Math resources Geometry

Congruence and similarity

Congruent triangles

# Congruent triangles

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

Students will first learn about congruent triangles as part of geometry in eighth grade and again in high school geometry.

## 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.

### 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.

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

### 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.

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

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^{\circ}.

## Common Core State Standards

How does this relate to 8 th grade and high school?

• Grade 8 – Geometry (8.G.A.2)
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

• High school – Similarity, right triangles and trigonometry (9-12.HSG-SRT.B.5)
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

## How to recognize congruent triangles

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

1. Check the corresponding angles and corresponding sides.
2. Decide if the polygons are congruent or not.
3. If the triangles are congruent, state which congruence condition fits the pair of triangles.

## Congruent triangles examples

### Example 1: recognize congruent triangles

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

1. Check the corresponding angles and corresponding sides.

Both triangles have sides 5~{cm} and 7~{cm}. They both have an angle of 95^{\circ}.

2Decide if the polygons are congruent or not.

The 95^{\circ} angles are in a corresponding position. The triangles are mirror images of each other. The triangles are congruent.

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

The triangles are congruent with the condition side-angle-side \textbf{(SAS)}.

### Example 2: recognize congruent triangles

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

Check the corresponding angles and corresponding sides.

Decide if the polygons are congruent or not.

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

### Example 3: recognize congruent triangles

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

Check the corresponding angles and corresponding sides.

Decide if the polygons are congruent or not.

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

### Example 4: recognize congruent triangles

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

Check the corresponding angles and corresponding sides.

Decide if the polygons are congruent or not.

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

## How to prove congruent triangles

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

1. Pair up the corresponding sides.
2. Pair up the corresponding angles.
3. 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.

Pair up the corresponding sides.

Pair up the corresponding angles.

State which congruence condition fits the pair of triangles.

### Example 6: prove triangles are congruent

Prove that triangle ABD is congruent to triangle BCD

Pair up the corresponding sides.

Pair up the corresponding angles.

State which congruence condition fits the pair of triangles.

### Teaching tips for congruent triangles

• Use models and manipulatives. Having students physically move triangles around to see how they match up, even when oriented differently, makes the concept more concrete. Tools like Geogebra, Desmos, geometric solids, and triangle cutouts facilitate hands-on triangle comparisons.

• Focus on corresponding parts. Students must deeply understand that for triangle congruence, it’s not about any three angles or sides overall being equal, but rather matching angles and sides specifically “corresponding” between the triangles that must be equivalent. Use coloring, labels, and worksheets to indicate these correlations.

• Incorporate real-world examples. Provide images of congruent objects students see everyday—folded paper, signs, architecture shapes. Ask questions like “How do you know these triangles within the structures are congruent?” to encourage practical applications.

• Use proper math language. Reinforce vocabulary terms like corresponding, congruent, transformation, orientation, etc. alongside having students articulate triangle congruence postulates and proofs clearly using if-then logic and proper notations. Saying “these triangles match” is too vague—accurate language is key!

### Easy mistakes to make

• Assuming three angles being equal is a condition for triangle congruence
A common mistake is incorrectly assuming that just because two triangles have the same three angle measurements, it means the triangles must be congruent. However, having identical angles is not sufficient – the corresponding sides of the triangles also need to be equal in length for them to be congruent.

• Not recognizing rotations or mirror images as congruent triangles
A triangle may be a rotation or a mirror image of the first shape (or both). The triangle may still be congruent. A common error is not realizing that congruent triangles can be in different orientations – they do not necessarily have to be positioned the same way.

• Incorrectly naming angles
When referring to angles within a triangle, it is important to use the proper mathematical notation instead of informal names. For instance, consider a triangle labeled ABC that contains a 50 degree angle. It would be wrong to call this " Angle B."

In mathematical terminology, the appropriate way to name this angle is " angle ABC" or " angle CBA," which clearly identifies the vertex point and the side endpoints that delineate the angle. Calling it by a simple letter risks ambiguity and reduces precision, which can lead to flawed conclusions about triangle congruence.

• Not giving enough details on proof questions
When attempting to prove two triangles congruent, it is essential to demonstrate the logic and evidence that verifies their equality. Simply stating the triangles are congruent without sufficient supportive details will likely result in lost credit.

For example, consider a proof problem requiring you to confirm ∆ABC ≅ ∆DEF. It is not enough to merely assert “These triangles have corresponding equal angles and sides, therefore they are congruent.”

### Practice congruent triangles questions

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

SAS

SSS

RHS

ASA

The triangles are right triangles. They have the same hypotenuse-leg and the same non-included side.

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

RHS

SSS

ASA

SAS

The triangles have two identical sides and an identical included angle.

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

RHS

SSS

ASA

SAS

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

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

RHS

SSS

SAS

ASA

The triangles have three identical sides.

5) Prove that triangle ABC is congruent to triangle DEF.

SAS

because

\begin{aligned} &AB = EF \\ &AC = DE\\ &\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} \ ABC= \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 ABC 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 = DE \\ &\text{angle} \ CAB= \text{angle} \ EFD\\ &\text{angle} \ CBA= \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 = EF \\ &\text{angle} \ A= \text{angle} \ F\\ &\text{angle} \ B= \text{angle} \ E \end{aligned}

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

## Congruent triangles FAQs

What are the triangle congruence criteria?

There are three main criteria for triangle congruence. If two triangles meet any one of these three criteria, they are considered congruent.

● Side-Side-Side (SSS) criterion: Two triangles are congruent if all three of their corresponding side lengths are equal.

● Side-Angle-Side (SAS) criterion: Two triangles are congruent if two of their corresponding side lengths and the angle between those sides are equal.

● Angle-Angle-Side (AAS) criterion: Two triangles are congruent if two of their corresponding angles and the side between those angles are equal.

In the case of right triangles, the Pythagorean theorem means that we can calculate the length of the third leg given the lengths of any other two.

So right triangles are a special case of the Side-Side-Side criterion where we only need two pairs of congruent side lengths.

● Hypotenuse-Leg (HL) criterion: Two right triangles are congruent if their hypotenuse lengths are equal and the lengths of one of their legs are equal.

We don’t need a Leg-Leg criterion for right triangles, because that would just be another case of Side-Angle-Side congruence.

How do the triangle congruence criteria help us work with other figures?

The triangle congruence criteria help us compare and work with other figures in several ways. First, they allow us to determine whether two triangles are congruent, which can be helpful when we want to determine whether two other figures are also congruent.

Second, the triangle congruence criteria can help us break down other figures into triangles, which can be easier to work with. For example, we might divide a quadrilateral into two triangles, determine whether the two triangles are congruent, and then use that information to infer something about the quadrilateral.

Lastly, the triangle congruence criteria can also help us solve problems involving other figures by allowing us to use properties of congruent triangles. For example, if we know that two triangles are congruent, we can use the corresponding parts of the congruent triangles to find missing measurements in one of the figures.

Why can’t I use Angle-Side-Side or Angle-Angle-Angle to show triangle congruence?

The simple answer is that these two methods don’t always work. Side-Side-Angle (SSA) seems like it could work, but it’s possible to have two triangles with the same angle and two side measurements that aren’t congruent.

The same goes for Angle-Angle-Angle (AAA). Two triangles can have the same three angles but not be congruent.

This is why we use the four methods that do work: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Hypotenuse-Leg (HL) for right triangles.

These methods have been proven to work every time, so we can rely on them to show that two triangles are congruent.

## The next lessons are

• Transformations
• Mathematical proof
• Area

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