Certainly! To determine which similarity criterion proves that two triangles are similar, we can analyze each of the similarity criteria: AA (Angle-Angle), AAS (Angle-Angle-Side), SSS (Side-Side-Side), and SAS (Side-Angle-Side).
Question 1: Similarity Criterion
A common criterion that is often sufficient to establish similarity is the AA Criterion. If two triangles have two corresponding angles that are congruent, then the triangles are similar. This is because if two angles of one triangle are the same as two angles of another triangle, the third angle must also be the same due to the angle sum property of a triangle.
Therefore, the answer is: AA~
Question 2: Reasoning and Proof
To prove that the two triangles are similar using the AA criterion, follow these steps:
-
Identify Corresponding Angles: Let the triangles be \(\triangle ABC\) and \(\triangle DEF\).
- Suppose we know the angles: \(\angle A \cong \angle D\) and \(\angle B \cong \angle E\).
-
Use the Angle Sum Property:
- Since the sum of the angles in a triangle is \(180^\circ\), we can deduce the third angles as follows: \[ \angle C = 180^\circ - \angle A - \angle B \] \[ \angle F = 180^\circ - \angle D - \angle E \]
- From the congruence of \(\angle A\) and \(\angle D\) and \(\angle B\) and \(\angle E\), it follows that: \[ \angle C = 180^\circ - \angle A - \angle B = 180^\circ - \angle D - \angle E = \angle F \]
-
Conclusion:
- Since all three angles in \(\triangle ABC\) match with the angles in \(\triangle DEF\), we can conclude: \[ \triangle ABC \sim \triangle DEF \quad \text{(by AA~)} \]
This proof establishes that the two triangles are similar based on the premise of two corresponding angles being congruent, thereby satisfying the AA similarity criterion.
Final Answer: The triangles are similar by the AA criterion due to the congruence of two corresponding angles in each triangle. This confirms our earlier choice for Question 1.
1 answer