Similar and Congruent Triangles

Similar and Congruent Triangles

1. Congruent Triangles

Two triangles are congruent () when they have the exact same shape and size — all three pairs of corresponding sides are equal, and all three pairs of corresponding angles are equal.

ABC543DEF543»=

Congruence Tests (How to Prove It)

You do NOT need to verify all 6 pairs. Any one of these shortcuts is sufficient:

Test What You Need Diagram Hint
SSS All 3 pairs of sides equal Three pairs of tick marks
SAS 2 sides + the angle between them Angle must be the included angle
ASA 2 angles + the side between them Side must connect the two known angles
AAS 2 angles + any side Since 2 angles fix the third, one side determines size
HL Hypotenuse + one leg (right triangles only) Must have a right angle
Invalid tests — these do NOT prove congruence
  • AAA — proves similarity, not congruence (same shape, but could be different sizes)
  • SSA (or ASS) — ambiguous case; two different triangles can have the same SSA
SASµabIncludedangleASA®¯cIncludedsideHLHypLegRighttriangleonly

2. Similar Triangles

Two triangles are similar () when they have the same shape but possibly different sizes — all corresponding angles are equal, and all corresponding sides are in the same ratio (the scale factor).

where is the scale factor.

ABC342»k=2DEF684

Similarity Tests (How to Prove It)

Test What You Need How Often on SAT
AA 2 pairs of equal angles Very common — this is the main one
SSS~ All 3 pairs of sides proportional Rare
SAS~ 2 pairs of sides proportional + included angle equal Occasional
Why AA is enough

Since the angles of any triangle sum to , knowing 2 angles automatically determines the third. So AA = AAA, and that's all you need for similarity.

3. Setting Up Proportions

Once you establish similarity, the key skill is setting up the proportion correctly. The SAT tests this directly.

The Rule

Corresponding sides are opposite corresponding angles

If , then the side opposite corresponds to the side opposite , and so on.

Worked Example

ABC85?DEF6x?»

Given: , , , . Find .

Common mistake — mismatching sides

Make sure you pair sides from the same triangle in each fraction. A proportion like (ratio within each triangle) also works. But never mix sides from different triangles in one fraction like .

4. Key Properties to Remember

Similar Triangles — Ratios

If the scale factor between two similar triangles is , then:

Quantity Ratio
Corresponding sides
Corresponding perimeters
Corresponding areas

Example: If two similar triangles have sides in ratio , their areas are in ratio .

Angle Sum Property

If you know two angles, you know the third. This is why AA is sufficient for similarity — knowing two angles determines the entire shape.

Exterior Angle Theorem

This is useful for finding hidden angle equalities that prove AA similarity.

5. Quick Decision Flowchart

When you see a triangle problem on the SAT, follow this checklist:

  1. Are there parallel lines? → Look for corresponding/alternate angles → AA similarity
  2. Do two triangles share a vertex? → That's one equal angle. Look for a second (right angle, vertical angle, or from a parallel line) → AA similarity
  3. Are two lines crossing between parallel lines? → Vertical angles + alternate interior angles → AA similarity
  4. Are you given all side lengths? → Check if they're proportional → SSS similarity
  5. Is it asking for a missing side? → Confirm similarity, then set up a proportion
  6. Is it asking for an area? → Use : square the side ratio

SAT Tips

  1. AA is king — on the SAT, virtually every similarity question is solved by finding two equal angles. Don't waste time looking for side ratios unless you have to.
  2. The order of letters matters — means , , . Set up proportions accordingly.
  3. Parallel lines = free similarity — any time a line is parallel to one side of a triangle and cuts the other two sides, you get similar triangles. This is the single most tested setup.
  4. Right angles are your friend — two right triangles only need one more equal angle to be similar (since the right angle is already shared).
  5. Altitude to the hypotenuse — in a right triangle, the altitude from the right angle to the hypotenuse creates three similar triangles (the original + two smaller ones). This is a classic SAT setup.
  6. Watch for "not drawn to scale" — don't judge similarity by how the triangles look. Use given angles and sides only.
  7. Congruence is rare on the SAT — similar triangles appear much more often. When congruence does appear, it's usually in the context of isosceles/equilateral triangles or transformations (reflections, rotations).
  8. On Desmos — if a problem gives you coordinates, plot the triangles. You can visually confirm similarity and measure side lengths.
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