8
The first test for similar triangles
Textbook: pp. 32–33
GoalKnow the first similarity test for triangles (AA) and use it to identify similar triangles and compute lengths.
New words
AA test (angle-angle) · BB alomaticommon angle · umumiy burchakparallel line · parallel to‘g‘ri chiziqshadow · soya
Explanation
First test (AA): if two angles of one triangle equal two angles of another, the triangles are similar. The third angles are then equal too, since the angle sum is 180°. This test makes checking similarity easy because no sides need to be measured. A line parallel to one side of a triangle cuts off a triangle similar to the given one, because there is a common angle and corresponding angles are equal. Likewise, objects standing in the sun at the same time and their shadows form similar right triangles.
Worked examples
Triangle angles 50°, 60° and 60°, 70°. The first's third angle is 70°, the second's is 50°; they share the angles 50°, 60°, 70°, so they are similar.
DE ∥ AC with D on AB and E on BC, BD = 3, DA = 6, AC = 15. △BDE ∽ △BAC with k = BD : BA = 3 : 9 = 1/3, so DE = 15 · 1/3 = 5.
Class activity
“Height from shadow”: on a sunny day measure the shadows of a schoolyard tree and of yourself (or a stick) and use similar triangles to calculate the tree's height. Do not look at the sun.
Practice
1
A pole 3 m tall casts a 2 m shadow. At the same time a tree casts an 8 m shadow. How tall is the tree (m)?
12
2
Two triangles have angles 40°, 75° and 65°, 40°. Are they similar? Why?
Yes: the first's third angle is 65° and the second's is 75°, so both have 40°, 65°, 75° (AA test).
3
MN ∥ BC with AM = 4, MB = 2 and BC = 12. How long is MN?
8
4
Why are the third angles equal when two angles are equal?
The angles add up to 180°, so the third angle is 180° minus the other two.