Question

Given that ∆ABC is an isosceles triangle, and AD is the median of ∆ABC, congruence postulate SSS can be used to prove which of the following?

a) ∆ABC is a right-angled triangle.
b) ∆ABC is an obtuse-angled triangle.
c) ∆ABC is an equilateral triangle.
d) ∆ABC is an acute-angled triangle.

Answer 1

The congruence postulate SSS can prove that two halves of an isosceles triangle split by its median are congruent, but it cannot prove whether the triangle is right, obtuse, or acute-angled, nor can it prove the triangle is equilateral without additional angle information.

Step-by-step explanation:

To determine whether the congruence postulate SSS (Side-Side-Side) can be used to prove certain characteristics of an isosceles triangle with a median, we should understand what each of these implies:

Firstly, we know that in an isosceles triangle, at least two sides are congruent. If AD is a median, it divides the triangle into two equal halves. Due to this symmetry, we can infer that the two triangles created (∆ABD and ∆ACD) are congruent to each other. This is because they have two sides equal due to the isosceles nature of ∆ABC and the median creates the third side of equal length in both triangles.

The SSS congruence postulate asserts that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. However, this postulate alone does not indicate whether ∆ABC is right-angled, obtuse-angled, or acute-angled, nor does it prove that ∆ABC is equilateral (which would require all three sides to be equal).

Therefore, while SSS can help prove that the two halves of ∆ABC are congruent when split by the median AD, it cannot prove any specific angle property of ∆ABC, such as being right, obtuse, or acute-angled, without additional information about the angles themselves.