Three sets of conditions that guarantee two triangles are similar: AA (two pairs of equal angles), SAS~ (two pairs of proportional sides with equal included angle), and SSS~ (all three pairs of sides in the same ratio).
Think of a photo and its enlargement. They look the same but are different sizes. For triangles, you only need to check that two angles match (AA)—if the angles are the same, the shape is the same, even if the size differs. It's like verifying two buildings have the same blueprint, even if one is a scale model.
Showing a random 20 of 50 problems.
Example 1
easy
What is the scale factor between two similar triangles with sides 4,5,6 and 8,10,12?
Example 2
medium
On a map with scale 1:50000, two towns are 8 cm apart. What is the real distance in km?
Example 3
medium
A flag pole's shadow at 3 PM is 20 ft. A nearby 5 ft mailbox casts a 4 ft shadow. How tall is the flag pole? Justify with a similarity criterion.
Example 4
easy
Are all isoceles right triangles similar to each other?
Example 5
medium
The altitude to the hypotenuse of a right triangle creates two smaller triangles. Why is each similar to the original?
Example 6
medium
△ABC∼△XYZ with AB=9,XY=6. If BC=12, find YZ.Triangle ABC — AB = 9, BC = 12
Example 7
easy
△ABC and △DEF share angle A=D=70°, and AB/DE=AC/DF=1.5. Which criterion proves similarity?
Example 8
hard
△ABC∼△DEF with area ratio 25:81. The longest side of △ABC is 10. Find the longest side of △DEF.
Example 9
hard
△ABC∼△DEF with sides in ratio 2:3. If the perimeter of △ABC is 24, what is the perimeter of △DEF?
Example 10
medium
△PQR has ∠P=40°,∠Q=60°. △STU has ∠S=40°,∠U=80°. Are the triangles similar? Which correspondence?Triangle PQR — angles 40°, 60°, 80°
Example 11
medium
Which is true: all isosceles triangles are similar to each other?
Example 12
medium
In △ABC: AB=6, BC=9, AC=12. In △DEF: DE=4, EF=6, DF=8. Are the triangles similar? State the criterion.
Example 13
easy
A 6-ft person casts a 4-ft shadow. A nearby tree casts a 20-ft shadow at the same time. Using similar triangles, how tall is the tree?
Example 14
easy
Why is checking just two angles enough to prove triangles are similar?
Example 15
easy
Name the three standard criteria that prove two triangles are similar.
Example 16
hard
In right triangle ABC with right angle at C, the altitude from C meets AB at D. Prove that △ACD∼△ABC.
Example 17
medium
In triangle ABC, DE is drawn parallel to BC with D on AB and E on AC. Why are triangles ADE and ABC similar?
Example 18
medium
In △ABC, D is on AB and E is on AC so that DE∥BC. Prove △ADE∼△ABC.
Example 19
hard
In a triangle, the angle bisector from A divides side BC at D. Given △ABD and △ACD share an angle at D — explain why they are NOT in general similar.
Example 20
medium
Triangle ABC has sides 5, 12, 13. Triangle DEF has sides 10, 24, 26. Prove the triangles are similar and state the criterion used.