2.1 Geometry of lines and triangles  (Page 2/5)

ΔAGE: A right–angle, the hypotenuse and another side were specified.

ΔNOH: One side and two angles were specified.

ΔAMP: Three sides were specified.

ΔBAT: Three angles were specified, but nothing said how big the triangle could be!

ΔMOD: Two sides and the angle not between the two sides were specified, so that it could happen that some learners drew a short OM side and others drew a longer OM side; because nothing was said how long OM had to be!

• When two triangles are identical in every way – size and shape – then we call them congruent . This means that if you cut one out, it can be placed exactly on top of the other. As you will see later, the word can be used for other identical shapes as well, but for now we will concern ourselves only with triangles.
• From the drawing exercise you saw that there are four different ways to ensure that triangles are congruent. Here they are, with helpful sketches:

Case 1: Two triangles with two sides and the included angle equal, will be congruent.

Case 2: Two right–angled triangles with the hypotenuse and another side equal, will be congruent.

Case 3: Two triangles with three sides equal will be congruent.

Case 4: Two triangles with two angles and corresponding sides equal, will be congruent. This means that the equal sides must be opposite corresponding angles.

Investigation:

The ne x t exercise shows 15 triangles, named A to O. They are all mixed–up and in strange orientations. Work in a group of 4 or 5 to decide whether any of them are congruent. Group the names of those that are congruent, with explanations and reasons. This is not a straight–forward exercise; it is much more like a puzzle. You will have to use all your experience, common-sense and logic. The sizes are not correct, so that you have to use the information given, and not measure anything.

Activity 3

To apply the four cases of congruence in problems

[LO 4.4, 3.3, 3.4]

• When you have shown that two triangles are congruent (as you had to do in the previous exercise) you have to do a number of things: First decide which case of congruence will apply. Then say why each of the three items is equal. Then write your conclusions down in the proper order. Here is an example of how it can be done. We use the symbol  to show congruence. So we can see that, if we know that three very special things are equal, we know that everything else must be equal too!

Prove that ΔABC and ΔDEF are congruent.

1. $\hat{A}$ = 60° because the angles of a triangle add up to 180°

Therefore, $\hat{A}=\hat{F}$ because both are 60°

2. $\hat{C}=\hat{E}$ because both are 50°

3. BC = DE because they are both 12 units, and they lie opposite equal angles. We have two equal angles and a corresponding equal side in each of the two triangles.

4. So we write: ΔABC  ΔDEF (AAS) which means: ΔABC is congruent to ΔDEF because two angles and a corresponding side are equal. This means that all other things must be equal too.

From the exercise in the previous section, do at least three congruencies in this way.

Exercise: