Mathematics Year 7: Similarity criteria for triangles
Mathematics, Year 7 (Portugal): Find out that two angles are enough to make triangles similar, meet the other two criteria, avoid misusing them and use them to work out measurements.
Three ways to be sure two triangles are similar
To decide whether two triangles are similar you do not have to compare everything. One of these checks is enough. AA criterion: two angles of one triangle equal two angles of the other. SAS criterion: two sides in proportion and the angle between those sides equal. SSS criterion: all three sides in proportion, with the same ratio. If one holds, the triangles are similar, and everything else (the third angle, the remaining sides) follows.
Why two angles are enough
The angles of a triangle add up to 180°. If two angles are 50° and 60°, the third must be 70°, in both triangles. If you build triangles with a ruler and protractor, with angles of 50° and 60° and a side between them of 4 cm, 6 cm or 9 cm, you get different triangles, all with the same shape: they are similar. The length of that side only decides the size, never the shape.
Measuring a height with a shadow
On level ground at the same time of day, a vertical pole 1.5 m tall casts a shadow of 2 m and a tree, also vertical, casts a shadow of 10 m. The sun's rays arrive at the same slope and each object makes a right angle with the ground, so the two triangles have two equal angles: they are similar (AA criterion). The ratio of the shadows is 10 ÷ 2 = 5, so the tree's height is 5 × 1.5 = 7.5 m. As a proportion: height ÷ 1.5 = 10 ÷ 2, height = 7.5 m.
The trap: two sides and an angle, but the wrong angle
The SAS criterion needs the angle to sit between the two sides. If the equal angle is elsewhere, it can fail. Draw a 30° angle and, from its vertex, a side of 10 cm. Open the compass to 6 cm and put its point on the other end: it cuts the other arm of the angle in two places, giving two different triangles. The angle opposite the 10 cm side is 56.4° in one and 123.6° in the other. Double one (20 cm and 12 cm): next to the other there are proportional sides and an equal angle, yet they are not similar.
Where you see this outside school
People who make maps, plans and buildings use similar triangles to work out what they cannot measure directly: the height of a building or the width of a river. They measure a small triangle within reach and, because it is similar to the big one, the ratios between the sides give the missing measurement. With a ruler, a tape measure and the sun, you can also measure the school flagpole.
Test what you learned
Which information is enough to be sure that two triangles are similar?
- a) They have the same perimeter
- b) They have one equal angle
- c) Two angles of one triangle equal two angles of the other
- d) They have the same area
Correct answer: c) Two angles of one triangle equal two angles of the other — Right, the AA criterion: the angles add up to 180°, so the third angles are equal too and the shapes are the same.
On level ground, a vertical pole 2 m tall casts a shadow of 3 m. At the same time a vertical tree casts a shadow of 12 m. How tall is the tree?
- a) 8 m
- b) 11 m
- c) 18 m
Correct answer: a) 8 m — Right: the triangles are similar (AA), the shadows are in ratio 12 ÷ 3 = 4, so the height is 4 × 2 = 8 m.
Triangle P has sides of 10 cm and 6 cm and a 30° angle opposite the 6 cm side. Triangle Q has sides of 20 cm and 12 cm and a 30° angle opposite the 12 cm side. Since 10 ÷ 20 = 6 ÷ 12, are P and Q certainly similar?
- a) Yes: two proportional sides and one equal angle are enough
- b) Not necessarily: the angle is not between the two sides, so the SAS criterion does not apply
- c) No, never: triangles with different side lengths cannot be similar
Correct answer: b) Not necessarily: the angle is not between the two sides, so the SAS criterion does not apply — Right: with 10 cm, 6 cm and 30° opposite the 6 cm side there are two possible triangles, with 56.4° or 123.6° opposite the 10 cm side.
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