Triangle geometry nastiness

In summary, the conversation discusses how to prove that in a triangle with two unequal sides, the angle opposite the shorter side will be smaller than the angle opposite the larger side. The participants suggest using the external angle theorem and trigonometry to prove this, but the person asking for help is not allowed to use trigonometry in their geometry test. The conversation ends with gratitude for the explanations and the opportunity to learn something new.
  • #1

steppenwolf

i might just be stupid and blind to a really obvious answer, but this just stopped me dead in our end of unit geometry test today:

prove that in a triangle with two unequal sides the angle opposite the shorter side will be smaller then angle opposite the larger side.

help! so geometry isn't my forté but what don't i see? even just a little hint would be welcome, I'm sure this won't be a challenge for most of you!
 
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  • #2
Here's the best I could come up with off the top of my head- it seems to me to be a bit awkward.


Call the angles (and vertices) A, B, C and suppose the side opposite A is shorter than the side opposite B. Striking an arc with center at C and radius the length of side BC strikes side AC inside the triangle ABC (BECAUSE BC is shorter than AC). Call this point D. Connecting BD gives isosceles triangle BCD. Call the base angles (in other words not angle C) of BCD "theta". The line BD divides angle B into two angles, one of which is theta. Call the other angle at B, "gamma". Then we have theta= A+ gamma and B= theta + gamma.
Those give B= A+ 2gamma. Since the measures of the angles are all positive, B> A.
 
  • #3
Let a < b, then we must prove [alpha] < [beta].
Let h be the height of the triangle with respect to c.
Then, h = b * sin [alpha] = a * sin [beta].
So, a/b = sin [alpha] / sin [beta].
If both [alpha] and [beta] are <=90°, then we're done.
If not, then it's obvious that only one of them can be >90°. I think it's easy to prove that this can only be true for [beta].
 
  • #4
hallsofivy that is just beautiful! thankyou so much, even if i lost 12% of my mark i have still learned something. thanks also arcnet, but unfortunately i don't think we were permitted to use trig at all, only axioms and very elementary theorems. i am frustrated as i used the external angle theorem to get through the vast majority of questions but didn't see its use here. thanks again!
 

What is "Triangle geometry nastiness"?

"Triangle geometry nastiness" refers to the complex and difficult concepts and equations involved in solving problems related to triangles and their properties.

What are the basic properties of a triangle?

A triangle is a 2-dimensional shape with three sides and three angles. The sum of the angles in a triangle is always 180 degrees. The lengths of the sides can vary, but the relationship between the angles and sides remains constant.

How do you find the missing angle in a triangle?

To find the missing angle in a triangle, you can use the fact that the sum of the angles is 180 degrees. If you know the measurements of two of the angles, you can subtract them from 180 to find the missing angle.

What is the Pythagorean Theorem?

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. It can be written as a^2 + b^2 = c^2, where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse.

How do you calculate the area of a triangle?

The area of a triangle can be calculated using the formula A = (1/2)bh, where A is the area, b is the base of the triangle, and h is the height of the triangle. The base and height can be any two sides of the triangle, as long as the height is perpendicular to the base.

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