Identifying Similar Triangles - Interactive Activity

Definitions and Main Ideas

There are three shortcuts for proving two triangles similar without checking every angle and every side. The AA Postulate needs only two pairs of congruent angles. SSS Similarity needs all three pairs of sides proportional, and SAS Similarity needs two pairs of sides proportional with the included angles congruent. The work of this lesson is choosing the right test and verifying its condition completely — especially for SSS, where all three sides must be checked.

Vocabulary

AA Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

SSS Similarity Theorem

If all three sides of one triangle are proportional to all three sides of another triangle, then the triangles are similar.

SAS Similarity Theorem

If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.

Included Angle

The angle formed by two sides of a triangle; for SAS Similarity the congruent angle must be the one between the two proportional sides.

Main Ideas

1. AA: two congruent angle pairs are enough, because the third angles are then congruent automatically (the angles of a triangle add to 180°).
2. SSS Similarity: form all three ratios of corresponding sides — the triangles are similar only if all three ratios are equal.
3. SAS Similarity: the congruent angle must be the included angle, between the two pairs of proportional sides.
4. When a pair of triangles fails any one check, the triangles are not similar, even if the other checks pass.
5. Write the similarity statement in corresponding order once similarity is established.

Three Tests Explorer

What to do: Choose one of the three tests below. The figure shows a pair of triangles that passes that test, and the readout underneath verifies the test's condition with the measures from the figure. Try all three tests and notice what each one checks.

Mode buttons: each button shows a pair of triangles that passes that test. Watch the readout change to verify that test's condition.

Name the Test

What to do: Look at the givens marked on the pair of triangles — angle measures, side lengths, and matching marks. Choose the one test that proves the triangles similar, or choose Not similar when no test's condition is fully met. Check every part of a test's condition before you choose.

Item 1 of 8

Answer choices: choose the test that fits all of the givens. After you choose, the figure highlights the parts that the test you chose relies on.

Next Item button: moves to the next pair of triangles in order. Watch the item counter change.

What Information Is Needed?

What to do: Each item names the test you must use and shows only part of the information that test needs. Choose the additional information that would complete that test's condition.

Item 1 of 6

Answer choices: choose the additional information the named test still needs. After you choose, the figure highlights the parts that the information you chose concerns.

Next Item button: moves to the next item in order. Watch the item counter change.

Build the Proof of Similarity

What to do: Work through three steps for each pair of triangles. Step 1 chooses the test that fits the givens, Step 2 chooses the correct verification of that test's condition, and Step 3 chooses the correct conclusion. A worked solution appears when all three steps are complete.

Problem 1 of 5

Step 1 of 3

Answer choices: choose the statement that is correct for this step. The figure highlights the parts your choice names while your choice is checked.

New Problem button: moves to the next pair of triangles in order. Watch the problem counter change.

Your Turn

What to do: Answer each question. A first miss gives a hint and another try, with the missed choice crossed out. The full explanation appears when you answer correctly or after a second miss. Your score counts first-try answers only.

First-try score: 0 of 0 answered

Answer choices: choose the one correct answer. After you choose, read the feedback underneath the choices.

Previous and Next buttons: move between questions. Watch the question strip and the score line change.