equal to 30 degrees.

## 2. How do you prove triangle angles are less than or equal to 30 degrees?

There are a few different methods for proving that triangle angles are less than or equal to 30 degrees. One common method is using the law of cosines or the law of sines to calculate the angles and show that they are all less than or equal to 30 degrees.

## 3. Why is it important to prove triangle angles are less than or equal to 30 degrees?

Proving that triangle angles are less than or equal to 30 degrees is important because it allows us to determine the shape and properties of a triangle. It also helps us to solve various geometric problems and understand the relationships between the sides and angles of a triangle.

## 4. Can we prove triangle angles are less than or equal to 30 degrees for any type of triangle?

Yes, we can prove that triangle angles are less than or equal to 30 degrees for any type of triangle, including equilateral, isosceles, and scalene triangles. The method for proving this may vary slightly depending on the type of triangle, but the concept remains the same.

## 5. What are some real-life applications of proving triangle angles are less than or equal to 30 degrees?

Proving triangle angles are less than or equal to 30 degrees can have practical applications in various fields such as architecture, engineering, and surveying. It can also be used in navigation and map-making to accurately determine distances and angles between different locations.