Constructing Triangles, Quadrilaterals Using Angle Measures and Side Angles (Definition, Basics, Branches, Facts, Examples) - BYJUS

# Constructing Triangles, Quadrilaterals using Angle Measures and Side Angles

We have learned the angle sum property of polygons. The sum of all angles inside a shape depends on the number of sides and angles in it. We can use this concept and use instruments to construct a shape by drawing angles and sides of known measurements....Read MoreRead Less

## Constructing Triangles and Quadrilaterals Using Angle Measures and Side Angles

When the sum of the angle measures is 180 degrees, you can draw a triangle with three given angle measures.

The three angles form 180° in a straight angle, such as ∠ A + ∠ B + ∠ C in the circle shown in the figure below. Congruent angle pairs are formed by the transversals created by the side lengths of the triangle. ∠ A and ∠ A, for example, are congruent because their interior angles are opposite. ∠ B and ∠ B are also congruent because their interior angles are opposite. We know that the interior angles ∠ A + ∠ B + ∠ C must also equal 180° because ∠ A = ∠ A and ∠ B = ∠ B. ∠ A straight angle, or essentially a straight line, is one that is 180°. The interior angles of a triangle must add up to 180°, which means that none of the angles can be 180° individually.

You can change the lengths of the sides while constructing a triangle, through angle measures to create a variety of triangles that meet the criteria. Triangles are identical if the sides and angles of the corresponding triangles are equal. There is only one triangle possible when using angles and sides to make a triangle. When the sum of the lengths of any two sides is greater than the length of the third side, you can draw a triangle with three given side lengths.

Only one triangle can be constructed with three given side lengths; you can use different side lengths, but the resulting triangle will be the same size and shape.

When the sum of the angle measures is 360°, you can draw a quadrilateral with four given angle measures.

## Examples of Constructing Triangles using Angle Measures

Example 1:  Draw a triangle with angle measures of 30°, 60°, and 90°, if it is possible.

Solution:

Since 30° + 60° + 90° = 180°, you can draw a triangle using the given angle measures.

Step 1: Drawing the 30° angle.

Step 2: Drawing the 60° angle.

Step 3: Measuring the remaining angle, the angle measure is 90°.

Example 2:  Draw a triangle with angle measures of 45°, 45°, and 90°,  if it is possible.

Solution:

Since 45° + 45° + 90° = 180°, you can draw a triangle using the given angle measures.

Step 1: Drawing the 45° angle.

Step 2: Drawing the 45° angle.

Step 3: Measuring the remaining angle, the angle measure is 90°.

Example 3:  Draw a triangle with angle measures of 100°, 55°, and 25°, if it is possible.

Solution:

Since 100° + 55° + 25° = 180°, you can draw a triangle using the given angle measures.

Step 1: Drawing the 100° angle.

Step 2: Drawing the 55° angle.

Step 3: Measuring the remaining angle, the angle measure is 25°.

Example 4: Draw a triangle with angle measures of 60°, 60°, and 60°, if it is possible.

Solution:

Because 60° + 60° + 60° = 180°. Now you can draw a triangle using the given angle measures.

Step 1: Drawing the 60° angle.

Step 2: Drawing the 60° angle.

Step 3: Measuring the remaining angle, the angle measure is 60°.

## Constructing Triangles using Angles and Sides

For side-angle-side triangles, you can use this construction to measure the length of the undefined side or angles.

Step-by-step procedure for the construction of triangles:

Step 1: Draw a line using one of the required lengths. If you choose the longest, it’s usually easier to adjust where the triangle will end up.

Step 2: Construct a line at the desired angle from one end of the first line.

Step 3: For the second side, stretch this line to the desired length. (or, if it’s already too long, make a mark along the length of it).

Step 4: Connect the ends of the two lines. This is the last and third side.

## Examples of Constructing Triangles using Angles and Sides

Example 1: Create a triangle with side lengths of 3 and 4 centimeters that meet at a 20° angle.

Solution:

Step 1: Drawing a 20° angle.

Step 2: Use a ruler to mark 3 centimeters on one ray and 4 centimeters on the other ray.

Step 3: Draw the third side to form the triangle.

Example 2: Make a triangle with 1-inch and 2-inch side lengths that meet at a 60° angle.

Solution:

Step 1: Draw a 60° angle.

Step 2: Mark 1 inch on one ray and 2 inches on the other ray with the help of a ruler.

Step 3: Draw the third side to form the triangle.

## Constructing Triangles using Side Lengths

For side-angle-side triangles, you can use this construction to measure the length of the undefined side or angles.

Step-by-step procedure for the construction of triangles:

Step 1: Draw a line using one of the required lengths. If you choose the longest, it’s usually easier to adjust where the triangle will end up.

Step 2: Construct a line at the desired angle from one end of the first line.

Step 3: For the second side, stretch this line to the desired length. (or, if it’s already too long, make a mark along the length of it).

Step 4: Connect the ends of the two lines. This is the last and third side.

## Examples of Constructing Triangles using Side Lengths

Example 1: Make a triangle with the given side lengths, if it is possible.

a) 4cm, 2cm, 3cm

Solution: The length of any two sides added together is greater than the length of the third side.

4cm + 2cm > 3cm

4cm + 3cm > 2cm

2cm + 3cm > 4cm

Step 1: Draw a 4 – centimeter side.

Step 2: Use a compass to calculate where the 2-centimeter side and the 3-centimeter side meet.

Step 3: At either intersection, the third vertex can be found. Make a triangle.

b) 2.5in, 1in, 1in

Solution:

Since 1in + 1in < 2.5 in, it is not possible to draw the triangle.

Try to draw a triangle for testing purposes. To demonstrate that the 1-inch sides cannot interact, draw a 2.5-inch side and use a compass.

c) 2cm, 2cm, 5cm

Solution:

Since 2cm + 2cm < 5cm, it is not possible to draw the triangle.

Try to draw a triangle for testing purposes. To demonstrate that the two-centimeter sides cannot interact, draw a 5-centimeter side using a compass.

d) 4cm, 3cm, 3cm

Solution:

The sum of the lengths of any two sides is greater than the length of the third side.

4cm + 3cm > 3cm

3cm + 4cm > 3cm

3cm + 3cm > 4cm

Step 1: Draw a 4 – centimeter side.

Step 2: Use a compass to determine where the 3 centimeter side and the 3 centimeter side meet.

Step 3: At either intersection, the third vertex can be found. Make a triangle.

In geometry, a quadrilateral is a closed polygon with four edges and four vertices. In other words, a quadrilateral is any figure with four sides. In a quadrilateral, quad means four, and lateral refers to the quadrilateral’s sides. As a result, quadrilaterals refer to all closed figures with four sides. Each of the quadrilateral’s sides can be equal, unequal, parallel, or irregular, resulting in a variety of shapes in these four-sided shapes. Whatever shape it is, every quadrilateral has four sides, four vertices, and a total angle of 360 degrees. It holds true for all quadrilaterals.

Steps to construct quadrilaterals using a protractor:

Step 1: Draw the first two angles on the boundary line using a protractor so that each angle has one side on a line.

Step 2: Draw the third angle formed by taking the projection of the second angle’s direction as the boundary line of the third angle.

Step 3: Measure the remaining angle using the protractor. The resulting angle should have to be equal to the fourth angle given in the question (or) ㄥD = 360° – (ㄥA + ㄥB + ㄥC).

Let’s understand these steps by using the examples below:

## Examples of Constructing a Quadrilateral

Example 1:  Make a quadrilateral with angle measures of 60°, 120°, 70° and 110°, if it is possible.

Solution:

Since 60° + 120° + 70° + 110° = 360°, you can construct a quadrilateral with the given angle measures.

Step 1: Draw a 60° angle and a 120° angle so that each has one side on a line.

Step 2: Draw the remaining side at a 70° angle.

Step 3: Measure the remaining angle. The angle measures 110°.

Example 2: Make a quadrilateral with angle measures of 100°, 90°, 65° and 105°, if it is possible.

Solution:

Since 100° + 90° + 65° + 105° = 360°, you can construct a quadrilateral with the given angle measures.

Step 1: Draw a 100° angle and a 90° angle so that each has one side on a line.

Step 2: Draw the remaining side at a 65° angle.

Step 3: Measuring the remaining angle. The angle measure resulted is 105°.

Example 3: Make a quadrilateral with angle measures of 100°, 40°, 20° and 20°, if it is possible.

Solution:

Since 100° + 40° + 20° + 20° is not equal to 360°, you cannot draw a quadrilateral with the given angle measures.

Step 1: Draw a 100° angle and a 40° angle so that each has one side on a line.

Step 2: Draw the remaining side at a 20° angle.

Step 3: Measure the remaining angle. The angle measures 200°.

## Finding the Area by Constructing a Triangle using Side Lengths

Various formulas can be used to calculate the area of a triangle. $$\frac{1}{2}\times{\text{base}}\times{\text{height}}$$ is the basic formula for calculating the area of a triangle, where the “base” is the side of the triangle upon which altitude is formed and the “height” is the length of the altitude mapped to the “base” from its opposite vertex.

## Examples of Finding the Area by Constructing Triangle using Side Lengths

Landscape boards with lengths of 2 yards, 5 yards, and 6 yards are used to enclose a flower bed. Calculate the size of the flowerbed.

Solution:

The lengths of boards used to enclose a triangular region are familiar to you.

You’ve been given the task of calculating the area of the triangular region.

• Using a 1 cm: 1 yd scale, draw a triangle with sides of 2 yards, 5 yards, and 6 yards. Estimate the base of the flower bed and its height by using the drawing. Then, to estimate the area, use the formula for a triangle’s area.
• Draw the triangle now.

• The flower bed appears to be in the shape of a right triangle, with a 5-yard base and a 2-yard height.

• As a result, the flower bed’s area is approximately,

$$A = \frac{1}{2}\times 5 \times 2= 5$$ square yards

Another method: The height from the largest angle to the 6 centimeter side is about 3.4 centimeters when measured with a ruler. As a result, the surface area is approximately.

$$A = \frac{1}{2}\times {3.4} \times 6= 10.2$$ square yards