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Isosceles triangle

Isosceles triangle

Here you will learn about isosceles triangles, including what an isosceles triangle is and the properties of isosceles triangles.

Students first learn about triangles in kindergarten and 1 st grade in geometry with their work in reason with shapes and their attributes. They expand their knowledge of triangles as they progress through elementary school and into secondary school.

What is an isosceles triangle?

An isosceles triangle is a unique type of triangle because it has two equal (congruent) sides and two equal (congruent) angles.

Properties of an isosceles triangle:

  • Two sides of equal length.
  • Two equal angles.

Isosceles Triangle image 1 US

The dashes on the sides and the angles of the triangle mean they are equal in measure or congruent.

The equal (congruent) angles of an isosceles triangle are called the base angles.

The base angles are opposite the equal (congruent) side lengths.

The unequal side (third side) of an isosceles triangle is often called the base of the triangle.

The unequal angle (third angle) of an isosceles triangle is called the vertex angle.

Isosceles Triangle image 2 US

There are three different types of isosceles triangles.

Obtuse isosceles triangleAcute isosceles triangleRight isosceles triangle
● Has 1 obtuse angle

Isosceles Triangle table image 1
● Has 3 acute angles

Isosceles Triangle table image 2
● Has 1 right angle
Isosceles Triangle table image 3

Symmetry

An isosceles triangle has \bf{1} line of symmetry. The line goes from the vertex to the exact middle of the base. The line of symmetry is perpendicular to the base of the isosceles triangle.

Isosceles Triangle image 4 US

What is an isosceles triangle?

What is an isosceles triangle?

Common Core State Standards

How does this apply to 4th grade and 5th grade math?

  • Grade 4 – Geometry (4.G.A.3)
    Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

  • Grade 5 – Geometry (5.G.B.3)
    Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

How to identify isosceles triangles

In order to identify isosceles triangles:

  1. Recall the definition.
  2. Explain why the triangle fits or does not fit the definition.

[FREE] Triangles Check for Understanding Quiz (Grade 4 to 5)

[FREE] Triangles Check for Understanding Quiz (Grade 4 to 5)

[FREE] Triangles Check for Understanding Quiz (Grade 4 to 5)

Use this quiz to check your grade 4 to 5 students’ understanding of triangles. 10+ questions with answers covering a range of 4th and 5th grade triangle topics to identify areas of strength and support!

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[FREE] Triangles Check for Understanding Quiz (Grade 4 to 5)

[FREE] Triangles Check for Understanding Quiz (Grade 4 to 5)

[FREE] Triangles Check for Understanding Quiz (Grade 4 to 5)

Use this quiz to check your grade 4 to 5 students’ understanding of triangles. 10+ questions with answers covering a range of 4th and 5th grade triangle topics to identify areas of strength and support!

DOWNLOAD FREE

Isosceles triangle examples

Example 1: classify isosceles triangles

Is the triangle below an isosceles triangle?

Isosceles Triangle image 5 US

  1. Recall the definition.

An isosceles triangle is a triangle with two equal (congruent) sides and two equal (congruent) angles.

2Explain why the triangle fits or does not fit the definition.

This triangle is an isosceles triangle because it has two side lengths that are 12 \, cm each. So, the triangle has two equal (congruent) sides.

Example 2: classify isosceles triangles

Is the isosceles triangle below a right isosceles triangle, obtuse isosceles triangle, or acute isosceles triangle?

Isosceles Triangle image 6 US

Recall the definition.

Explain why the triangle fits or does not fit the definition.

Example 3: classify isosceles triangles

Choose the triangle that is a right isosceles triangle.

Isosceles Triangle image 7 US

Recall the definition.

Explain why the triangle fits or does not fit the definition.

Example 4: symmetry isosceles triangle

How many lines of symmetry does the triangle have?

Isosceles Triangle image 8 US

Recall the definition.

Explain why the triangle fits or does not fit the definition.

Example 5: classify isosceles triangle

What type of isosceles triangle is shown in this image?

Isosceles Triangle image 10 US

Recall the definition.

Explain why the triangle fits or does not fit the definition.

Example 6: classifying isosceles triangles

What type of isosceles triangle does this photo represent?

Isosceles Triangle image 12 US

Recall the definition.

Explain why the triangle fits or does not fit the definition.

Teaching tips isosceles triangles

  • Incorporate drawing activities where students can use rulers and protractors to create isosceles triangles.

  • Incorporate origami activities that are creative and help students understand the properties of triangles, specifically isosceles triangles.

  • Although worksheets have their place for practicing skills, using digital platforms for game playing is fun for students and helps them to practice skills. Check out free platforms such as desmos, quizlet, blooket, and quizziz.

Easy mistakes to make

  • Thinking that isosceles triangles can be classified as equilateral triangles
    A question may ask students to explain if an isosceles triangle can be equilateral. The easy mistake to make is stating that isosceles triangles can be classified as equilateral triangles. Understanding that properties of isosceles triangles and equilateral triangles can help with questions like this.

    Isosceles triangles only have two equal (congruent) sides and angles and cannot be classified as equilateral. However, equilateral triangles have three equal (congruent) sides and angles and can be classified as isosceles.

  • Angle facts
    A common mistake when classifying triangles is mixing up the definitions of acute angle and obtuse angle. So when students classify the triangles, they wind up classifying them incorrectly. Knowing angle facts will help with this.
    • Acute angles are greater than 0^{\circ} and less than 90^{\circ}.
    • Right angles equal 90^{\circ}.
    • Obtuse angles are greater than 90^{\circ} and less than 180^{\circ}.

  • Base angles
    The common mistake is thinking that the base of the angles are always on the bottom of the isosceles triangle. Seeing the triangles in different positions will help with this understanding.

    For example, here is a picture where the base angles of an isosceles triangle are on the top.

    Isosceles Triangle image 14 US

  • Sides of equal length
    The common mistake is identifying the wrong sides as the equal (congruent sides). Make sure that you get the equal sides and angles in the correct position. It is the 2 sides which are opposite the 2 equal base angles which are equal in length.

    Isosceles Triangle image 15 US

Practice isosceles triangle questions

1. Which of these diagrams shows an isosceles triangle?

Isosceles Triangle image 16 US

GCSE Quiz False

Isosceles Triangle image 17 US

GCSE Quiz False

Isosceles Triangle image 18 US

GCSE Quiz True

Isosceles Triangle image 19 US

GCSE Quiz False

The dash marks on the sides of the triangle indicate which lengths are equal. You need the triangle which has the same mark on two sides.

2. How many lines of symmetry does an isosceles triangle have?

0
GCSE Quiz False

1
GCSE Quiz True

2
GCSE Quiz False

3
GCSE Quiz False

An isosceles triangle only has one line of symmetry that cuts the triangle into two equal parts that are mirror images of each other.

 

The line of symmetry goes from the vertex angle to the middle point of the base.

 

Isosceles Triangle image 20 US

3. Select the triangle that represents an obtuse isosceles triangle.

Isosceles Triangle image 21 US

GCSE Quiz False

Isosceles Triangle image 22 US

GCSE Quiz False

Isosceles Triangle image 23 US

GCSE Quiz False

Isosceles Triangle image 24 US

GCSE Quiz True

An obtuse isosceles triangle is an isosceles triangle that has 1 obtuse angle.

 

Isosceles Triangle image 25 US

4. If a triangle has side measurements of 14 inches, 14 inches, and 19.8 inches and one angle that is 90^{\circ}, what type of triangle is it?

Isosceles triangle

GCSE Quiz False

Right isosceles triangle

GCSE Quiz True

Acute isosceles triangle

GCSE Quiz False

Right scalene triangle

GCSE Quiz False

Here is a drawing of the triangle with the given information.

 

Isosceles Triangle image 26 US

 

The triangle has two side lengths that are equal in measure, and it also has a 90^{\circ}. So, the triangle is a right isosceles triangle.

5. Which photo represents an isosceles triangle?

Isosceles Triangle image 27 US

GCSE Quiz True

Isosceles Triangle image 28 US

GCSE Quiz False

Isosceles Triangle image 29 US

GCSE Quiz False

Isosceles Triangle image 30 US

GCSE Quiz False

An isosceles triangle has two equal sides and two equal angles.

 

This photo of the roof appears to be an isosceles triangle because of the two equal sides and the two equal angles.

 

Isosceles Triangle image 31 US

6. Joey drew a picture of an isosceles triangle. Which set of angle measurements can be the angles of the triangle she drew?

90^{\circ}, 55^{\circ}, 35^{\circ}
GCSE Quiz False

100^{\circ}, 45^{\circ}, 35^{\circ}
GCSE Quiz False

90^{\circ}, 60^{\circ}, 30^{\circ}
GCSE Quiz False

70^{\circ}, 55^{\circ}, 55^{\circ}
GCSE Quiz True

An isosceles triangle has two equal (congruent) sides and two equal (congruent) angles.

 

So the angle measurements of the triangle Joey drew could be 70^{\circ}, 55^{\circ}, 55^{\circ}.

Isosceles triangle FAQs

How do you find the area of isosceles triangles?

You can use the formula for finding the area of a triangle, which is a=\cfrac{1}{2} \cdot b \cdot h. The height of an isosceles triangle is a line segment that starts at the vertex angle and is perpendicular to the opposite side. The height is also known as the altitude of the triangle.

How do you find the perimeter of the isosceles triangle?

You can find the perimeter of an isosceles triangle the same way you would find the perimeter of any polygon, add up all the side lengths.

Does an isosceles triangle have a hypotenuse?

Only right isosceles triangles will have a hypotenuse. The hypotenuse is the longest side of a right triangle, so only right triangles have hypotenuses.

What is trigonometry?

Trigonometry is the study of triangles where you will learn ratio relationships of right triangles such as sine, cosine, and tangent ratios. You will learn about trigonometry in high school.

Can you use the Pythagorean theorem on isosceles triangles?

You can only use the Pythagorean theorem on right isosceles triangles because the Pythagorean theorem only applies to right triangles.

Still stuck?

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