MATHEMATICS

Grade 9

ALGEBRA AND GEOMETRY

Module 9

CONGRUENCY

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:

Figure 1

Quadrilateral APEK  Quadrilateral CDNM

ACTIVITY 2

To practically determine what the conditions for congruent triangles are

1.1 Study page 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 always 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).

Figure 2

2.1 Again study page 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 page 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.

3.3 If you only receive the information as in the sketches below, can you say with certainty that the two triangles are always congruent?

Figure 3

DEFρCBAρ3.4 Will the following two triangles be congruent? Why?

4.1 Study page A-4 of the accurately constructed triangles. All the triangles on page A-4 were constructed by using two sides and the angle not between the two given sides, (ss) 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.

4.2 There are two triangles, which, although the two sides and the angle are equal, are not congruent. Name them.

4.3.1 Do you think that, if two sides and the angle not between the two sides, are used to construct triangles they would alwaysbe congruent?

4.3.2 What condition must the given sides satisfy for the triangles to be congruent?

4.4.4 If you only receive the information as in the sketches below, can you with certainty say that the two triangles are alwayscongruent? (Remember you now do not know what the lengths of the two given sides).

Figure 4

4.5.1 There are four triangles on page A-4 where the given angle is 90°. If the angle not between the two given sides is equal to 90°, do you think that the two triangles will always be congruent? (rhs)

4.5.2 If you only receive the information like in the two sketches below, can you with certainty say that the two triangles are always be congruent?

Figure 5

5. On page A-5 there are triangles of which the three angles of the one triangle are equal to the three angles of the other triangle. ()

5.1 Are the triangles constructed like this always necessarily congruent?

5.2 If you only receive the information like in the two sketches below, can you with certainty say that the two triangles are always be congruent?

Figure 6

6. Now give the combinations of sides and angles for triangles to be congruent. Illustrate each combination as in the example below:

1.

Figure 7

Homework assignment

1. State whether the following pairs of triangles are congruent or not. Do each number like the example below.

Example:

Figure 8

 A =  D; B = E and C = F

N.B. If the triangles are not necessarily congruent, only write ΔABC  ΔDEF and then write down why you say so.

Figure 11

Figure 12

Figure 13

2. In each of the following pairs of triangles two pairs of equal elements are marked. In each case write down another pair of equal elements for the triangles to be congruent. Give the congruency test which you used and also give all the possibilities without repeating a congruency test.

Figure 14

Figure 15

Figure 16

Figure 17

Assessment