What you’ll learn
Understand altitude/right-triangle similarity.
Key idea
An altitude to the hypotenuse gives smaller right triangles with an acute angle equal to an original acute angle.
Pair the correct acute angles as well as right angles; similar shapes need not be congruent.
Worked example
Classify examples and non-examples for the concept 'Understand altitude/right-triangle similarity'.
Triangle ABC has C=90, A=50, B=40 degrees. D lies on hypotenuse AB and CD is perpendicular to AB. Complete the similarity proof for the original and both smaller triangles.
- In ACD, A=50, D=90, so C=40. In CBD, B=40, D=90, so C=50.
- ABC, ACD and CBD have respective ordered angles (50,40,90); each pair is similar by AA.
Answer: All three are similar by AA — each smaller triangle has a right angle and a corresponding acute angle of the original.
Quick Check
Classify examples and non-examples for the concept 'Understand altitude/right-triangle similarity'.
Triangle ABC has C=90, A=20, B=70 degrees. D lies on hypotenuse AB and CD is perpendicular to AB. Complete the similarity proof for the original and both smaller triangles.