---
title: "Similar Triangles and Proving Similarity"
description: "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."
canonical: https://lightmysky.com/learn/mathematics/similar-triangles-and-proving-similarity-mt_NPRVlr4lrs
source: https://lightmysky.com/learn/mathematics/similar-triangles-and-proving-similarity-mt_NPRVlr4lrs.md
retrieved: 2026-09-12
---

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# 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.

Subject: Mathematics · Area: Geometry · Ages 14 to 15
Page: https://lightmysky.com/learn/mathematics/similar-triangles-and-proving-similarity-mt_NPRVlr4lrs

## Ready when they can

- 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

## Lesson: Same shape, new size

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.

**Example.** 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.

**Tip.** 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.

**Recap.** Match two angle pairs to prove similarity, pair the vertices, then scale.

## Practice

13 questions on this page, each with its working shown.

## Needs first

- [Scale drawings and maps](https://lightmysky.com/learn/mathematics/scale-drawings-and-maps-mt_1badik7iKJ)
- [Angles with parallel lines](https://lightmysky.com/learn/mathematics/angles-with-parallel-lines-mt_H_pNJ3ZI_S)

## Opens up

- [Trigonometry basics](https://lightmysky.com/learn/mathematics/trigonometry-basics-mt_KB_Czd7RQH)
- [Similar Shapes: Area and Volume Scale Factors](https://lightmysky.com/learn/mathematics/similar-shapes-area-and-volume-scale-factors-mt_v37c_AeLQa)