SAS Similarity

High School

Definition

Stands for side-angle-side similarity. If two triangles have a corresponding angle that is equal and two sides are the same proportion to the other triangle's corresponding sides, you can use SAS to prove that the two are similar.

Worked examples

\(\triangle ABC \sim \triangle DEF\) because \(\angle A = \angle D\) and \(\frac{AB}{DE} = \frac{AC}{DF} = 2\)

The angle between the two proportional sides must match for SAS similarity to work.

\(\frac{PQ}{XY} = \frac{3}{6} = \frac{1}{2}\), \(\frac{PR}{XZ} = \frac{4}{8} = \frac{1}{2}\), \(\angle P = \angle X = 50^\circ \Rightarrow \triangle PQR \sim \triangle XYZ\)

Check that the equal angle is between the two pairs of proportional sides, then conclude similarity.

Common mistakes

\(\frac{AB}{DE} = \frac{BC}{EF}\) and \(\angle C = \angle F\) → The angle must be between the two proportional sides If the angle is not included between the proportional sides, you cannot use SAS similarity.
\(\frac{AB}{DE} = 2\) and \(\frac{AC}{EF} = 2\) with \(\angle A = \angle D\) → \(\frac{AB}{DE} = 2\) and \(\frac{AC}{DF} = 2\) with \(\angle A = \angle D\) Match corresponding sides: the sides around angle A must compare to the sides around angle D.
Two sides proportional means the triangles are similar → You need the included angle equal and two sides proportional SAS requires both the side proportions and the angle between them; sides alone are not enough.

Where you'll use it next

SAS similarity is essential for proving triangles similar in geometry proofs, solving indirect measurement problems, and working with scale models and maps in trigonometry and advanced geometry.

Found in 1 StudyPug lesson

Grade 9 Math

Unlock the power of the Side-Angle-Side Similarity Theorem to prove triangle similarity. Learn to identify corresponding sides, angles, and apply this crucial concept in geometry and real-world scenarios.

9th Grade9th