To demonstrate that two triangles are similar using the triangle similarity criteria, we can choose either the SAS (Side-Angle-Side) criteria or the SSS (Side-Side-Side) criteria. For this explanation, let’s assume we have two triangles, Triangle ABC and Triangle DEF.
Using the SAS (Side-Angle-Side) Similarity Criterion:
Step 1: Show Corresponding Angles Are Equal
Suppose we know the following:
- Angle A = Angle D
- Angle B = Angle E
Since two angles of Triangle ABC are equal to two angles of Triangle DEF, by the Angles-Angle (AA) Similarity Postulate, we can conclude that the third angles must also be equal (Angle C = Angle F, since the sum of angles in a triangle is 180 degrees).
Step 2: Show Corresponding Sides Are Proportional
Next, let’s assume we measure the sides of the two triangles:
- Side AB = 4 cm
- Side AC = 6 cm
- Side BC = 5 cm
And for Triangle DEF:
- Side DE = 8 cm
- Side DF = 12 cm
- Side EF = 10 cm
Now, we will check the ratios of the corresponding sides.
Calculation of Ratios:
For the triangles, we calculate the ratios of the lengths of corresponding sides:
- Ratio of AB to DE: \( \frac{AB}{DE} = \frac{4}{8} = \frac{1}{2} \)
- Ratio of AC to DF: \( \frac{AC}{DF} = \frac{6}{12} = \frac{1}{2} \)
- Ratio of BC to EF: \( \frac{BC}{EF} = \frac{5}{10} = \frac{1}{2} \)
All three ratios are equal: \( \frac{1}{2} \).
Conclusion:
Since we have shown that:
- Two angles in Triangle ABC are equal to two angles in Triangle DEF (by AA criterion).
- The ratios of the corresponding sides are equal (by SAS criterion).
Therefore, by the SAS similarity criterion, Triangle ABC is similar to Triangle DEF.
The conclusion is that triangles are similar if two sides are in proportion and the included angle is equal, which in this case holds true.