Similar and Congruent Figures 1 Definition of Similar
- Slides: 11
Similar and Congruent Figures 1
Definition of Similar Triangles • Two triangles are called similar if they both have the same three angle measurements. • The two triangles shown are similar. • Similar triangles have the same shape but possibly different sizes. • You can think of similar triangles as one triangle being a magnification of the other. 2
Similar Triangle Notation • The two triangles shown are similar because they have the same three angle measures. • The symbol for similarity is Here we write: • The order of the letters is important: corresponding letters should name congruent angles. C B A D E F 3
• Let’s stress the order of the letters again. When we write note that the first letters are A and D, and The second letters are B and E, and The third letters are C and F, and We can also write: C B A D E F 4
Proving Triangles Similar • To prove that two triangles are similar you only have to show that two pairs of angles have the same measure. C • In the figure, • The reason for this is that the unmarked angles are forced to have the same measure because three angles of any triangle always add up to P D T A B 5
• When trying to show that two triangles are similar, there are some standard ways of establishing that a pair of angles (one from each triangle) have the same measure: • They may be given to be congruent. • They may be vertical angles. • They may be the same angle (sometimes two triangles share an angle). • They may be a special pair of angles (like alternate interior angles) related to parallel lines. • They may be in the same triangle opposite congruent sides. • There are numerous other ways of establishing a congruent pair of angles. 6
Example • In the figure, • Show that A • First, note that because these are alternate interior angles. • Also, because these are alternate interior angles too. • This is enough to show the D triangles are similar, but notice the remaining pair of angles are vertical. B C E 7
Proportions from Similar Triangles • Suppose • Then the sides of the triangles are proportional, which means: F E D C • Notice that each ratio consists of corresponding segments. A B 8
Example • Given that if the sides of the triangles are as marked in the figure, find the missing sides. • First, we write: F 8 D • Then fill in the values: 6 7 C 12 E 9 • Then: A 10. 5 B 9
Example • In the figure, are right angles, and Find • First note that since and since the triangles share angle C. • Let x denote AB. Then: D E 12 9 A x B 6 C 10
Definition of Congruent Triangles • Two triangles are congruent if one A can be placed on top of the other for a perfect match (they have the same size and shape). • In the figure, is congruent to B In symbols: D • Just as with similar triangles, it is important to get the letters in the correct order. For example, since A and D come first, we are saying that when the triangles are made to E coincide, A and D will coincide. C F 11