congruent triangle problem solving

Pairs Level 1 Level 2 Level 3 Similar Shapes Exam-Style Description Help More Geometry

This is level 1; Determining whether two triangles are congruent and finding the reason. The diagrams are not drawn to scale. Lengths and angles are not always shown in proportion.

For each pair of triangles select the correct description of congruency.

This is Congruent Triangles level 1. You can also try: Level 2 Level 3

For Students:

• Times Tables
• TablesMaster
• Investigations
• Exam Questions

For Teachers:

• Starter of the Day
• Shine+Write
• Random Names
• Maths Videos
• Laptops in Lessons
• Maths On Display
• Class Admin
• Create An Account
• About Transum
• Privacy Policy

©1997-2024 WWW.TRANSUM.ORG

© Transum Mathematics 1997-2024 Scan the QR code below to visit the online version of this activity.

https://www.Transum.org/go/?Num=421

Description of Levels

Pairs - The classic pairs game with simple congruent shapes.

Level 1 - Determining whether two triangles are congruent and finding the reason

Level 2 - Further questions on recognising congruency ordered randomly

Level 3 - Use your knowledge of congruent triangles to find lengths and angles

Similar Shapes - Similarity is a related concept.

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Geometry including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

Log in Sign up

Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Quick Reference

The following conditions guarantee congruence:

SSS - Given the lengths of all three sides.

SAS - Given two sides and the included angle.

ASA or AAS - Given two angles and any side.

The following is a set of conditions that does not guarantee congruence:

ASS - Given two sides and an angle that is not included between them. However congruence could be confirmed if the angle is a right angle (RHS - right-angle, hypotenuse, side) or an obtuse angle or if the side not adjacent to the angle is longer than the side that is adjacent.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

7.1.3: Triangles

• Last updated
• Save as PDF
• Page ID 62470

• The NROC Project

Learning Objectives

• Identify equilateral, isosceles, scalene, acute, right, and obtuse triangles.
• Identify whether triangles are similar, congruent, or neither.
• Identify corresponding sides of congruent and similar triangles.
• Find the missing measurements in a pair of similar triangles.
• Solve application problems involving similar triangles.

Introduction

Geometric shapes, also called figures, are an important part of the study of geometry. The triangle is one of the basic shapes in geometry. It is the simplest shape within a classification of shapes called polygons . All triangles have three sides and three angles, but they come in many different shapes and sizes. Within the group of all triangles, the characteristics of a triangle’s sides and angles are used to classify it even further. Triangles have some important characteristics, and understanding these characteristics allows you to apply the ideas in real-world problems.

Classifying and Naming Triangles

A polygon is a closed plane figure with three or more straight sides. Polygons each have a special name based on the number of sides they have. For example, the polygon with three sides is called a triangle because “tri” is a prefix that means “three.” Its name also indicates that this polygon has three angles. The prefix “poly” means many.

The table below shows and describes three classifications of triangles. Notice how the types of angles in the triangle are used to classify the triangle.

The sum of the measures of the three interior angles of a triangle is always 180 o . This fact can be applied to find the measure of the third angle of a triangle, if you are given the other two. Consider the examples below.

A triangle has two angles that measure 35 o and 75 o . Find the measure of the third angle.

The third angle of the triangle measures 70 o .

One of the angles in a right triangle measures 57 o . Find the measurement of the third angle.

The third angle of the right triangle measures 33 o .

There is an established convention for naming triangles. The labels of the vertices of the triangle, which are generally capital letters, are used to name a triangle.

You can call this triangle \(\ A B C\) or \(\ \triangle A B C\) since \(\ A\), \(\ B\), and \(\ C\) are vertices of the triangle. When naming the triangle, you can begin with any vertex. Then keep the letters in order as you go around the polygon. The triangle above could be named in a variety of ways: \(\ \triangle A B C\), or \(\ \triangle C B A\). The sides of the triangle are line segments \(\ A B\), \(\ AC\), and \(\ CB\).

Just as triangles can be classified as acute, obtuse, or right based on their angles, they can also be classified by the length of their sides. Sides of equal length are called congruent sides. While we designate a segment joining points \(\ A\) and \(\ B\) by the notation \(\ \overline{A B}\), we designate the length of a segment joining points \(\ A\) and \(\ B\) by the notation \(\ AB\) without a segment bar over it. The length \(\ AB\) is a number, and the segment \(\ \overline{A B}\) is the collection of points that make up the segment.

Mathematicians show congruency by putting a hash mark symbol through the middle of sides of equal length. If the hash mark is the same on one or more sides, then those sides are congruent. If the sides have different hash marks, they are not congruent. The table below shows the classification of triangles by their side lengths.

To describe a triangle even more specifically, you can use information about both its sides and its angles. Consider this example.

Classify the triangle below.

This is an isosceles right triangle.

Classify the triangle shown below.

• acute scalene
• right isosceles
• obtuse scalene
• obtuse isosceles
• Incorrect. This triangle has one angle (angle \(\ Q\)) that is between 90 o and 180 o , so it is an obtuse triangle. It is also scalene because all the sides have different lengths. The correct answer is obtuse scalene.
• Incorrect. This triangle does not contain a right angle. It has one angle (angle \(\ Q\)) that is somewhere between 90 o and 180 o , so it is an obtuse triangle. It is also scalene because all the sides have different lengths. The correct answer is obtuse scalene.
• Correct. This triangle has vertices \(\ P\), \(\ Q\), and \(\ R\), one angle (angle \(\ Q\)) that is between 90 o and 180 o , and sides of three different lengths.
• Incorrect. Although this triangle is obtuse, it does not have two sides of equal length. Its three sides are all different lengths, so it is scalene. The correct answer is obtuse scalene.

Identifying Congruent and Similar Triangles

Two triangles are congruent if they are exactly the same size and shape. In congruent triangles, the measures of corresponding angles and the lengths of corresponding sides are equal. Consider the two triangles shown below:

Since both \(\ \angle B\) and \(\ \angle E\) are right angles, these triangles are right triangles. Let’s call these two triangles \(\ \triangle A B C\) and \(\ \triangle D E F\). These triangles are congruent if every pair of corresponding sides has equal lengths and every pair of corresponding angles has the same measure.

The corresponding sides are opposite the corresponding angles.

A double headed arrow means “corresponds to”

\(\ \begin{array}{l} \angle B \leftrightarrow \angle E \\ \angle A \leftrightarrow \angle D \\ \angle C \leftrightarrow \angle F \\ \overline{A B} \leftrightarrow \overline{D E} \\ \overline{A C} \leftrightarrow \overline{D F} \\ \overline{B C} \leftrightarrow \overline{E F} \end{array}\)

\(\ \triangle A B C\) and \(\ \triangle D E F\) are congruent triangles as the corresponding sides and corresponding angles are equal.

Let’s take a look at another pair of triangles. Below are the triangles \(\ \triangle A B C\) and \(\ \triangle R S T\).

These two triangles are surely not congruent because \(\ \triangle R S T\) is clearly smaller in size than \(\ \triangle A B C\). But, even though they are not the same size, they do resemble one another. They are the same shape. The corresponding angles of these triangles look like they might have the same exact measurement, and if they did they would be congruent angles and we would call the triangles similar triangles.

Congruent angles are marked with hash marks, just as congruent sides are.

We can also show congruent angles by using multiple bands within the angle, rather than multiple hash marks on one band. Below is an image using multiple bands within the angle.

If the corresponding angles of two triangles have the same measurements, they are called similar triangles. This name makes sense because they have the same shape, but not necessarily the same size. When a pair of triangles is similar, the corresponding sides are proportional to one another. That means that there is a consistent scale factor that can be used to compare the corresponding sides. In the previous example, the side lengths of the larger triangle are all 1.4 times the length of the smaller. So, similar triangles are proportional to one another.

Just because two triangles look similar does not mean they are similar triangles in the mathematical sense of the word. Checking that the corresponding angles have equal measure is one way of being sure the triangles are similar.

Corresponding Sides of Similar Triangles

There is another method for determining similarity of triangles that involves comparing the ratios of the lengths of the corresponding sides.

If the ratios of the pairs of corresponding sides are equal, the triangles are similar.

Consider the two triangles below.

\(\ \triangle A B C\) is not congruent to \(\ \triangle D E F\) because the side lengths of \(\ \triangle D E F\) are longer than those of \(\ \triangle A B C\). So, are these triangles similar? If they are, the corresponding sides should be proportional.

Since these triangles are oriented in the same way, you can pair the left, right, and bottom sides: \(\ \overline{A B}\) and \(\ \overline{D E}\), \(\ \overline{B C}\) and \(\ \overline{E F}\), \(\ \overline{A C}\) and \(\ \overline{D E}\). (You might have called these the two shortest sides, the two longest sides, and the two leftover sides and arrived at the same ratios). Now we will look at the ratios of their lengths.

\(\ \frac{A B}{D E}=\frac{B C}{E F}=\frac{A C}{D F}\)

Substituting the side length values into the proportion, you see that it is true:

\(\ \frac{3}{9}=\frac{4}{12}=\frac{6}{18}\)

If the corresponding sides are proportional, then the triangles are similar. Triangles \(\ A B C\) and \(\ D E F\) are similar, but not congruent.

Let’s use this idea of proportional corresponding sides to determine whether two more triangles are similar.

Determine if the triangles below are similar by seeing if their corresponding sides are proportional.

\(\ \triangle A B C\) and \(\ \triangle D E F\) are similar.

The mathematical symbol ~ means “is similar to.” So, you can write \(\ \triangle A B C\) is similar to \(\ \triangle D E F\) as \(\ \triangle A B C \sim \triangle D E F\).

Determine whether the two triangles are similar, congruent, or neither.

• \(\ \triangle A B C \text { and } \triangle D E F\) are congruent.
• \(\ \triangle A B C \text { and } \triangle D E F\) are similar.
• \(\ \triangle A B C \text { and } \triangle D E F\) are similar and congruent.
• \(\ \triangle A B C \text { and } \triangle D E F\) are neither similar nor congruent.
• Incorrect. Congruent triangles have corresponding sides of equal length and corresponding angles of equal measure. They are the same exact size and shape. \(\ \triangle A B C\) is equilateral and \(\ \triangle D E F\) is isosceles, so they are not the same exact shape. The correct answer is \(\ \triangle A B C\) and \(\ \triangle D E F\) are neither similar nor congruent.
• Incorrect. The ratios of the corresponding sides are not equal, so the triangles cannot be similar: \(\ \frac{6.5}{5}=\frac{6.5}{5} \neq \frac{5}{5}\). The correct answer is \(\ \triangle A B C\) and \(\ \triangle D E F\) are neither similar nor congruent.
• Incorrect. All congruent triangles are similar, but these triangles are not congruent. Congruent triangles have corresponding sides of equal length and corresponding angles of equal measure. \(\ \triangle A B C\) is equilateral and \(\ \triangle D E F\) is isosceles, so they are not the same exact shape. The correct answer is \(\ \triangle A B C\) and \(\ \triangle D E F\) are neither similar nor congruent.
• Correct. The corresponding angle measures are not known to be equal as shown by the absence of congruence marks on the angles. Also, the ratios of the corresponding sides are not equal: \(\ \frac{6.5}{5}=\frac{6.5}{5} \neq \frac{5}{5}\).

Finding Missing Measurements in Similar Triangles

You can find the missing measurements in a triangle if you know some measurements of a similar triangle. Let’s look at an example.

\(\ \triangle A B C\) and \(\ \triangle X Y Z\) are similar triangles. What is the length of side \(\ B C\)?

The missing length of side \(\ BC\) is 8 units.

This process is fairly straightforward, but be careful that your ratios represent corresponding sides, recalling that corresponding sides are opposite corresponding angles.

Solving Application Problems Involving Similar Triangles

Applying knowledge of triangles, similarity, and congruence can be very useful for solving problems in real life. Just as you can solve for missing lengths of a triangle drawn on a page, you can use triangles to find unknown distances between locations or objects.

Let’s consider the example of two trees and their shadows. Suppose the sun is shining down on two trees, one that is 6 feet tall and the other whose height is unknown. By measuring the length of each shadow on the ground, you can use triangle similarity to find the unknown height of the second tree.

First, let’s figure out where the triangles are in this situation! The trees themselves create one pair of corresponding sides. The shadows cast on the ground are another pair of corresponding sides. The third side of these imaginary similar triangles runs from the top of each tree to the tip of its shadow on the ground. This is the hypotenuse of the triangle.

If you know that the trees and their shadows form similar triangles, you can set up a proportion to find the height of the tree.

When the sun is at a certain angle in the sky, a 6--foot tree will cast a 4-foot shadow. How tall is a tree that casts an 8-foot shadow?

The tree is 12 feet tall.

Triangles are one of the basic shapes in the real world. Triangles can be classified by the characteristics of their angles and sides, and triangles can be compared based on these characteristics. The sum of the measures of the interior angles of any triangle is 180 o . Congruent triangles are triangles of the same size and shape. They have corresponding sides of equal length and corresponding angles of the same measurement. Similar triangles have the same shape, but not necessarily the same size. The lengths of their sides are proportional. Knowledge of triangles can be a helpful in solving real-world problems.

One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

In order to access this I need to be confident with:

This topic is relevant for:

Congruent Triangles

Here we will learn about congruent triangles, including how to identify them and prove the congruence of triangles.

There are also congruent triangles worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are congruent triangles?

Congruent triangles are triangles that are exactly the same size and shape.

There are 4 conditions to prove congruency in triangles.

See also: Congruent shapes

Side-side-side (SSS)

When two triangles have all three sides the same, they are congruent triangles.

The second triangle may be a rotation or a mirror image of the first triangle (or both).

Right angle, hypotenuse and one other side (RHS)

When two triangles are right-angled triangles and have the hypotenuse and one of the shorter sides the same, they are congruent triangles.

The second triangle may be a rotation or a mirror image of the first triangle (or both). The third side would also be identical and this can be checked using Pythagoras’ theorem.

Side-angle-side (SAS)

When two triangles have two sides and the included angle the same, they are congruent triangles. The included angle is the angle in between the two sides.

Angle-side-angle (ASA)

When two triangles have two angles and the included side the same, they are congruent triangles. The included side is the side in between the two angles.

This can also be known as angle-angle-side (AAS) as if two angles in a triangle are known, the third angle can be worked out using the angle fact that the sum of interior angles in a triangle is 180° .

Congruent is used to describe shapes such as quadrilaterals or polygons which are exactly the same shape and the same size.

Similar is used to describe shapes such as quadrilaterals or polygons which are the same shape, but different sizes.  A scale factor is involved.

How to recognise congruent triangles

In order to recognise if a pair of triangles are congruent:

Check the corresponding angles and corresponding sides.

Decide if the shapes are congruent or not.

• If the triangles are congruent, state which congruence condition fits the pair of triangles.

Congruent triangles worksheet

Get your free congruent triangles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on congruence and similarity

Congruent triangles  is part of our series of lessons to support revision on  congruence and similarity . You may find it helpful to start with the main congruence and similarity lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Congruence and similarity
• Congruent shapes
• Similar shapes

Congruent triangles examples

Example 1: recognise congruent triangles.

Decide whether this pair of triangles are congruent. If they are congruent, state why:

Both triangles have sides 5cm and 7cm .  They both have an angle of 95° .

2 Decide if the shapes are congruent or not.

The 95° angles are in a corresponding position. The triangles are mirror images of each other. The triangles are congruent.

3 If the triangles are congruent, which congruence condition fits the pair of triangles.

The triangles are congruent with the condition side-angle-side (SAS) .

Example 2: recognise congruent triangles

Both triangles have sides 6.3cm, 8.1cm and 10.2cm .

The triangles are the same shape and the same size – they are congruent.

If the triangles are congruent, which congruence condition fits the pair of triangles.

The triangles are congruent with the condition side-side-side (SSS) .

Example 3: recognise congruent triangles

Both triangles have sides 8cm .  

Both triangles have a 50° angle. 

But their second angles are different.

The triangles look like they are different shapes BUT the third angle can be worked out.

Using the angle fact that the sum of interior angles of a triangle is 180° we can work out the missing angle in both triangles.

The 8cm side is in between the 30° and 50° angles in both triangles. The triangles are the same shape and the same size.

They are congruent triangles.

The triangles are congruent with the condition angle-side-angle (ASA).

Example 4: recognise congruent triangles

Both triangles are right-angled triangles.

They both have a short side of 9cm .

But the hypotenuse of each triangle is different.

The triangles look like they are the same shape, but they are not. The triangles are not congruent.

The triangles are not congruent.

The right angle, hypotenuse and one other side (RHS) condition was close to being satisfied, but not quite.

How to prove congruent triangles

In order to prove that a pair of triangles are congruent:

Pair up the corresponding sides.

Pair up the corresponding angles.

State which congruence condition fits the pair of triangles.

Proving congruent triangles examples

Example 5: prove triangles are congruent.

Prove that triangle ABC is congruent to triangle DEF .

State which sides are identical, here there are two pairs of corresponding sides.

State which angles are identical, here there is one pair of equal angles.

You need to use the correct notation.

\text{angle} \ CAB = \text{angle} \ DEF

Triangle ABC is congruent to triangle DEF because they fit the side-angle-side (SAS) condition.

Example 6: prove triangles are congruent

Prove that triangle ABD is congruent to triangle BCD

State which sides are identical, here there is one pair of corresponding sides. It is the common side.

BD is common

State which angles are identical, here there are two pairs of equal angles.

Triangle ABD is congruent to triangle BCD because they fit the angle-side-angle (ASA) condition.

Common misconceptions

• AAA – all three angles

AAA – all three angles being equal is not a condition for triangle congruence. These two triangles have identical angles, but the second triangle is an enlargement of the first triangle. They are similar triangles not congruent triangles.

• Remember – triangles can be congruent but rotations or mirror images

The second triangle may be a rotation or a mirror image of the first shape (or both). The triangle may still be congruent.

• Use the correct notation

If we wanted to write about the 50° angle we shouldn’t just call it Angle B .

It is much better to use the notation angle ABC (or angle CBA ).

• For exam questions, check how many marks it is worth

Some exam questions ask you to explain why two triangles are congruent and are only worth one mark, here you only need to state the congruence condition (RHS, SSS, SAS or ASA). 

If the exam question asks you to prove that two triangles are congruent and are worth several marks. You will need to match up the 3 pairs of equal sides/angles and state the congruence condition.

• Give details for proof questions (higher)

Some questions asking to prove that two triangles are congruent may need more explanations in the details. For example you may need to use an angle fact, such as, “alternate angles are equal”.

Practice congruent triangles questions

1. Here is a pair of congruent triangles. Which congruence condition is satisfied?

The triangles are right-angled triangles. They have the same hypotenuse and the same short side.

3. Here is a pair of congruent triangles. Which congruence condition is satisfied?

The triangles have two identical angles and an identical included side.

4. Here is a pair of congruent triangles. Which congruence condition is satisfied?

The triangles have three identical sides.

5. Prove that triangle ABD is congruent to triangle DEF

SAS Because \begin{aligned} AB = EF \\ AC = DF\\ \text{angle} \ CAB= \text{angle} \ DEF\\ \end{aligned}

SAS Because \begin{aligned} AB = EF \\ AC = DF\\ \text{angle} \ A= \text{angle} \ E\\ \end{aligned}

SAS Because \begin{aligned} AB = EF \\ BC = DE\\ \text{angle} \ CAB= \text{angle} \ DEF\\ \end{aligned}

SAS Because \begin{aligned} AB = EF \\ AC = DF\\ \text{angle} \ ABC= \text{angle} \ DEF\\ \end{aligned}

The correct sides need to be paired up. The correct notation needs to be used for the angles.

6. Prove that triangle ABD is congruent to triangle DEF

SAS Because \begin{aligned} AB = EF \\ \text{angle} \ CAB= \text{angle} \ DFE\\ \text{angle} \ BAC= \text{angle} \ DEF\\ \end{aligned}

ASA Because \begin{aligned} AB = EF \\ \text{angle} \ CAB= \text{angle} \ DFE\\ \text{angle} \ CBA= \text{angle} \ DEF\\ \end{aligned}

ASA Because \begin{aligned} AB = DE \\ \text{angle} \ CAB= \text{angle} \ EFD\\ \text{angle} \ CBA= \text{angle} \ DEF\\ \end{aligned}

ASA Because \begin{aligned} AB = EF \\ \text{angle} \ A= \text{angle} \ F\\ \text{angle} \ B= \text{angle} \ E\\ \end{aligned}

Congruent triangles GCSE questions

1. Explain why these two triangles are congruent.

Side-angle-side (correct answer only)

Angle EDF = 43^{\circ} For finding the angle EDF

AC=DE For matching up the identical sides

\begin{aligned} \text{angle} \ BAC= \text{angle} \ DEF\\ \text{angle} \ BCA= \text{angle} \ EDF\\ \end{aligned} For matching up the identical angles

Triangle ABC is congruent to Triangle DEF because angle-side-angle (ASA) For giving the correct condition

3. ABCD is a rectangle. AC is the diagonal of the rectangle. Prove triangle ABC is congruent to triangle ACD .

Angle ABC = angle ADC = 90^{\circ} For identifying the right angles

The hypotenuse is AC and is common to both triangles For identifying the hypotenuse

Side AD = side BC (as opposite sides of a rectangle are equal) For identifying a pair of equal sides

Triangle ABC is congruent to Triangle DEF because right angle, hypotenuse, side (RHS) For identifying the right angles

Learning checklist

You have now learned how to:

• Identify congruent triangles and the condition of congruence
• Prove two triangles are congruent

The next lessons are

• Transformations
• Loci and construction

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths tuition programme.

Privacy Overview

• Parallelogram
• Quadrilateral
• Parallelepiped
• Tetrahedron
• Dodecahedron
• Fraction Calculator
• Mixed Fraction Calculator
• Greatest Common Factor Calulator
• Decimal to Fraction Calculator
• Whole Numbers
• Rational Numbers
• Place Value
• Irrational Numbers
• Natural Numbers
• Binary Operation
• Numerator and Denominator
• Order of Operations (PEMDAS)
• Scientific Notation

By Subjects

• Trigonometry

Congruent Triangles Worksheets

If you know the congruency theorems well, you wouldn’t face much trouble in doing these worksheets. They help you determine the dimensions of an unknown triangle provided it is congruent to another triangle whose dimensions are known.

Related Materials

• Privacy Policy

Join Our Newsletter

© 2024 Mathmonks.com . All rights reserved. Reproduction in whole or in part without permission is prohibited.

• PowerPoints

Congruent Triangles

Identical Twins have the exact same size and shape, and we instantly recognise that the two of them are exactly the same.

When two items have the exact same size and shape, we say that they are “Congruent”.

This lesson is all about “Congruent Triangles”, eg. pairs of Triangles which have the exact same size and shape.

Congruent Triangles are an important part of our everyday world, especially for reinforcing many structures.

Here are a typical pair of Congruent Triangles

Two triangles are congruent if they are completely identical.

This means that the matching sides must be the same length and the matching angles must be the same size.

The Identical (eg.”Congruent”) Triangles can be in different positions, (or orientations), and still be the exact same size and shape.

The position of the matching Triangles does not affect the fact that they are identical, or “Congruent”.

Introductory Video About Congruency

The following Video by Mr Bill Konst about Congruence, covers the “SSS Rule for Triangles”, as well as covering Quadrilaterals and some interesting optical illusions.

Shortcut Rules for Congruent Triangles

It turns out that we do not have to check all the sides and angles of two Triangles to work out that they are Congruent.

There are FOUR “Shortcut Rules” for Congruent Triangles that we will be covering in this lesson.

The first of these “Shortcut Rules” is the “Side Side Side”, or “SSS” Rule.

SSS – Side Side Side Rule for Triangles

We can actually use just the three sides to work out if two triangles are congruent.

This is called “SSS” or the “Side Side Side Rule”.

In diagrams, the actual values of the sides are sometimes not given.

Instead we have markers to show where the matching same length sides are on the two triangles.

For these Triangles we can apply the “SSS” rule, as long as we have all three sides matching each other on the two triangles.

Symbols Used in Congruency

There is a special symbol we use to indicate that Triangles are Congruent, that is like an equals sign with an extra line added on top of it.

This triple ine symbol is what we use in Australia; however some other countries use an equals sign with a squiggly “tilda” line added to the top of it.

In addition, Triangles are usually labelled with capital alphabet letters.

When we say the Triangles are Congruent using their letters, we need to make sure the order of the letters matches the path around the two triangles correctly.

This is shown in the following example.

We need to be careful with the labelling when our Triangles are in different positions.

Eg. We need to make sure the order of the letters matches going around the two triangles in the same order of sides. Going A to B to C should be the exact same path as D to E to F.

SAS – Side Angle Side Rule for Triangles

Two Triangles will be congruent if two matching sides have equal lengths and the angle included by these sides is the same.

The following video covers the “SSS” and “SAS” Rules for Congruent Triangles.

The American teacher doing the videos does not always use the most correct language, but he is enthusiastic and explains his examples well.

AAS – Angle Angle Side Rule for Triangles

Two triangles are congruent if two matching angles are equal and a matching side is equal in length.

There is also an old “ASA” Angle Side Angle Rule; however this has been brought in to be part of the “AAS” Rule.

It is quite okay to use the “ASA” Rule if the order of the items is “ASA” as shown in the above diagram.

However, most people these days just use “AAS”, so that there is one less congruency rule to memorise.

The following “YayMath” 26 minute gives a comprehensive lesson on the ASA and AAS Rules.

RHS – Right Angle Hypotenuse Rule

Any two Right-Angled 90 degree Triangles are congruent if the hypotenuse and one pair of matching sides are equal in length.

This Rule works because of Pythagoras Theorem for 90 Degree Triangles.

Pythagoras Rule means that the missing side lengths have to be equal, so we are indirectly using the “SSS” Rule here.

Why the SSA Rule Does Not Exist

The following video shows why there is not an SSA Rule for congruent triangles.

Triangle Rules Summary

There are four rules that we use to determine if Triangles are congruent: SSS, SAS, AAS, and RHS.

These are shown in the diagram below:

Congruent Triangles Music Videos

Here is a quick little tune by Abbie about the Triangle Rules, mentioning exclusion of the invalid Angle Side Side “Donkey Rule”.

This next one is a heavy metal Parody (sounds a bit like a Joan Jet song):

Congruent Triangles Examples

The following examples show the required working out for demonstrating that a pair of Triangles are identical.

Composite Shapes Examples

The following video shows some more complex examples, where triangle sides are joined together to other triangles and shapes.

In the next two examples, Congruent Triangles are found within the given Geometric Shapes, which allows side lengths to be proven as equal.

Rules for Angles in Parallel Lines are also used, in particular, the following Alternate Interior Angles Rule:

This first example is the classic “Bow Tie” shaped question for joined congruent triangles.

The next example involves two triangles sharing the diagonal of a Parallelogram.

Congruent Triangles Worksheets

The following worksheet has basic multiple choice questions on Congruent Triangles.

(There are answers on the last page of the PDF document).

The following video has an worked example to get you started.

Click the link below to do this worksheet:

Basic Level Worksheet

The following worksheet has medium level of difficulty multiple choice questions on Congruent Triangles.

Medium Level Worksheet

This last worksheet has challeging multiple choice questions on Congruent Triangles.

Challenging Worksheet

Congruent Triangles Games

The above game is a matching game with several levels of difficulty.

It is more like a Quiz than a Game, but will help you learn congruent triangles.

Click the link below to play this game.

http://www.mangahigh.com/en_au/maths_games/shape/congruence/congruent_triangles

This next game involves dropping the shapes into the correct boxes using the arrow keys.

It also contains multiple choice questions that need to be answered.

http://www.classzone.com/books/geometry_concepts/page_build2.cfm?CFID=15754374&ch=5&id=game

The following is an activity where we get to build congruent triangles based on the congruency rule we pick to work with.

http://illuminations.nctm.org/ActivityDetail.aspx?ID=4

This final game is a Jeapordy Game, but is very slow to load up.

http://www.superteachertools.com/jeopardyx/jeopardy-review-game.php?gamefile=1322685871

Related Items

Classifying Triangles Angle Sum in a Triangle Exterior Angle of a Triangle Angles and Parallel Lines Pythagoras and Right Triangles Jobs that use Geometry

If you enjoyed this lesson, why not get a free subscription to our website. You can then receive notifications of new pages directly to your email address.

Go to the subscribe area on the right hand sidebar, fill in your email address and then click the “Subscribe” button.

To find out exactly how free subscription works, click the following link:

How Free Subscription Works

If you would like to submit an idea for an article, or be a guest writer on our website, then please email us at the hotmail address shown in the right hand side bar of this page.

If you are a subscriber to Passy’s World of Mathematics, and would like to receive a free PowerPoint version of this lesson, that is 100% free to you as a Subscriber, then email us at the following address:

Please state in your email that you wish to obtain the free subscriber copy of the “Congruent Triangles” Powerpoint.

Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.

Like Us on Facebook

Help Passy’s World Grow

Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.

Help us to maintain this free service and keep it growing.

Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!

PayPal does accept Credit Cards, but you will have to supply an email address and password so that PayPal can create a PayPal account for you to process the transaction through. There will be no processing fee charged to you by this action, as PayPal deducts a fee from your donation before it reaches Passy’s World.

Enjoy, Passy

8 Responses to Congruent Triangles

Pingback: Similar Triangles | Passy's World of Mathematics

Pingback: Geometry in the Animal Kingdom | Passy's World of Mathematics

Pingback: Jobs With Geometry | Passy's World of Mathematics

Pingback: Trigonometry – Labeling Triangles | Passy's World of Mathematics

Pingback: The Sine Ratio | Passy's World of Mathematics

Pingback: The Cosine Ratio | Passy's World of Mathematics

Pingback: Trigonometric Ratios | Passy's World of Mathematics

Pingback: The Tangent Ratio | Passy's World of Mathematics

Leave a Reply Cancel reply

You must be logged in to post a comment.

• Search for:

Translate Website

Popular lessons.

• Exponents in the Real World
• Three Circle Venn Diagrams
• Online Graph Makers
• Mean Median Mode for Grouped Data
• Sydney Harbour Bridge Mathematics

Latest Lessons

• Sorry – Our Site Got Hacked
• Mathematics of Photography
• Free Exponents Posters
• Exponential Population Growth
• Ocean Mathematics

Our Facebook Page

• AFL Football
• Area Formulas
• Area of Circle
• Area of Composite Shapes
• Area of Ellipse
• Area of Simple Shapes
• Circumference
• Directed Numbers
• eBay Math Problem
• Equations Games
• Equations One Step
• Equations Three Step
• Equations Two Step
• Fibonacci Sequence in Music
• Free Online Math Games
• Free Online Math Tests
• Free Online Math Videos
• Free Online Math Worksheets
• How to Subscribe
• Jobs that use Geometry
• Math Applications
• Math in Music
• Math in the Real World
• Math Videos
• Mathematics of Aircraft
• Mathwarehouse Interactives
• Measurement
• Measurement Formulas
• Microsoft Mathematics
• My Virtual Home
• Off-Road Algebra
• Olympic Games
• Online Calculators
• Online Math Games
• Online Math Lessons
• Online Math Resources
• Online Math Tests
• Online Math Tools
• Online Math Videos
• Online Math Worksheets
• Online Maths Tests Review
• Online Tests and Quizes
• Percentages
• Polynomials
• Probability
• Rounding Numbers
• Straight Line Graphs
• Sydney Harbour Bridge
• Tall Buildings
• Topology and Networks
• Trigonometry
• Tsunami Mathematics
• Uncategorized
• Venn Diagrams
• Wall Charts and Posters
• Weight Training Math
• Word Problems

Who has visited Map

Linking to us, visitor statistics, user feedback.

• Entries RSS
• Comments RSS
• WordPress.org

Donate to Passy’s World

Like us on facebook.

• 1st Grade Math
• 2nd Grade Math
• 3rd Grade Math
• 4th Grade Math
• 5th Grade Math
• 6th Grade Math
• 7th Grade Math
• 8th Grade Math
• Knowledge Base
• Math for kids

Triangle Congruence Worksheets

Created: December 19, 2022

Last updated: June 6, 2023

Congruent triangles may seem like a piece of cake on the surface. However, this isn’t always the case with young learners. For a student who’s just learning about congruence, it may take a while to get a hang of this concept. Fortunately, a congruent triangles worksheet is here to save the day. Designed for a curious young learner, it equips students with all the knowledge and skills they need to ace this topic. Let’s find out more about this worksheet, shall we?

What Are Congruent Triangles?

Before we delve into the crux of the matter, it’s important to first establish what congruent triangles are. Put simply, congruent triangles are triangles that have the same size and shape. It means that the corresponding sides are equal. In the same vein, the corresponding angles have equal value too.

Triangle Congruence Worksheet Answers PDF

Congruent Triangles Worksheet Answers

Geometry Worksheets Congruent Triangles

Answer Key Triangle Congruence Worksheet Answers

Geometry Worksheet Congruent Triangles Answer

Triangle Congruence Worksheet PDF

Now, here’s the big question that often stumps young learners: how can you tell or prove that two triangles are congruent?

There are four rules that help us prove triangle congruence namely: the SSS rule, SAS rule, ASA rule, and AAS rule.

The Side-Side-Side Rule

This rule states that if three sides of one triangle are equal to three sides of another triangle, then both triangles are congruent.

The Side-Angle-Side Rule

The Side-Angle-Side rule is a tad different. According to this rule, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.

1:1 Math Lessons

The Angle-Side-Angle Rule

According to the Angle-Side-Angle rule, if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then both triangles are congruent.

The Angle-Angle-Side Rule

This rule states that if two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then there is congruence.

About the Proving Triangles Congruent Worksheet

This worksheet helps students learn more about triangle congruence and explains what this concept entails. It contains a wide range of exercises that are arranged in a progressive fashion, from simple to complicated. This approach will ease students into the concept, gradually building their skills as they ascend.

Triangle Congruence Worksheets PDF

Triangle Congruence Worksheet

Congruent Triangles Worksheet

Proving Triangles Congruent Worksheet

Triangle Congruence Worksheet Answer Key

It also contains the answers to the exercises. Wondering why? Using triangle congruence worksheet answer keys, kids can have self-guided learning sessions where they cross-check what they’ve done against the answers. The congruent triangles worksheet answers aren’t cheat sheets. They’re simply guides.

Benefits of the Triangle Congruence Worksheet Answers PDF

The triangle congruence worksheet is an excellent choice for young learners delving into this topic. These tools are interactive and loaded with sufficient visual stimulations to keep students engaged and ease the learning process. They will help a learner develop problem-solving skills and over time, master the concept of triangle congruence.

Most importantly, this worksheet is free!

More Worksheets

• Special Right Triangles Worksheets
• Similar Triangles Worksheets
• Classifying Triangles Worksheets
• Triangle Sum Theorem Worksheets
• Area Of A Triangle Worksheets
• Angles In A Triangle Worksheets

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.

Need help with Geometry?

• Does your child struggle to grasp geometry?
• Try studying with an online tutor.

Kid’s grade

Is your child having trouble understanding the concept of geometry? An online tutor could provide the necessary guidance.

After-School Math Program

Related worksheets, equations and inequalities worksheets.

Equations and inequalities are mathematical statements that show the relationship between two expressions. For example, while linear inequality comes from a linear function, an equation is a statement asserting the equivalence of two terms. Some students may be unable to make this distinction, so they need help from resources. Kids will find problems with inequalities […]

Trapezoid Worksheets

Any quadrilateral that has at least two parallel sides is called a trapezoid. Some people call this shape trapezium, but they mean the same thing. A trapezoid worksheet can help your kids learn more about the trapezoid and its calculations. Benefits of using the trapezoid worksheet Trapezoid worksheets offer preschoolers many benefits, three of which […]

Color by Number Worksheets for Kindergarten

If you want your child to have fun and at the same time improve their fine motor skills, color by number worksheets for kindergarten are the way to go. When kids color by number, there is always a number system provided on the worksheet so they can follow the prompt while coloring images: an exciting […]

We use cookies to help give you the best service possible. If you continue to use the website we will understand that you consent to the Terms and Conditions. These cookies are safe and secure. We will not share your history logs with third parties. Learn More