2.1 Geometry of lines and triangles

Mathematics

Grade 9

Algebra and geometry

Module 8

Geometry of lines and triangles

Activity 1

To learn conventions of naming sides and angles in triangles

• Refer to the triangle alongside to understand the terms.
• The three sides of the triangle are called AB, AD and BD.
• We often name a side in a triangle by using the small letter of the name of the opposite corner. The corners (vertices) of the triangle are named by using capital letters.
• When A or $\hat{A}$ is used, it refers to the a ngle made by side BA and side AD.
• For the angle one can also say BAD or $B\hat{A}D$ .
• But if I say A=38°, it is clear what I mean.
• We refer to triangle ABD or ΔABD, writing the letters in alphabetical order.

Exercise:

To show that you understand the naming

conventions, draw the following triangle in the space

to the right:

Draw ΔQRT with q=4cm, T=65° and QT=5,5cm.

You should notice that you don’t need to be told

the length of t, nor the sizes of $\hat{Q}$ or $\hat{R}$ . First

draw a rough sketch and fill the details in on that

sketch to help you plan your drawing.

Assignment:

You should already know that triangles are classified according to their shape. Make an A4–sized poster for your own use, clearly showing the characteristics of the following types of triangle: equilateral, right–angled, isosceles and scalene. Name the vertices and sides according to the conventions above. You must work as accurately and neatly as you can.

Measure the sides and the angles of your triangles and fill these measurements in on your poster.

Activity 2

To develop the principle of congruence in triangles

[LO 4.4, 3.3, 3.5]

• In the previous exercise you drew ΔQTR from specifications given to you. Ask the other learners who did this exercise to show you their drawings, and check whether their triangles agree perfectly with the sizes given in the question. Measure the side and the two angles not specified, to see whether they also agree with yours.
• You should find that all triangles drawn by anybody according to the instructions, are always identical. In fact, it is impossible to draw that triangle so that it is different! Write down, with a partner, why you think this is the case.
• Here are more descriptions of triangles. See whether the same happens with them – in other words, that it is impossible to draw different triangles that fit the same description. Again, write down your view of each situation. The last one is quite difficult to draw – try it!

• Draw ΔAGE with a=4cm, E=90° and AG=5cm.
• Draw ΔNOH with HN=4cm, H=56° and O=72°.
• Draw ΔBAT with B=48°, T=65° and A=67°.
• Draw ΔMOD with m=5,5cm, O=65° and DM=4cm.
• Draw ΔAMP with a=4,2cm, m=5cm and p=5,6cm.
  • In each of the above triangles, only three of the six sizes (three sides and three angles) were specified in the question. And sometimes that was enough to ensure that everyone drew identical triangles. But in ΔBAT and ΔMOD the three items were not enough to ensure identical triangles from everyone.
  • So, when is it enough? Maybe you have already discovered the secrets:

ΔQRT: Two sides and the angle between them were specified.