Pengi Editor's Note: This article was originally published by Think Academy. We're sharing it here for educational value. Think Academy is a leading K-12 math education provider.
Similar Triangles Explained: Geometry Rules & Examples
In geometry, "similar" means two figures share the same shape but not always the same size. Similar triangles have equal angles and proportional sides—one is a scaled version of the other. Common mistakes include confusing similar with congruent, misunderstanding side proportions, and mixing up the area ratio. Mastering these ideas is a key part of Common Core Grade 7 math and lays the groundwork for geometry proofs and real-world problem solving.
What Does "Similar" Mean?
In geometry, the word similar means exactly the same in shape, but might be different in size. Two figures are similar if one can be transformed into the other by flipping, turning, sliding, with resizing.
Similar vs. Congruent
Congruent shapes are a special kind of similar shapes. When two shapes are congruent, they have exactly the same in size and shape. That means we don't need to resize, just slide, flip, or turn.
What Are Similar Triangles?
Similar triangles have the same angles and their matching sides are in the same ratio. One triangle is a resized version of the other.
This idea is often written as:
Example Problems: Similar Triangles
Example 1
Problem: Given that (\triangle ABC \sim \triangle DEF, \quad m{\large \angle}A = 45^\circ, \quad m{\large \angle}B = 90^\circ). What is the measure of ({\large \angle}F , ?)?
Solution:
(∵ ; \triangle ABC \sim \triangle DEF, ) all corresponding angles are equal.
[∴ ; m{\large \angle}C = m{\large \angle}F]
[∵ ; m{\large \angle}C = 180^\circ – (m{\large \angle}A + m{\large \angle}B) = 180^\circ – (45^\circ + 90^\circ) = 45^\circ]
[∴ ; m{\large \angle}F = m{\large \angle}C = 45^\circ]
Example 2
Problem: Given that △XYZ ~ △MNO, and the ratio of their sides is 2:3. The area of △XYZ is 20 cm². What is the area of △MNO?
Solution:
We use the area ratio, which is the square of the side ratio:
[\left( \frac{2}{3} \right)^2 = \frac{4}{9}]
This means △XYZ is (\frac{4}{9}) the area of △MNO, so:
[\frac{20 ,\text{cm}^2}{A_{\triangle MNO}} = \frac{4}{9}]
[A_{\triangle MNO} = 45 ,\text{cm}^2]
Summary: Key Takeaways About Similar Triangles
- Similar means same shape, but not always the same size. One figure can be flipped, turned, slid, and then resized to match the other.
- If △ABC ~ △DEF, then:
- ( m{\large \angle} A = m{\large \angle} D, ; m{\large \angle} B = m{\large \angle} E, ; m{\large \angle} C = m{\large \angle} F )
- ( \frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF} )
- ( \text{Area ratio} = \left(\frac{AB}{DE}\right)^2 )
- Similar vs. Congruent
- Congruent: same shape and same size
- Similar: same shape, size may be different
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