Conditions for Similarity
Conditions for similarity define when two geometric figures are mathematically similar in Grade 8 math (Yoshiwara Core Math). Two figures are similar if and only if their corresponding angles are equal AND their corresponding sides are proportional — both conditions must hold simultaneously. A 3×5 rectangle and a 6×10 rectangle are similar (scale factor 2). A square and a non-square rhombus are not similar despite proportional sides, because their angles differ. Similarity is foundational for scale drawings, indirect measurement, and proportional reasoning throughout Grade 8.
Key Concepts
Property Two figures are similar if, and only if: Their corresponding angles are equal, and Their corresponding sides are proportional. Both conditions must be true for figures to be similar.
Examples A square and a non square rhombus both have proportional sides, but their angles are not equal. Therefore, they are not similar. A 3x5 rectangle and a 6x10 rectangle are similar. All their angles are $90^\circ$ (equal), and their corresponding sides are proportional ($\frac{6}{3} = \frac{10}{5} = 2$). Two trapezoids are similar. One has bases of length 4 and 10 and a height of 3. If the similar trapezoid has a height of 9, its scale factor is $\frac{9}{3}=3$. Its bases will be $4 \times 3=12$ and $10 \times 3=30$.
Explanation For any two shapes to be similar, they must pass two tests. First, all their matching angles must be equal. Second, the ratios of all their matching sides must be the same. Failing either test means they are not similar.
Common Questions
What are the two conditions for similarity?
Two figures are similar if (1) all corresponding angles are equal and (2) all corresponding sides are proportional. Both must hold.
Are a square and a rhombus similar?
No. A rhombus has angles that differ from 90°, so angles are not equal to a square's angles.
How do you check if two rectangles are similar?
Check that ratios of corresponding sides are equal. A 3×5 and 6×10 rectangle: 3/6 = 5/10 = 1/2. ✓
What is the difference between similar and congruent figures?
Congruent figures have equal angles AND equal sides. Similar figures have equal angles and proportional (not necessarily equal) sides.
Why must both conditions hold for similarity?
Each condition alone is insufficient. Figures can have equal angles but wrong proportions, or proportional sides but wrong angles.