Write the corners in matching order
Δ ABC ~ Δ DEF lists the corners so that A↔D, B↔E and C↔F.
A big triangle, turned round
Triangle PQR is small. Triangle XYZ is a bigger copy, but it is turned round, so its corners are not in the same places.
Corners with the same number of tick marks have equal angles.
Which of these says correctly that the two triangles are similar?
What this lesson covers
The idea
Two triangles are similar if their corresponding angles are equal and corresponding sides are in the same ratio; this is written Δ ABC ~ Δ DEF, with vertices listed in corresponding order (A↔D, B↔E, C↔F).
A big triangle, turned round
Triangle PQR is small. Triangle XYZ is a bigger copy, but it is turned round, so its corners are not in the same places.
Corners with the same number of tick marks have equal angles.
Which of these says correctly that the two triangles are similar?
- Δ PQR ~ Δ XYZ
- Δ PQR ~ Δ YZX
- Δ PQR ~ Δ ZXY
Match the corners
Each statement must list the big triangle's corners in matching order: the first letter is the corner equal to the first letter, and so on. Tap the three letters in order.
Matching order
Δ ABC ~ Δ DEF lists the corners in matching order: A↔D, B↔E and C↔F.
Two triangles are similar if (i) their corresponding angles are equal: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F (ii) their corresponding sides are in the same ratio: AB/DE = BC/EF = CA/FD The symbol ~ means "is similar to".
The order matters. Not allowed: Δ ABC ~ Δ EDF Allowed: Δ BAC ~ Δ EDF
Notes
Δ ABC ~ Δ DEF lists the corners in matching order: A↔D, B↔E and C↔F. Then the angles are equal and the sides are in the same ratio.
Check yourself
Δ ABC ~ Δ PQR. Which angle is equal to ∠B?
Δ ABC ~ Δ PQR. Which ratio is equal to AB/PQ?
In Δ ABC, ∠A = 40°, ∠B = 75° and ∠C = 65°. In Δ XYZ, ∠X = 65°, ∠Y = 40° and ∠Z = 75°. Which statement is correct?
Δ ABC ~ Δ DEF with ∠A = 40° and ∠B = 75°. Find ∠F, in degrees.
Answer: 65
∠C = 180° − 40° − 75° = 65°. F matches C, so ∠F = ∠C = 65°.
- ∠P. A is the first letter in Δ ABC and P is the first letter in Δ PQR, so ∠P = ∠A.
- ∠Q — correct. Yes! B is the second letter, and Q is the second letter. Matching corners are B↔Q.
- ∠R. A, B, C match P, Q, R in that order. R is the third letter, so ∠R = ∠C.
- BC/QR — correct. Yes! B↔Q and C↔R, so BC matches QR. All three ratios are equal.
- BC/PR. BC matches QR, not PR. PR matches AC.
- AC/QR. AC matches PR. QR matches BC.
- Δ ABC ~ Δ XYZ. That would match ∠A = 40° with ∠X = 65°. They are not equal.
- Δ ABC ~ Δ YZX — correct. Yes! A↔Y (40°), B↔Z (75°), C↔X (65°). Every pair is equal.
- Δ ABC ~ Δ ZXY. That would match ∠A = 40° with ∠Z = 75°. They are not equal.