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# Properties of geometric figures

## 10.8 Properties of geometric figures

In this section, we will investigate the properties of triangles and quadrilaterals by constructing different types of triangles, including equilateral triangles and isosceles triangles. We will also investigate the properties of different quadrilaterals, including squares, rectangles, rhombuses and parallelograms. For most of our investigations, we will focus on the sum of the interior angles of shapes and some of the special relationships between sides and angles.

### Finding the sum of the interior angles of a triangle

1. What do you know about the sum of angles on a straight line? Complete the sentence: The sum of angles on a straight line …
3. Use a ruler and a pencil to construct any $$\triangle ABC$$. You can choose any length for the sides of the triangle and any angles for the interior angles of the triangle.
4. Label the interior angles $$\hat{A}, \hat{B}$$ and $$\hat{C}$$. An example is shown below.

5. Cut the triangle into three pieces.

6. Draw a straight line on a piece of paper. Mark a point in the middle of the line. Can you arrange the three vertices of $$\triangle ABC$$ at the point on the straight line? An example is shown below.

7. What can you deduce about the sum of angles in a triangle? Complete the sentence: The sum of angles in a triangle …

## Worked Example 10.17: Calculating the size of a missing angle in a triangle

$$\triangle ABC$$ has three angles of different sizes. This means it is a scalene triangle. Calculate the size of $$\hat{B}$$ in $$\triangle ABC$$.

### Write an expression for the sum of angles in a triangle.

We know that the sum of the interior angles of a triangle is equal to $$180^{\circ}$$.

$$\hat{A} + \hat{B} + \hat{C} = 180^{\circ}$$.

### Calculate the size of $$\hat{B}$$.

From the information given in the diagram, we know that $$\hat{A} = 81^{\circ}$$ and $$\hat{C} = 53^{\circ}$$.

So

\begin{align} \hat{A} + \hat{B} + \hat{C} &= 180^{\circ} (\text{sum } \angle \text{s in } \triangle) \\ 81^{\circ} + \hat{B} + 53^{\circ} &= 180^{\circ} \\ \hat{B} &= 180^{\circ} − 53^{\circ} − 81^{\circ} \\ \therefore \hat{B} &= 46^{\circ} \end{align}
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#### Equilateral triangles

An equilateral triangle is a triangle with all sides equal in length. The following activity is an investigation into the interior angles of an equilateral triangle.

equilateral triangle
a triangle with all sides equal in length

### Investigating the interior angles of an equilateral triangle

1. What do you know about the sides of an equilateral triangle? Write a short sentence and include a diagram in your answer.
2. We have already learned how to construct a triangle given the lengths of all three sides. You might need to go back and revise the method for constructing a triangle given all three sides. Then, construct $$\triangle PQR$$ such that $$PQ = QR = RP$$. You can choose any length for the sides of the triangle.
3. Check the accuracy of your construction by measuring the length of each side of the triangle.
4. Use a protractor to measure the size of angles $$\hat{P}$$, $$\hat{Q}$$ and $$\hat{R}$$.
5. What do you notice?
6. What can you deduce about the interior angles of an equilateral triangle?
7. Complete the sentence: The interior angles of an equilateral triangle are … and the size of each angle is $$\ldots^{\circ}$$.
8. Do you think the following statement is true or false? A triangle with longer sides than $$\triangle PQR$$ would have bigger interior angles and a triangle with shorter sides than $$\triangle PQR$$ would have smaller interior angles.

#### Isosceles triangles

An isosceles triangle is a triangle with two equal sides.

isosceles triangle
a triangle with two sides that are equal

The diagram below shows isosceles triangle $$\triangle DEF$$ with $$DE = FD$$.