Question

Invent measurements that would show triangle XYZ is similar to triangle NLM using the Side-Angle-Side Triangle Similarity Theorem.

Answer 1

Triangle XYZ will be similar to triangle NLM if two pairs of sides are proportional and the angle between these sides is congruent in both triangles, according to the Side-Angle-Side Triangle Similarity Theorem.

Step-by-step explanation:

To establish that triangle XYZ is similar to triangle NLM using the Side-Angle-Side (SAS) Triangle Similarity Theorem, we need to verify two conditions. First, that two pairs of corresponding sides are proportional, and second, that the angle between these sides in one triangle is congruent to the angle between the corresponding sides in the other triangle.

For example, if the lengths of sides XY and XZ in triangle XYZ are in the same ratio as the lengths of sides NL and NM in triangle NLM (i.e., XY/NL = XZ/NM), and the angle formed by sides XY and XZ in triangle XYZ (angle XYZ) is equal in measure to the angle formed by sides NL and NM in triangle NLM (angle NLM), then triangles XYZ and NLM are similar by SAS similarity.

Furthermore, consider the application of the SAS similarity in triangulation, for example, to measure the distance to an inaccessible object. This demonstrates the practical use of triangle similarity in calculating properties of a triangle to deduce other distances or measures, such as in the measurement of the width of the Moon or Earth-bound observations.

Answer 2

By verifying that the ratios of corresponding sides are equal and the corresponding angles have the same measure, we can demonstrate the similarity between triangle XYZ and triangle NLM based on the Side-Angle-Side Triangle Similarity Theorem.

How to show that triangle XYZ is similar to triangle NLM

To show that triangle XYZ is similar to triangle NLM using the Side-Angle-Side (SAS) Triangle Similarity Theorem, identify corresponding sides and angles that are proportional in both triangles.

Here are the measurements that would demonstrate the similarity:

Side: Measure the length of side XY and side NL. Ensure that the ratio of the lengths of these sides is equal in both triangles.

For example, if XY measures 4 units and NL measures 8 units, the ratio is 4:8 or 1:2.

Angle: Measure the angle formed by sides XY and YZ and the angle formed by sides NL and LM. Ensure that these angles have the same measure in both triangles.

For example, if angle XYZ measures 60 degrees, angle NLM should also measure 60 degrees.

Side: Finally, measure the length of side YZ and side LM.

Again, ensure that the ratio of the lengths of these sides is equal in both triangles.

Using the previous example, if YZ measures 6 units, LM should measure 12 units to maintain the ratio of 1:2.

By verifying that the ratios of corresponding sides are equal and the corresponding angles have the same measure, we can demonstrate the similarity between triangle XYZ and triangle NLM based on the Side-Angle-Side Triangle Similarity Theorem.

Find the missing triangles in the attached image.