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Learn · Mathematics · Grade 9 · Triangles

Congruent and similar triangles

SSS, SAS, ASA for congruence; AA, SAS, SSS for similarity; the similarity ratio for lengths and for areas; Thales and the intercept theorem.

High school Lesson 7 of 31 Interactive figure Practice

First, the rules

Same shape — same size, or scaled

Congruent triangles are copies of each other; similar triangles have the same angles and sides in one fixed ratio. Both are decided by three pieces of information.

★ Congruence criteria
Example 1
SSS three sides · SAS two sides and the angle between them · ASA two angles and the side between them (AAS works too). Not SSA in general, and never AAA.
What congruence gives you
Example 1
Every matching side and angle is equal — write the vertices in matching order (ABC ≅ DEF means A ↔ D, B ↔ E, C ↔ F).
★ Similarity criteria
Example 2
AA two equal angles · SAS two sides in the same ratio and the included angle equal · SSS three sides in the same ratio.
★ Ratios
Example 2
lengths × k, perimeters × k, areas × k²
k is the similarity ratio (scale factor).
★ Intercept theorem (Thales)
Example 3
|AD||DB| = |AE||EC|, |AD||AB| = |DE||BC|
DE ∥ BC with D on AB, E on AC: the small triangle ADE is similar to ABC. The midsegment is the case k = ½.

See it

Scale a triangle by k

Drag the corners of ABC and change k. The copy keeps every angle and multiplies every side by k; the perimeter follows k, the area follows k².

Step by step

Worked examples

1

Proving two triangles congruent

SAS

M is the midpoint of both AB and CD (the segments cross at M). Show that triangles AMC and BMD are congruent and conclude that AC = BD.

ABCDM

Ask yourself: Which three pieces match? Two sides come from the midpoints; what about the angle between them?

Show the solution
  1. |AM| = |MB| and |CM| = |MD| (M is the midpoint of each).
  2. Angle AMC = angle BMD (vertical angles).
  3. Two sides and the included angle: SAS, so AMC ≅ BMD, and the third sides match: |AC| = |BD|.

Answer SAS ⇒ AMC ≅ BMD ⇒ |AC| = |BD|

2

Similar triangles and area

AA · k²

Triangles ABC and DEF have A = D and B = E. |AB| = 6, |DE| = 9 and the area of ABC is 20. Find the similarity ratio and the area of DEF.

Ask yourself: Two equal angles — which criterion is that? Then areas scale with the square.

Show the solution
  1. AA: the triangles are similar with ABC ↔ DEF.
  2. Ratio k = |DE||AB| = 96 = 32.
  3. Areas scale by k²: 20 · 94 = 45.

Answer k = 32, area 45

3

A line parallel to the base

intercept theorem

In triangle ABC, D is on AB and E on AC with DE ∥ BC. |AD| = 4, |DB| = 6 and |BC| = 15. Find |DE| and the ratio of the areas of ADE and ABC.

ABCDE4615?

Ask yourself: ADE is similar to ABC. What is the ratio of AD to the whole side AB?

Show the solution
  1. |AB| = 4 + 6 = 10, so k = |AD||AB| = 410 = 25.
  2. |DE| = k · |BC| = 25 · 15 = 6.
  3. Areas: k² = 425, so A(ADE) : A(ABC) = 4 : 25.

Answer |DE| = 6; 4 : 25

Practice: similar triangles

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Scale a side by the ratio, scale an area by its square, or use the intercept theorem. Your best score is kept in this browser only.

About this lesson

Congruent and similar triangles: the SSS, SAS and ASA congruence criteria, the AA, SAS and SSS similarity criteria, the similarity ratio for lengths and the squared ratio for areas, the intercept (Thales) theorem for a line parallel to one side, and the midsegment as a special case. Worked examples and practice questions.

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