Use congruence and similarity concepts to solve for side lengths/angles. | Lines, angles, and triangles | SAT Math

Know when triangles are congruent or similar: congruent triangles match exactly, while similar triangles keep angles equal and sides proportional.

Core Idea

Similar triangles have the same angles and proportional sides. If two angles of one triangle equal two angles of another (AA), the triangles are similar, and you can set up a proportion to find unknown sides.

Understanding

Rule: Congruent triangles match exactly; similar triangles keep the same shape.

Congruent means equal sides and equal angles.
Similar means equal angles and proportional sides.
Match corresponding vertices before you write the proportion.

Once the pairing is right, cross-multiply and solve.

Step by Step

Confirm similarity: check that two pairs of angles match (AA).
Label corresponding vertices in order (e.g., $△ 𝐴 𝐵 𝐶 \sim △ 𝐷 𝐸 𝐹$ ).
Write the proportion using corresponding sides: $𝐴 𝐵 𝐷 𝐸 = 𝐵 𝐶 𝐸 𝐹 = 𝐴 𝐶 𝐷 𝐹$ .
Plug in known lengths, cross-multiply, and solve.

Misconceptions

Mixing up which sides correspond — always match by the angle they're opposite.
Assuming triangles are similar without verifying at least two equal angles.
Setting up the proportion with non-corresponding sides (e.g., pairing the shortest side of one with the longest of the other).

Question

Worked Example

Triangles $𝑃 𝑄 𝑅$ and $𝑆 𝑇 𝑈$ are similar, with $𝑃 𝑄$ corresponding to $𝑆 𝑇$ . If $𝑃 𝑄 = 6$ , $𝑆 𝑇 = 9$ , and $𝑄 𝑅 = 10$ , what is the length of $𝑇 𝑈$ ?

Select an answer to see the explanation