Triangle Similarity Criteria

AA, SAS, SSS and Proportional Sides

In this lesson:

• Decide when triangles are similar using AA, SAS, SSS
• Set up proportions to find missing sides
• Use parallel lines, shadows, and scale factor

What You Will Learn Today

1. Find a missing side from corresponding-side proportions, including overlapping-triangle, parallel-line and shadow setups
2. Establish similarity by AA, SAS or SSS
3. State the definition of similar triangles; explain how it differs from congruence
4. Use scale factor for perimeter () and area () ratios

Same Shape but Different Size

Similar: equal corresponding angles, proportional corresponding sides. Congruent: the special case where every ratio is 1.

Two Matching Angles Guarantee Similarity

AA criterion: two pairs of equal angles make the triangles similar.

• : , , so
• : ,
• Angles sum to , so too
• and , so
• One equal angle is not enough

AA Example With Angle Matching

: angles , ,
: angles , ,

Match by angle measure, not by letter position:

• , ,
• So

In , A matches D, B matches E, C matches F.

Quick Check on Angle-Angle Similarity

Two triangles have these angles:

• : , ,
• : , ,

Are they similar?

Find each missing angle, then decide...

SAS Similarity Requires the Included Angle

SAS: two proportional sides and equal included angles.

, , ; , ,

Similar by SAS. Non-example: same ratios but instead. Angle B is not between AB and AC, so SAS does not apply.

SSS Similarity Checks All Side Ratios

SSS: all three pairs of sides proportional.

and : , so similar.

and : every ratio is , so similar.

If even one ratio differs, SSS does not apply.

Identify Corresponding Sides From Similarity

In :

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

Worked Example for Finding Missing Sides

, with , , , .

Find and , keeping on top:

Scale Factor Links Perimeter and Area

The scale factor is the ratio of corresponding sides.

• Perimeters are in ratio
• Areas are in ratio

Sides in ratio , so . Perimeter becomes . Area becomes .

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 : angle is shared and , so by AA.

Overlapping Triangles With Parallel Lines

, , , , . Then .

Find EC: , so , , .

Find DE: , so , .

Why Pairing AD With DB Fails

would give . But DB is a side of neither triangle. Correct: , so .

Bowtie and Shadow Similar Triangles

Crossing segments: , crossing at . Vertical angles and alternate interior angles give . With , , : , .

Shadows: parallel sun rays make similar right triangles. , so ft.

Scale Factor Applied to Two Triangles

Two similar triangles have perimeters and . The smaller has area .

• Area ratio:
• Larger area:

Perimeters give ; areas need .

Key Takeaways and Common Mistakes

• AA: two matching angles make triangles similar
• SAS and SSS: proportional sides (SAS adds the included angle)
• Proportions: same triangle on top
• Scale factor: perimeter , area

Watch out: one angle is not enough; with , DB is a side of neither triangle

Coming Up Next in Triangle Topics

Up next: Properties of isosceles triangles

• What equal sides tell you about angles
• Using the altitude from the apex
• Finding unknown angles and sides