2.3 Congruency

Mathematics

Grade 9

Algebra and geometry

Module 9

Congruency

• The term congruency means that two figures are identical in every way. It therefore means that all the sides of the one figure are equal to all the sides of the other figure and that all the angles of one figure are equal to all the angles of the other figure. If the figures are cut out, they will fit precisely on one another.

ACTIVITY 1

To understand what the term congruency in general means

[LO 3.2.1]

Study the figures on the grid (A-1) and decide which of them are congruent. Then give each pair of congruent figures by writing them down with the letters in order of the sides and angles which are equal. The symbol for congruency is .

For example:

Quadrilateral APEK  Quadrilateral CDNM

• A triangle has six elements; namely three angles and three sides. Only three of these elements are needed to construct a triangle:

• Combinations of the three elements are:

• 3 sides (sss)
• 2 sides and the angle between them (ss)
• 2 angles and a side (s)
• 2 sides and the angle not between them (ss)
• 3 angles ()
• A 90° - angle, a side and the hypotenuse (90°ss or rhs)

ACTIVITY 2

To practically determine what the conditions for congruent triangles are

• You are given four pages with accurately constructed triangles.

1.1 Study p a ge A -2 of the accurately constructed triangles. Study the triangles which were constructed by using three given sides and write down all the pairs of triangles which are congruent (sss). Remember that, as in activity 1, the triangles must be written down in order of the sides which are equal to each other.

1.2 Will two triangles of which the sides of the one triangle are equal to the sides of the other triangle a lw a ys be congruent to each other?

1.3 If you only receive the information as in the sketches below, can you always with certainty say that the two triangles will be congruent? (Remember no real lengths are given).

2.1 Again study p a ge A -2 of the accurately constructed triangles. Now look at the triangles constructed by using two sides and the angle between the two given sides, (s  s) , and write down all the pairs of triangles which are congruent. Again remember to write down the triangles in order of the side, angle, side which are equal.

2.2 Will two triangles of which two sides and the angle between them are equal, always be congruent?

2.3 If you only receive the information as in the sketches below, can you always with certainty, say that the two triangles will be congruent? (Remember no real lengths or magnitudes of angles are given).

3.1 On p a ge A -3 of the accurately constructed triangles two angles and a side (  s) are used to construct the triangles. Study these triangles and write down the pairs of triangles, which are congruent. Again remember to write down the triangles in order of the elements, which are equal.

3.2 In ΔDOM and ΔLOC DM = OC, D = O en M = L, but these two triangles are not congruent. Why is that so? Give a general rule by completing the following sentence:

Two triangles are congruent if angle, angle, side of the one triangle are equal to angle, angle and the ……………………..side of the other triangle.