Geometry Theorems — Thales' Theorem and Triangle Similarity
Geometry Theorems — Thales' Theorem and Triangle Similarity. Practice questions to deepen understanding of Thales' theorem and triangle similarity. Online math practice with full solutions and step-by-step explanations.
Thales' theorem and similarity practice — Thales, equal ratios, similarity theorems AA/SSS/SAS, similarity ratio, ratio of areas.
Thales' theorem: parallel lines → equal ratios. Thales in a triangle: DE ∥ third side.
⫽ Thales theorem:
If three parallel lines intersect two transversals, then:
If three parallel lines intersect two transversals,
then the ratios between the segments are equal!
AB/BC = DE/EF ✓
△ Thales in a triangle:
If a line is parallel to a side in a triangle, then:
If DE ∥ BC in triangle ABC,
then: AD/DB = AE/EC ✓
The parallel line divides the sides in the same ratio!
↔️ Converse theorem:
If a line divides two sides of a triangle in the same ratio, then:
If in triangle ABC:
AD/DB = AE/EC
then DE ∥ BC ✓
This is a way to prove parallelism!
DE ∥ BC ⇔ AD/DB = AE/EC
(both directions are true!)
△ Definition:
Two triangles are similar if:
Two triangles are similar if:
1. All corresponding angles are equal ✓
2. The ratio of corresponding sides is constant ✓
Notation: △ABC ~ △DEF
📐 Similarity ratio:
In similar triangles with similarity ratio k=2, the ratio of perimeters is:
If the similarity ratio = k
then:
• Ratio of lengths (sides, perimeter) = k ✓
• Ratio of areas = k² ✓
• Ratio of volumes = k³ ✓
k = 2
Ratio of perimeters = 2:1 ✓
Ratio of areas = 4:1
∠ AAA similarity:
Two triangles are similar if:
If three corresponding angles are equal,
then the triangles are similar! ✓
∠A = ∠D, ∠B = ∠E, ∠C = ∠F
⇒ △ABC ~ △DEF
Two equal angles (AA)!
The third follows from the 180° sum ✓
━ SSS similarity:
Two triangles are similar if:
If the ratio of three corresponding sides is constant,
then the triangles are similar! ✓
AB/DE = BC/EF = AC/DF = k
⇒ △ABC ~ △DEF
The 3-4-5 triangle is similar to the 6-8-10 triangle
Ratio: 3/6 = 4/8 = 5/10 = 1/2 ✓
∠ SAS similarity:
Two triangles are similar if:
If two sides have the same ratio and the included angle between them is equal,
then the triangles are similar! ✓
AB/DE = AC/DF and ∠A = ∠D
⇒ △ABC ~ △DEF
The angle must be included (between the two sides)!
🔢 Computation:
In a triangle, DE ∥ BC. If AD=4, DB=2, AE=6, then EC equals:
DE ∥ BC
AD = 4, DB = 2
AE = 6, EC = ?
Solution:
Thales theorem:
AD/DB = AE/EC
4/2 = 6/EC
2 = 6/EC
EC = 3 ✓
🔢 Computation:
Two similar triangles with sides 3-4-5 and 6-8-10. The similarity ratio is:
Triangle 1: 3-4-5
Triangle 2: 6-8-10
Solution:
Ratio = small side / large side
3/6 = 4/8 = 5/10 = 1/2 ✓
Similarity ratio: 1:2
📐 Areas:
In similar triangles with similarity ratio 3:1, the ratio of areas is:
If the similarity ratio = k
then the ratio of areas = k² ✓
Similarity ratio = 3:1
k = 3
Ratio of areas = 3² : 1² = 9:1 ✓
∠ Angle bisector:
The angle bisector in a triangle divides the opposite side in the ratio of:
The angle bisector divides the opposite side in the ratio of the sides!
If AD bisects ∠A,
then BD/DC = AB/AC ✓
This is an extension of Thales theorem!
📏 Heights:
In similar triangles with similarity ratio k, the ratio of heights is:
In similar triangles:
The ratio of all lengths (sides, heights, medians, radii) = k ✓
Everything in the same ratio!
If similarity ratio = 2:1
then:
• Ratio of heights = 2:1
• Ratio of medians = 2:1
• Ratio of perimeters = 2:1
⊿ Right triangle:
The altitude to the hypotenuse in a right triangle creates:
The altitude to the hypotenuse in a right triangle divides it into two triangles
And all three (the original + the 2 smaller ones) are similar to each other! ✓
△ABC ~ △ACD ~ △CBD ✓
🔍 Identification:
A triangle with angles 50°-60°-70° is similar to a triangle with angles:
Triangles are similar if all angles are equal!
50°-60°-70° is similar only to 50°-60°-70° ✓
Is 25°-30°-35° half of 50°-60°-70°?
No! Angles do not work in ratios!
The exact same angles are needed ✗
(Also: 25°+30°+35° = 90° ≠ 180°!)
🔢 Computation:
A large triangle is similar to a small triangle in ratio 4:1. If the area of the small one is 5 cm², the area of the large one is:
Similarity ratio = 4:1
Small area = 5 cm²
Large area = ?
Solution:
Ratio of areas = (similarity ratio)²
Ratio of areas = 4² : 1² = 16:1
Large area = 5 × 16 = 80 cm² ✓
⫽ Extended Thales:
A line parallel to one of the triangle sides and passing through the midpoint of a second side:
If a line is parallel to a triangle side and passes through the midpoint of a second side,
then it passes through the midpoint of the third side! ✓
This is a midsegment!
By Thales: AD/DB = AE/EC
If AD = DB (midpoint),
then also AE = EC (midpoint) ✓
△ Relationship:
Congruent triangles are also:
Congruence = a special case of similarity!
Congruence ⊂ similarity ✓
Congruent triangles = similar with ratio 1:1
• Congruence: same shape and same size
• Similarity: same shape (different sizes allowed)
🔢 Application:
In a right triangle with legs 6 and 8, the altitude to the hypotenuse equals:
Hypotenuse: c = √(6²+8²) = 10
Area: S = ½×6×8 = 24
Also: S = ½×c×h = ½×10×h
24 = 5h
h = 4.8 ✓
h = (a×b)/c = (6×8)/10 = 48/10 = 4.8 ✓
📚 Summary:
Which of the following theorems is not true?
"In similar triangles the ratio of angles is constant"
This is false! ✗
Angles are not in a ratio - they are exactly equal!
∠A = ∠D (not in ratio) ⚠️
✓ All corresponding angles are equal (not in ratio!)
✓ Ratio of corresponding sides is constant = k
✓ Ratio of lengths = k
✓ Ratio of areas = k²
✓ Ratio of volumes = k³
✓ Thales theorem: line ∥ to side ⇒ same ratio division
✓ Similarity theorems: AA, SSS, SAS
✓ Angle bisector: BD/DC = AB/AC
✓ Congruence = similarity with k=1