Congruence If shapes are identical in shape and size then we say they are congruent. Congruent shapes can be mapped onto each other using translations, rotations and reflections. The following triangles are congruent because AB = PQ, BC = QR, A B P Q C R C A B P Q R C A B P Q R C A B P Q R C A B P Q R A B C R P Q C A B P Q R Stress that if two shapes are congruent their corresponding lengths and angles are the same. and AC = PR. A = P, B = Q, and C = R.

Congruent triangles Two triangles are congruent if they satisfy the following conditions: Side, side, side (SSS) 1) The three sides of one triangle are equal to the three sides of the other. Stress that the conditions outlined on the following slides are the minimum conditions required to be sure that two triangles are congruent. If two triangles satisfy these minimum conditions, we can deduce that all the sides and all the angles are equal. We call this first rule side, side, side or SSS for short.

Congruent triangles Two triangles are congruent if they satisfy the following conditions: 2) Two sides and the included angle in one triangle are equal to two sides and the included angle in the other. Side, angle, side (SAS) We call this rule side, angle, side or SAS for short.

Congruent triangles Two triangles are congruent if they satisfy the following conditions: Angle, angle, side (AAS) 3) Two angles and one side of one triangle are equal to the corresponding two angles and side in the other. We call this rule angle, angle, side or AAS for short. This can be written as ASA if the corresponding side is between the two angles.

Congruent triangles Two triangles are congruent if they satisfy the following conditions: 4) The hypotenuse and one side of one right-angled triangle is equal to the hypotenuse and one side of another right-angled triangle. Stress that both triangles must have a right angle. We call this rule right angle, hypotenuse, side or RHS for short. Link: S9.1 Constructing triangles. Right angle, hypotenuse, side (RHS)

Similar shapes If one shape is an enlargement of the other then we say the shapes are similar. The angles in two similar shapes are the same size and the lengths of their corresponding sides are in the same ratio. A similar shape can be a reflection or a rotation of the original. The following triangles are similar because Give an example of what is meant by the corresponding side lengths being in the same ratio. For example, if the lengths of all the sides in the second shape are double the lengths of the corresponding sides in the first shape then the ratios of the corresponding side lengths will all simplify to 2. This number is the scale factor for the enlargement. A C R P B Q A P B C R Q C R A B P Q C R A B P Q B C R Q A P A B C R P Q A B P Q C R A =  P, B = Q, and C = R. PQ AB = QR BC = PR AC

Similar shapes Which of the following shapes are always similar? Any two rectangles? Any two squares? Any two isosceles triangles? Any two circles? Any two equilateral triangles? Establish that any two regular polygons with the same number of sides are mathematically similar as are any two circles. Any two regular polyhedra (of the same number of sides) are also similar. Any two cubes? Any two cylinders?

Finding the scale factor of an enlargement We can find the scale factor for an enlargement by finding the ratio between any two corresponding lengths. Scale factor = length on enlargement corresponding length on original The two corresponding lengths can be found using a ruler. Always make sure that the two lengths are written using the same units before dividing them.

Finding the scale factor of an enlargement The following rectangles are similar. What is the scale factor for the enlargement? 6 cm 9 cm The scale factor for the enlargement is 9/6 = 1.5

Finding the lengths of missing sides The following shapes are similar. What is the size of each missing side and angle? 3.6 cm c 3 cm 37° 6 cm 37° a 4.8 cm 5 cm 4 cm e 53° b Make sure that pupils realize that all corresponding angles in similar shapes are equal. Point out that to scale from the left-hand shape to the right-hand shape we multiply by 6/5 (or 1.2). In other words, we multiply by 6 and divide by 5. To scale from right-hand shape to the left-hand shape we multiply by the reciprocal 5/6 (or divide by 1.2). in other words we multiply by 5 and divide by 6. 53° 6 cm 7.2 cm d The scale factor for the enlargement is 6/5 = 1.2