Isosceles Triangle Practice Problems and Exercises

Master isosceles triangles with step-by-step practice problems. Learn to identify equal sides, calculate base angles, and solve vertex angle problems.

📚Master Isosceles Triangle Properties Through Practice
  • Identify equal sides and legs in isosceles triangles
  • Calculate base angles using the equal angles property
  • Find vertex angles when given base angle measurements
  • Classify isosceles triangles as acute, right, or obtuse
  • Apply the angle sum property to solve triangle problems
  • Distinguish between base angles and vertex angles in examples

Understanding Isosceles triangle

Complete explanation with examples

Definition of isosceles triangle

The isosceles triangle is a type of triangle that has two sides (legs) of equal length.

A consequence of having two sides of equal length implies that also the two angles opposite these sides measure the same.

Detailed explanation

Practice Isosceles triangle

Test your knowledge with 20 quizzes

What kind of triangle is given in the drawing?

999555999AAABBBCCC

Examples with solutions for Isosceles triangle

Step-by-step solutions included
Exercise #1

Calculate the size of angle X given that the triangle is equilateral.

XXXAAABBBCCC

Step-by-Step Solution

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

x+x+x=180 x+x+x=180

3x=180 3x=180

We divide both sides by 3:

x=60 x=60

Answer:

60

Video Solution
Exercise #2

What is the size of each angle in an equilateral triangle?

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Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify that an equilateral triangle has all sides of equal length, which implies its angles are also equal.
  • Step 2: Utilize the property that the sum of angles in any triangle is 180180^\circ.
  • Step 3: Since each angle is equal in an equilateral triangle, divide the total sum of 180180^\circ by 3.

Now, let's work through each step:
Step 1: In an equilateral triangle, all angles are equal in size.
Step 2: The sum of angles in any triangle is always 180180^\circ.
Step 3: Divide 180180^\circ by 3.

Calculating 180÷3=60180^\circ \div 3 = 60^\circ.

Therefore, the size of each angle in an equilateral triangle is 6060^\circ.

Answer:

60

Video Solution
Exercise #3

Which kind of triangle is given in the drawing?

666666666AAABBBCCC

Step-by-Step Solution

As we know that sides AB, BC, and CA are all equal to 6,

All are equal to each other and, therefore, the triangle is equilateral.

Answer:

Equilateral triangle

Video Solution
Exercise #4

Given the size of the 3 sides of the triangle, is it an equilateral triangle?

12-X12-X12-XAAABBBCCC2X

Step-by-Step Solution

To determine if the triangle is equilateral, we need to check if all three sides of the triangle are equal.

The given side lengths are 2X2X, 12X12 - X, and 12X12 - X.

For the triangle to be equilateral, we must have the equality:

  • 2X=12X2X = 12 - X

Let's solve this equation:

2Xamp;=12X2X+Xamp;=123Xamp;=12Xamp;=123Xamp;=4 \begin{aligned} 2X &= 12 - X \\ 2X + X &= 12 \\ 3X &= 12 \\ X &= \frac{12}{3} \\ X &= 4 \end{aligned}

Substitute X=4X = 4 back into the expressions for the sides:

  • 2X=2(4)=82X = 2(4) = 8

  • 12X=124=812 - X = 12 - 4 = 8

  • The third side, also 12X=812 - X = 8.

The three calculated side lengths are 88, 88, and 88.

Since all three sides are equal, the triangle is an equilateral triangle.

Therefore, the answer is Yes, the triangle is equilateral.

Answer:

Yes

Video Solution
Exercise #5

Is the triangle in the drawing an acute-angled triangle?

Step-by-Step Solution

To determine if the triangle is an acute-angled triangle, we need to understand the nature of its angles. In an acute-angled triangle, all three angles are less than 9090^\circ. However, we do not have explicit angle measures or side lengths shown in the drawing. Instead, we assess the probable nature of the depicted triangle.

Given that an acute-angled triangle must have its largest angle smaller than 9090^\circ, comparison property of triangle sides through Pythagorean type logic suggests that an acute triangle inequality c2<a2+b2c^2 < a^2 + b^2 (for sides aa, bb, and hypotenuse cc) must hold.

In our problem, the depiction ultimately leads us to infer the implied relations among the triangle's angles. The given solution and analysis indicate it does not meet this criterion.

Hence, the triangle in the given drawing is not an acute-angled triangle, confirming the choice: No.

Answer:

No

Video Solution

Frequently Asked Questions

How do you identify an isosceles triangle?

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An isosceles triangle has two sides of equal length called legs. You can identify it by looking for two equal sides or two equal angles opposite those sides.

What are base angles in an isosceles triangle?

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Base angles are the two angles adjacent to the base (the unequal side) of an isosceles triangle. These angles are always equal to each other and are always acute angles (less than 90°).

How do you find the vertex angle of an isosceles triangle?

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To find the vertex angle: 1) Add the two base angles together, 2) Subtract this sum from 180°, 3) The result is your vertex angle. Remember that all triangle angles sum to 180°.

Can an isosceles triangle have a right angle?

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Yes, an isosceles triangle can have a right angle (90°) at the vertex. In this case, the two base angles would each measure 45°, since 180° - 90° = 90°, and 90° ÷ 2 = 45°.

What is the difference between legs and base in isosceles triangles?

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The legs are the two equal sides of an isosceles triangle, while the base is the third side that has a different length. The vertex angle lies between the two legs.

How do you solve isosceles triangle angle problems?

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Use these key properties: 1) Base angles are equal, 2) All angles sum to 180°, 3) If you know one base angle, the other base angle is the same, 4) Subtract the sum of base angles from 180° to find the vertex angle.

Can base angles in an isosceles triangle be obtuse?

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No, base angles in an isosceles triangle cannot be obtuse (greater than 90°). If they were obtuse, their sum would exceed 180°, which is impossible since all three angles must sum to exactly 180°.

What are the types of isosceles triangles?

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Isosceles triangles are classified by their vertex angle: 1) Acute isosceles - vertex angle less than 90°, 2) Right isosceles - vertex angle equals 90°, 3) Obtuse isosceles - vertex angle greater than 90°.

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