explain how the SAS criterion can be used to prove similarity in two triangles

Answer 1

The SAS (Side-Angle-Side) criterion is a method used in geometry to determine the similarity of two triangles. According to the SAS criterion for similarity, if in two triangles:

1. Two sides of one triangle are proportional to two sides of the other triangle.
2. The included angle between those two sides in one triangle is congruent to the included angle between the corresponding sides in the other triangle.

Then, the two triangles are similar.

Steps to Use the SAS Criterion for Similarity:

1. Identify Two Triangles: Start with two triangles, say triangle ABC and triangle DEF.

2. Measure Two Sides: Measure the lengths of two sides of triangle ABC (let’s say AB and AC) and two sides of triangle DEF (DE and DF). Ensure you have the lengths of the corresponding sides.

3. Check Proportions: Establish that the ratios of the lengths of the corresponding sides are equal. Mathematically, this can be expressed as:

   \[ \frac{AB}{DE} = \frac{AC}{DF} \]

   If this proportion holds, it means that the lengths of the sides of the two triangles are proportional.

4. Measure the Included Angles: Measure the included angle between the two sides in triangle ABC (angle A) and the included angle between the two sides in triangle DEF (angle D).

5. Check Angle Congruence: Verify that these angles are congruent:

   \[ \angle A \cong \angle D \]

6. Conclusion: If both the proportional sides condition and the angle congruence condition are satisfied, it follows from the SAS criterion that the triangles ABC and DEF are similar. This can be denoted as:

   \[ \triangle ABC \sim \triangle DEF \]

Important Note:

• The similarity of triangles means that their corresponding angles are equal, and their corresponding sides are proportional. However, the SAS criterion specifically focuses on the condition of two corresponding sides being proportional and the included angle being congruent.
• It's crucial to confirm that you're comparing the correct corresponding angles and sides, as establishing similarity depends on the right correspondence between the triangles.

Example:

• Consider triangles ABC and DEF where \(AB=4\), \(AC=6\), \(DE=8\), and \(DF=12\). The angles are such that \(\angle A = \angle D\).
• Check the proportions: \(\frac{AB}{DE} = \frac{4}{8} = \frac{1}{2}\) and \(\frac{AC}{DF} = \frac{6}{12} = \frac{1}{2}\).
• Since both ratios are equal and \(\angle A \cong \angle D\), we conclude that \( \triangle ABC \sim \triangle DEF \) by the SAS criterion.