AA Similarity Criterion (Motivate) MCQs Quiz | Class 10

Understanding AA Similarity Criterion

The Angle-Angle (AA) Similarity Criterion is a fundamental concept in geometry that provides a straightforward way to determine if two triangles are similar. Similar triangles have the same shape but not necessarily the same size. This means their corresponding angles are equal, and their corresponding sides are in proportion.

What is AA Similarity?

The AA Similarity Criterion states that if two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. The third angles must also be equal because the sum of angles in a triangle is always 180 degrees.

For example, if in triangle ABC and triangle DEF, angle A equals angle D, and angle B equals angle E, then triangle ABC is similar to triangle DEF (denoted as triangle ABC ~ triangle DEF). This implies that angle C must also equal angle F.

Consequences of Similarity: Proportional Sides

A crucial consequence of two triangles being similar is that their corresponding sides are proportional. This means the ratio of the lengths of corresponding sides is constant. For similar triangles ABC and DEF (where A corresponds to D, B to E, and C to F), we have:

AB / DE = BC / EF = AC / DF = k (where k is the constant ratio of similarity)

This proportionality of sides is incredibly useful for finding unknown side lengths in similar figures.

Key Takeaways for AA Similarity:

• Two Angles Are Enough: You only need to prove that two pairs of corresponding angles are equal. The third pair automatically becomes equal.
• Shape Preserved: Similar triangles maintain their shape, only their size changes (scaling up or down).
• Applications: Used extensively in geometry for proofs, finding heights and distances (e.g., using shadows), and understanding scaling.
• Real-World Relevance: Architects, engineers, and designers use principles of similarity for scaling models and designs.

Steps to Apply AA Similarity:

1. Identify two triangles that you suspect might be similar.
2. Look for pairs of corresponding angles that are equal. These could be given, vertically opposite angles, alternate interior angles (if lines are parallel), or common angles.
3. If you find two such pairs, declare the triangles similar by AA Similarity Criterion.
4. Once similarity is established, you can then state that their corresponding sides are proportional to solve for unknown lengths or prove other geometric properties.

Quick Revision List:

• AA stands for Angle-Angle.
• If angle A = angle D and angle B = angle E, then triangle ABC ~ triangle DEF.
• The ratio of corresponding sides is equal: AB/DE = BC/EF = AC/DF.
• Similarity is a relationship between shapes, not sizes.

Practice Questions:

1. In triangles PQR and XYZ, if angle P = 50 degrees, angle Q = 70 degrees, angle X = 50 degrees, and angle Y = 70 degrees, are the triangles similar? By what criterion?
2. Two triangles have angles (60, 80, 40) and (60, 40, 80) respectively. Are they similar? Explain.
3. If triangle MNO ~ triangle RST, and MN = 4 cm, RS = 6 cm, MO = 6 cm, what is the length of RT?
4. A vertical pole of length 6 meters casts a shadow 4 meters long on the ground. At the same time, a tower casts a shadow 28 meters long. Find the height of the tower.
5. In triangle ABC, a line DE is drawn parallel to BC, intersecting AB at D and AC at E. Is triangle ADE similar to triangle ABC? Justify your answer using AA Similarity.