Triangles | Activity

Question 4

Draw two line segments BC and EF of two different lengths, say 3 cm and 5 cm respectively. Then, at the points B and C respectively, construct angles PBC and QCB of some measures, say, 60° and 40°. Also, at the points E and F, construct angles REF and SFE of 60° and 40° respectively (see Fig. 6.23).

Solution

We will find the third angle in each triangle. Then we will compare the two triangles.

Step 1 — Find angle A

The sum of angles in a triangle is $180^\circ$ . We know $\angle B$ is $60^\circ$ . We know $\angle C$ is $40^\circ$ . Let's find $\angle A$ .

$\angle A + \angle B + \angle C = 180^\circ$

$\angle A + 60^\circ + 40^\circ = 180^\circ$

$\angle A + 100^\circ = 180^\circ$

$\angle A = 180^\circ - 100^\circ$

$\boxed{\angle A = 80^\circ}$

Diagram 1

Step 2 — Find angle D

The sum of angles in a triangle is $180^\circ$ . We know $\angle E$ is $60^\circ$ . We know $\angle F$ is $40^\circ$ . Let's find $\angle D$ .

$\angle D + \angle E + \angle F = 180^\circ$

$\angle D + 60^\circ + 40^\circ = 180^\circ$

$\angle D + 100^\circ = 180^\circ$

$\angle D = 180^\circ - 100^\circ$

$\boxed{\angle D = 80^\circ}$

Diagram 2

Step 3 — Compare the triangles

Both triangles have angles $80^\circ$ , $60^\circ$ , and $40^\circ$ . This means they are similar. For $\triangle ABC$ , side $BC$ is $3 \text{ cm}$ . For $\triangle DEF$ , side $EF$ is $5 \text{ cm}$ . These sides are between the $60^\circ$ and $40^\circ$ angles. Since $BC \neq EF$ , the triangles are not congruent.

Answer

(i) $\angle A = 80^\circ$ and $\angle D = 80^\circ$ . (ii) No, the triangles $\triangle ABC$ and $\triangle DEF$ are not congruent. (iii) Yes, the triangles $\triangle ABC$ and $\triangle DEF$ are similar.