Triangle Similarity Criteria

AA, SAS, SSS and Proportional Sides

In this lesson:

• Identify similar triangles using AA, SAS, SSS
• Set up proportions to find missing sides
• Solve common ACT similarity problems

What You Will Learn Today

After this lesson, you will:

1. Determine similarity using the AA criterion
2. Apply SAS and SSS similarity criteria
3. Set up proportions for missing side lengths
4. Solve overlapping triangle and shadow problems
5. Relate scale factor to perimeter and area ratios

Same Shape but Different Size

Similar triangles have the same shape but may differ in size.

Similar Triangles Have Matching Angles

Similar means equal corresponding angles and proportional corresponding sides.

• Notation:
• A matches D, B matches E, C matches F
• Letter order specifies the correspondence
• Congruence is similarity with scale factor

Two Matching Angles Guarantee Similarity

The AA criterion: if two angles of one triangle equal two angles of another, the triangles are similar.

• Angles in a triangle sum to
• Two matching angles force the third to match
• AA is the most common ACT similarity test

AA Example With Angle Matching

: angles , ,
: angles , ,

Match by angle measure:

• and
• So

The letter order reflects the correspondence.

Quick Check on Angle-Angle Similarity

Two triangles have these angles:

• : , ,
• : , ,

Are they similar?

Find each missing angle, then decide...

SAS Similarity Requires Proportional Sides

SAS: proportional sides with equal included angle.

• , ,
• , ,

Ratios equal + included angles equal similar by SAS.

SSS Similarity Checks All Side Ratios

SSS Similarity: all three pairs of sides proportional.

Sides: and

All three ratios equal similar by SSS.

If even one ratio differs, not similar.

Quick Check on SSS Similarity

Triangles with sides and .

Are they similar by SSS?

Compute all three ratios...

Identify Corresponding Sides From Similarity

In :

• , ,
• Match shortest to shortest, or by opposite angles
• Always write the similarity statement first

Worked Example for Finding Missing Sides

, with , , .

Find EF:

Your Turn to Find a Missing Side

, , , .

Find DF:

Cross-multiply and solve...

Scale Factor Links Perimeter and Area

The scale factor is the common ratio of corresponding sides.

• Perimeters are in ratio
• Areas are in ratio

If : perimeters scale by , areas by .

Quick Check on Scale Factor

Two similar triangles have corresponding sides cm and cm.

Scale factor . The smaller has area .

What is the area of the larger triangle?

Remember: area scales by , not ...

Parallel Lines Create Similar Triangles

When , then by AA.

Overlapping Triangles With Parallel Lines

In , , , , .

Find EC:

Shadow Problems Use Parallel Sun Rays

A -ft student casts a -ft shadow. A tree casts an -ft shadow.

The parallel sun rays create two similar right triangles.

ACT Practice on Triangle Similarity

Problem: In , . , , . Find .

Set up, cross-multiply, and solve...

Solution to the ACT Practice Problem

Scale factor: , and confirms.

Key Takeaways and Common Mistakes

• AA: two matching angles similar (most tested)
• SAS/SSS: proportional sides + angle, or all ratios
• Proportions: same triangle always in numerator

Watch out:

• One angle is not enough — AA needs two
• Area scales by , not
• Identify corresponding sides before computing

Coming Up Next in ACT Geometry

Up next: Congruence criteria for triangles

• SSS, SAS, ASA, AAS — the congruence tests
• How congruence proofs differ from similarity
• Common ACT congruence problem setups