Functional Skills

# Triangles ABC and CDE are Mathematically Similar?

Many people ask, “triangles ABC and CDE are mathematically Similar or not?”

If you are also one of them and want to discover the truth behind the triangles ABC and CDE’s similarity, you are at the right place. By checking out the popular triangle-related issues in this blog for Functional Skills levels 1 and 2, you can learn the basics of the triangle and prove similar triangles.

You’ll understand different types of triangles and some common similar triangle problems. In addition, this article will significantly assist you in determining whether triangles ABC and CDE are mathematically similar shapes or not.

Table of Content

## What is a Triangle?

A triangle is a two-dimensional closed shape with three sides, three angles, and three vertices. A triangle is a polygon as well.

A polygon is a two-dimensional closed object that is flat or plane and has straight sides in geometry. There are no curved sides to it. The edges of a polygon are also known as the sides. The vertices (or corners) of a polygon are the spots where two sides meet.

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### Examples of triangles

Sandwiches, traffic signs, fabric hangers, and a billiards rack are all examples of triangles in real life.

### Components of Triangle

- There are three sides to a triangle. The sides of the triangle ABC are AB, BC, and CA.
- The angle of the triangle, indicated by the symbol, is the angle created by any two sides of a triangle.
- There are three angles in a triangle.
- ABC, BCA, and CAB are the three angles of the triangle ABC. These angles are also known by the letters B, C, and A.
- A vertex is a point at which any two sides of a triangle intersect.
- There are three vertices in a triangle. The vertices of the triangle ABC are A, B, and C.

### Characteristics of Triangle

- The sum of a triangle’s three interior angles is always equal to 1800.
- When the lengths of two triangle sides are added together, the length of the third side is always bigger.
- A triangle’s area is equal to half of its base plus two times its height.

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### Different Types of Triangles

The many varieties of triangles are categorised based on the length of their sides and the angles’ measurements.

Triangles are categorised into the following types based on their side lengths:

**Triangle with equal sides**

An equilateral triangle is one in which all three sides of a triangle are the same length.

**Triangle of Isosceles**

When the two sides of a triangle are equal or congruent, an isosceles triangle is generated.

**Triangle of Scalene**

A scalene triangle is one in which none of the sides of the triangle are equal.

Triangles are categorised into the following types based on their angles:

**Acute Triangle**

An acute-angled triangle or acute triangle is formed when all of the angles of a triangle are acute; that is, they measure less than 90 degrees.

**A Right-Angled Triangle or Right Triangle**

A right triangle is formed when one of the triangle’s angles is 90 degrees.

**Triangle with an Obtuse Angle**

An obtuse-angled triangle, also known as an obtuse triangle, is a triangle with an obtuse angle. It is one in which one of the triangle’s angles is more than 90 degrees.

The following are the different sorts of triangles, as well as their sides and angles:

**Equilateral or Equiangular Triangle**

An equilateral or equiangular is one in which all sides and angles are equal.

**Right Triangle Isosceles**

A right triangle with two equal sides and one 90° angle is called an isosceles right triangle. In an isosceles right triangle, two sides and two acute angles are comparable.

**Triangle of Obtuse Isosceles**

A triangle having two equal sides and one obtuse angle is known as an obtuse isosceles triangle.

**Triangle of Isosceles Acute**

All three angles of an acute isosceles triangle are acute, and two sides are the same length.

## Triangles ABC and CDE are Mathematically Similar

We can solve the following problem (with similar triangles formula) to show that triangles ABC and CDE are mathematically similar or identical.

If AC is 8 cm and BC is 5 cm for the ABC triangle and CD is 18 cm and CE is 24 cm for the CDE triangle then:

a. Work out the length of DE and

b. Work out the length of AD.

**Solution**: From the given information in the diagram

a) The length of DE is 15cm and

b) The length of AB is 6cm

From the question,

Triangles ABC and CDE are mathematically similar.

a) To determine the length of DE

Since ΔABC is similar to ΔCDE

By similar triangles theorem, we can write that

\frac{/BC/}{/DE/}=\frac{/CA/}{/EC/}

/DE/

/BC/

=

/EC/

/CA/

From the diagram

/BC/ = 5cm

/CA/ = 8cm

/EC/ = 24 cm

/DE/ = ?

Putting these values into

\frac{/BC/}{/DE/}=\frac{/CA/}{/EC/}

/DE/

/BC/

=

/EC/

/CA/

We get

\frac{5}{/DE/}=\frac{8}{24}

/DE/

5

=

24

8

∴ 8 \times /DE/ = 5 \times 248×/DE/=5×24

Now, divide both sides by 8

\frac{8 \times /DE/}{8}= \frac{5 \times 24}{8}

8

8×/DE/

=

8

5×24

/DE/ = \frac{120}{8}/DE/=

8

120

/DE/ = 15cm

b) To work out the length of AB

Also, by similar triangles theorem, we can write that

\frac{/AB/}{/CD/}=\frac{/CA/}{/EC/}

/CD/

/AB/

=

/EC/

/CA/

From the diagram

/AB/ = ?

/CD/ = 18cm

/CA/ = 8cm

/EC/ = 24 cm

Putting these values into

\frac{/AB/}{/CD/}=\frac{/CA/}{/EC/}

/CD/

/AB/

=

/EC/

/CA/

We get

\frac{/AB/}{18}=\frac{8}{24}

18

/AB/

=

24

8

Now, multiply both sides by 18, that is

18 \times \frac{/AB/}{18}=\frac{8}{24} \times 1818×

18

/AB/

=

24

8

×18

/AB/ = \frac{144}{24}/AB/=

24

144

/AB/ = 6cm

Hence,

a) The length of DE is 15cm and

b) The length of AB is 6cm

### Some Common Similar Triangle Problems

The two triangles are similar if all three sides of one triangle are proportional to the three sides of another triangle. If AB/XY = BC/YZ = AC/XZ, then ABC XYZ is the result.

### Download Some Common Similar Triangle Problems

Visit Math Work Sheets to download some common and similar triangles and identify smaller triangles and calculate side lengths of similar triangles.

## Common Triangle Related Problems for Functional Skills Levels 1 & 2

One of the main goals of the Functional Skills mathematics standards is for students to achieve confidence, fluency, and a good attitude toward mathematics.

When students can exhibit a solid grasp of mathematical knowledge and skills and use it to solve mathematical problems, they will demonstrate their confidence in using mathematics.

The following information explores Functional Skills Levels 1 & 2 problems.

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### Maths Level 1 Problem

John thinks he is spending too much money on his car’s fuel.

Using the formula, he calculates the amount of fuel used in a year in litres.

Fuel used in litres = Distance travelled in miles × 4.5 ÷ Miles per gallon.

Last 12 months he travelled 12 000 miles. His vehicle did 50 miles according to the gallon. How many litres of gasoline did John use final 12 months?

The average cost of petrol last year was £2.33 per litre.

How much did it cost John last year for 100 litres of petrol?

John takes a loan out to buy a car.

He borrows £5000. He will pay 8% interest on the loan.

Calculate £9000 increased by 6%.

John looks for a car to buy.

There are 450 cars advertised.

40% of them are diesel cars. What is 40% of 450?

John finds 30 cars that he could buy.

He writes down how many miles each one has travelled.

48 567 | 65 721 | 99 140 | 72 223 | 62 496 |

47 110 | 98 750 | 73 245 | 87 228 | 69 312 |

59 250 | 73 600 | 84 750 | 65 005 | 91 605 |

73 250 | 58 600 | 80 500 | 92 575 | 91 189 |

Use the data to complete the frequency table below.

Mileage |
Frequency |

40 000 to 49 999 | 4 |

50 000 to 59 999 | 4 |

60 000 to 69 999 | 9 |

70 000 to 79 999 | 3 |

### Math Level 2 Problem

Olivia wants to keep chickens in her yard.

She finds this information about some different types of chickens.

Type of Chickens |
Numbers of eggs per year |

Buff Orpington | 150 |

Rhode Island Red | 160 |

Leghorn | 170 |

Plymouth Rock | 130 |

Speckled Sussex | 250 |

Olivia wants to fence the lawn where chickens can walk around safely.

She has 25 m of fencing. She reads that the minimum area for each chicken to run about safely is 1.7 m2 Olivia decides to make a square area of grass.

What is the greatest number of chickens she could keep?

Olivia buys 15 chickens.

She reads and finds out that the chickens will be happy and healthy if the average annual number of eggs per chicken is 260 or higher.

Olivia records how many eggs her 15 chickens lay in the first 6 weeks. The table shows her results:

Week 1 |
Week 2 |
Week 3 |
Week 4 |
Week 5 |
Week 6 |

48 | 50 | 60 | 74 | 62 | 73 |

D. In a recent report, it was estimated that 50 000 people keep Chikens. Write 50000 in words.

E. 64% of the eggs produced in the UK in 2018 were free-range. Write 64% as a fraction.

### FAQ

#### Why are triangles ABC and DEF similar?

Triangle DEF is similar to triangle ABC as its three angles are equal. To get the sides of triangle ABC, multiply the length of each side in triangle DEF by the same number, 3.

#### What triangles are similar to triangle ABC?

Equiangular triangles have the same shape but may have different sizes. So, equiangular triangles are also known as similar triangles. Triangle DEF, for example, is comparable to triangle ABC since its three angles are the same.

#### Which pair of triangles are similar?

The triangles are comparable or similar if two pairs of corresponding angles in a pair of triangles are congruent.

#### Are triangles ABC and XYZ similar?

Triangle ABC is the same as triangle XYZ. It signifies that the parts that make up the whole are the same.

#### Are all triangles similar?

All triangles are not similar because all triangles do not have equal angles or sides in ratio.

#### Are all right triangles similar?

No. Not all right triangles are similar.

#### Are all equilateral triangles similar?

Because the angles of every equilateral triangle are 60 degrees, the AAA Postulate states that all equilateral triangles are similar.

#### What type of triangles is always similar?

Equilateral triangles are all the same.

#### Why all triangles are not similar?

All triangles are not similar because all triangles do not have equal angles or sides in ratio.

#### Are all circles congruent or similar?

Congruent is defined as similar in shape but a different size. The sizes of the circles can be the same or different. All of the circles are congruent and similar.

## Conclusion

Triangles are particularly useful because they can be split into arbitrary polygons (with 4, 5, 6, or n sides). Understanding the fundamental properties of triangles allows for a more in-depth examination of these larger polygons.

It’s also crucial to learn and comprehend the various forms of triangles, as well as to identify which triangle problems are frequent and comparable. I hope this article assisted you in discovering the truth behind the triangles of ABC and CDE similarity.

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