Similar Triangles and Proving Similarity
Decide whether two triangles are similar from equal angles or matching side ratios, set the reason out in one line, and use the scale factor to find a missing length.
What a learner can do afterwards
- Show two triangles are similar by naming the pairs of equal angles
- Match corresponding vertices before writing any ratio
- Find a missing side using the scale factor between matching sides
1 · Read
Two triangles are similar when all their angles match, like a photo and its enlargement. The shortcut is the angle-angle test: if two pairs of angles match, the third must match too, since the angles always sum to 180 degrees. Write it in one line, naming the equal pairs.
Before you write any ratio, match the vertices. List them in matching order so each side pairs with its true partner. Shortest goes with shortest and longest with longest. If the diagram looks twisted, relabel so matching angles share letters.
Take a small triangle with a 3 cm side matching a 12 cm side. The scale factor is 12 divided by 3, which is 4, so a 5 cm side matches 5 times 4, which is 20 cm. With sides 4, 6 and 8 against a shortest side of 10, the factor is 10 divided by 4, which is 2.5.
Equal angles sit opposite matching sides, which gives a cross-check. If the side between the 50 and 60 degree angles pairs up in both triangles, your pairing is right. Five seconds of checking prevents most ratio errors.
Match two angle pairs to prove similarity, pair the vertices, then scale.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.