SAS Similarity Criterion (Motivate) MCQs Quiz | Class 10

Understanding SAS Similarity Criterion

The SAS (Side-Angle-Side) Similarity Criterion is a fundamental theorem in geometry used to prove that two triangles are similar. It provides a powerful tool for comparing triangles when we have information about two sides and the included angle.

What is SAS Similarity Criterion?

The SAS Similarity Criterion states that if one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

• Condition 1: Equal Included Angle – A corresponding angle in both triangles must be equal. For example, if we consider triangles ABC and DEF, then angle A must be equal to angle D.
• Condition 2: Proportional Including Sides – The two sides that form (include) the equal angle must be proportional. So, if angle A = angle D, then the ratio of the sides including angle A (AB and AC) must be equal to the ratio of the sides including angle D (DE and DF). That is, AB/DE = AC/DF.

If both these conditions are met, then triangle ABC is similar to triangle DEF (written as triangle ABC ~ triangle DEF).

Motivation and Proof Sketch

The term “motivate” in the topic suggests understanding the reasoning behind this criterion. The SAS Similarity Criterion can be logically derived from the AA (Angle-Angle) Similarity Criterion. One way to visualize this is by placing the smaller triangle inside the larger one such that the equal angles coincide. Due to the proportional sides, a line segment can be drawn parallel to one side of the larger triangle, creating a smaller similar triangle within, which confirms the AA similarity.

Steps to Apply SAS Similarity Criterion

1. Identify Corresponding Angles: Find a pair of corresponding angles in the two triangles that are equal.
2. Check Including Sides: Identify the two sides that include these equal angles in each triangle.
3. Verify Proportionality: Calculate the ratio of the lengths of the corresponding sides. If the ratios are equal, then the sides are proportional.
4. Conclude Similarity: If both the angle is equal and the including sides are proportional, then the triangles are similar by SAS.

Example

Consider two triangles, ABC and PQR:

• In triangle ABC: AB = 4 cm, AC = 6 cm, angle A = 50 degrees.
• In triangle PQR: PQ = 8 cm, PR = 12 cm, angle P = 50 degrees.

Here, angle A = angle P = 50 degrees. Now, let’s check the proportionality of the sides including these angles:

AB/PQ = 4/8 = 1/2

AC/PR = 6/12 = 1/2

Since angle A = angle P and AB/PQ = AC/PR, triangle ABC is similar to triangle PQR by the SAS Similarity Criterion.

Properties of Similar Triangles (Resulting from SAS Similarity)

Once two triangles are proven similar by SAS (or any other criterion), they possess certain properties:

• All corresponding angles are equal.
• All corresponding sides are in the same proportion.
• The ratio of their corresponding altitudes, medians, and angle bisectors is equal to the ratio of their corresponding sides.
• The ratio of their perimeters is equal to the ratio of their corresponding sides.
• The ratio of their areas is equal to the square of the ratio of their corresponding sides.

Quick Revision Points

• SAS Criterion: One equal angle, proportional *including* sides.
• Order Matters: When stating similarity (e.g., ABC ~ DEF), the order of vertices indicates corresponding parts.
• Similarity vs. Congruence: Similarity means same shape, different size (proportional). Congruence means same shape, same size (equal).
• Applications: Used in various real-world problems involving scaling, maps, and architectural designs.

Extra Practice Questions

1. If angle X = angle D and XY/DE = XZ/DF, then triangle XYZ is similar to triangle DEF by:
   (a) AA Similarity
   (b) SSS Similarity
   (c) SAS Similarity
   (d) RHS Congruence
2. In triangles GHI and JKL, angle H = 60 degrees, GH = 5, HI = 10. In triangle JKL, angle K = 60 degrees, JK = 7.5, KL = 15. Are they similar by SAS?
   (a) Yes
   (b) No, because 5/7.5 is not 10/15
   (c) No, because the angles are not 90 degrees
   (d) Cannot be determined without more information
3. Which pair of values for sides A and B would make triangle MNO (MN=9, MO=12, angle M=70) similar to triangle PQR (PQ=A, PR=B, angle P=70)?
   (a) A=3, B=4
   (b) A=18, B=24
   (c) A=4.5, B=6
   (d) All of the above
4. The SAS similarity criterion is a condition that guarantees two triangles have:
   (a) Equal area
   (b) Equal perimeter
   (c) Proportional corresponding sides and equal corresponding angles
   (d) All sides equal
5. If the ratio of the areas of two triangles similar by SAS is 4:9, what is the ratio of their corresponding sides?
   (a) 2:3
   (b) 4:9
   (c) 16:81
   (d) Cannot be determined