Proving $\angle ABC$ is Acute: Inside the Triangle $ABC$

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In summary, the given conversation discusses a triangle ABC with point P inside, where BP and CP are greater than AP. The goal is to prove that angle ABC is acute. Using the given information, it can be shown that angle ABC is less than 90 degrees. However, the triangle provided by Dan does not fit the given criteria.
  • #1
maxkor
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Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.
 
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  • #2
triineq.png

We are given that $x>y$ and $w>v$ therefore $x+w > y+v = \angle ABC$

Now $\angle CAB > x$ and $\angle ACB >w$ so $\angle CAB + \angle ACB > x+w > y+v = \angle ABC$

But $\angle CAB + \angle ACB = 180 - \angle ABC$ so $180 - \angle ABC > \angle ABC$ and $\angle ABC < 90$
 
  • #3
I don't get it. Why doesn't the triangle below fit the ciiteria?

-Dan
 

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  • #4
@topsquark
Why doesn't the triangle fit the criteria?
This triangle fits the criteria.
 
  • #5
maxkor said:
@topsquark
Why doesn't the triangle fit the criteria?
This triangle fits the criteria.
Aaahh! I was reading the problem wrong. I thought it was asking to show that the triangle ABC was acute. My bad.

-Dan
 

Related to Proving $\angle ABC$ is Acute: Inside the Triangle $ABC$

1. How do you prove that an angle inside a triangle is acute?

To prove that an angle inside a triangle is acute, you can use the Pythagorean Theorem, the Law of Sines, or the Law of Cosines. These mathematical methods involve using the lengths of the sides of the triangle to determine the measure of the angles.

2. What is an acute angle?

An acute angle is an angle that measures less than 90 degrees. In a triangle, an acute angle is one that is less than 90 degrees and is formed by two sides of the triangle.

3. Can an angle inside a triangle be both acute and obtuse?

No, an angle inside a triangle can only be either acute or obtuse, but not both. An acute angle measures less than 90 degrees, while an obtuse angle measures greater than 90 degrees. Therefore, they cannot be the same angle.

4. What is the difference between an acute angle and a right angle?

An acute angle measures less than 90 degrees, while a right angle measures exactly 90 degrees. In a triangle, a right angle is formed by the intersection of two sides that are perpendicular to each other.

5. Why is it important to prove that an angle inside a triangle is acute?

Proving that an angle inside a triangle is acute is important because it helps us understand the properties and relationships of the angles in a triangle. It also allows us to make accurate measurements and calculations in various fields such as mathematics, engineering, and physics.

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