Congruent and Similar Triangles Similar and Congruent Figures

Congruent and Similar Triangles

Similar and Congruent Figures • Congruent polygons have all sides congruent and all angles congruent. • Similar polygons have the same shape; they may or may not have the same size.

Examples These figures are similar and congruent. They’re the same shape and size. These figures are similar but not congruent. They’re the same shape, but not the same size.

Another Example These figures are neither similar nor congruent. They’re not the same shape or the same size. Even though they’re both triangles, they’re not similar because they’re not the same shape triangle. Note: Two figures can be similar but not congruent, but they can’t be congruent but not similar. Think about why!

Congruent Figures When 2 figures are congruent ( 2 figures have the same shape and size) §Corresponding angles are equal §Corresponding sides are equal §Symbol :

Congruent Triangles A X B Z C Y • AB = XY, BC = YZ, CA = ZX • A = X , B = Y, C = Z Note : Corresponding vertices are named in order.

THE ANGLE MEASURES OF A TRIANGLE AND CONGRUENT TRIANGLES • The sum of the angle measures of a triangle is 180 o Example ? 65 o ? = • 30 o 85 o Congruent triangles are triangles with the same shape and size Angle = 60 o; side = 5 cm Example 5 cm 60 o ? 90 o ?

Isosceles triangles • An isosceles triangle is the triangle which has at least two sides with the same length • In an isosceles triangle, angles that are opposite the equal-length sides have the same measure ? Example 82 cm ? 52 o The side = 82 cm, the angle = 76 o

Equilateral triangles • An equilateral triangle has three sides of equal length • In an equilateral triangle, the measure of each angle is 60 o Example 60 o 100 cm ? ? Angle = 60 o, side = 100 cm

Right triangles and Pythagorean theorem • A right triangle is the triangle with one right angle Hypotenuse c Leg • Pythagorean theorem b c 2 = a 2 + b 2 Leg Example a 60 o ? 3 cm ? 4 cm c 2 = 42 + 32 = 25 C=5

Tests for Congruency Ways to prove triangles congruent : • SSS ( Side – Side ) • SAS ( Side – Angle – Side ) • ASA ( Angle – Side – Angle ) or AAS ( Angle – Side ) • RHS ( Right angle – Hypotenuse – Side )

SSS ( Side –Side ) • Three sides on one triangle are equal to three sides on the other triangle. A B C X • AB = XY, • BC = YZ, • CA = ZX Y (SSS) Z

SAS ( Side – Angle – Side ) • Two pairs of sides and the included angles are equal. A B C X Y • AB = XY, • BC = YZ, • ABC = XYZ ( included angle ) (SAS) Z

ASA ( Angle – Side – Angle ) AAS ( Angle – Side ) • Two pairs of angles are equal and a pair of corresponding sides are equal. A B X C • AB = XY, • ABC = XYZ • BAC = YXZ Y (ASA) From given diagram, ACB = XZY Z (AAS)

RHS ( Right angle – Hypotenuse – Side ) A • Right-angled triangle with the hypotenuse equal and one other pair of sides equal. C B Z • ABC = XYZ = 90° ( right angle) • AC = XZ ( Hypotenuse) • BC = YZ (RHS) Y X