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Two concentric isosceles triangles are drawn from the common point upper A. The smaller triangle has equal side lengths of 3 units and a base of 5 units. The larger triangle has equal side lengths of 6 units and a base of 10 units.

Which statement correctly explains the criterion that proves the triangles are similar?

Statement #1: AA because all three angles are similar since the two angles at angle A
are the same.

Statement #2: SSS because the two side lengths of AD
to AB
and DE
to BC
are proportional, meaning the other side length is also proportional.

Statement #3: SAS because the given side lengths AD
to AB
and DE
to BC
are proportional; additionally, angle B
is the same measure as angle D
when translated.

(1 point)
Statement #
correctly explains the criterion that proves the triangles are similar.

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
2 hours ago

To determine which statement correctly explains the criterion that proves the triangles are similar, let's analyze each statement:

  1. Statement #1 (AA): This statement asserts that all three angles are similar because the two angles at point A are the same. While it is true that if two angles are the same, the triangles are similar, it does not explicitly address the angles formed by the bases of the triangles. However, it gives the implication of angle-angle similarity based on the property of congruent angles at point A.

  2. Statement #2 (SSS): This statement claims that the side lengths AD and AB, as well as DE and BC, are proportional, meaning all corresponding sides of the triangles are proportional. This statement is unclear because it doesn’t explicitly show the side lengths’ ratios, although proportionality of all sides is indeed a requirement for SSS similarity.

  3. Statement #3 (SAS): This statement implies that two sides are proportional (AD to AB and DE to BC) and that the included angles (angle B and angle D) are equal. This would constitute the Side-Angle-Side similarity criterion, which requires that two sides of one triangle are proportional to two sides of another triangle and the included angle between those sides is equal.

Given these descriptions, Statement #1 (AA) is the most straightforward and correct method to demonstrate similarity based purely on angle similarity; thus, it is the most appropriate choice for proving the triangles are similar.

The best answer is: Statement #1 correctly explains the criterion that proves the triangles are similar.