Isosceles Triangle Practice Problems with Solutions

Master identifying isosceles triangles with step-by-step practice problems. Learn to recognize equal angles, heights, medians, and angle bisectors in triangles.

📚What You'll Master in This Practice Session
  • Identify isosceles triangles using equal angle conditions
  • Apply height and angle bisector coincidence rules
  • Recognize when median and height are the same line
  • Solve problems involving isosceles triangle properties
  • Distinguish between isosceles and other triangle types
  • Use triangle identification methods in real geometry problems

Understanding Identification of an Isosceles Triangle

Complete explanation with examples

When we have a triangle, we can identify that it is an isosceles if at least one of the following conditions is met:

1) If the triangle has two equal angles - The triangle is isosceles.
2) If in the triangle the height also bisects the angle of the vertex - The triangle is isosceles.
3) If in the triangle the height is also the median - The triangle is isosceles.
4) If in the triangle the median is also the bisector - The triangle is isosceles.

Detailed explanation

Practice Identification of an Isosceles Triangle

Test your knowledge with 20 quizzes

Is the triangle in the drawing a right triangle?

Examples with solutions for Identification of an Isosceles Triangle

Step-by-step solutions included
Exercise #1

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

Step-by-Step Solution

An acute-angled triangle is defined as a triangle where all three interior angles are less than 90∘90^\circ.

In examining the visual depiction of the triangle provided in the problem, we need to see if it appears to satisfy this property. The assessment relies on observing the triangle's structure shown in the drawing and noting any geometric indications suggesting angle types.

Given the information from the drawing, if all angles seem to satisfy the condition of being less than 90∘90^\circ, then by definition, the triangle is an acute-angled triangle.

Conclusively, the answer to whether the triangle is acute-angled based on provided visual assessment and inherent assumptions in its illustration is: Yes.

Answer:

Yes

Video Solution
Exercise #2

In an isosceles triangle, the angle between ? and ? is the "base angle".

Step-by-Step Solution

An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."

Therefore, the correct choice is Side, base.

Answer:

Side, base.

Exercise #3

What kind of triangle is given in the drawing?

404040707070707070AAABBBCCC

Step-by-Step Solution

As all the angles of a triangle are less than 90° and the sum of the angles of a triangle equals 180°:

70+70+40=180 70+70+40=180

The triangle is isosceles.

Answer:

Isosceles triangle

Video Solution
Exercise #4

Given the values of the sides of a triangle, is it a triangle with different sides?

9.19.19.19.59.59.5AAABBBCCC9

Step-by-Step Solution

As is known, a scalene triangle is a triangle in which each side has a different length.

According to the given information, this is indeed a triangle where each side has a different length.

Answer:

Yes

Video Solution
Exercise #5

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

Frequently Asked Questions

Everything you need to know about Identification of an Isosceles Triangle

How do you identify an isosceles triangle in geometry?

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An isosceles triangle can be identified by checking if: 1) Two angles are equal, 2) The height bisects the vertex angle, 3) The height is also the median, or 4) The median is also the angle bisector. If any of these conditions are met, the triangle is isosceles.

What are the 4 ways to prove a triangle is isosceles?

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The four methods are: 1) Show two angles are equal, 2) Prove the height bisects the vertex angle, 3) Demonstrate the height equals the median, 4) Show the median equals the angle bisector. These conditions stem from the fundamental property that isosceles triangles have two equal sides.

Why do equal angles prove an isosceles triangle?

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Equal angles prove an isosceles triangle because of the angle-side relationship: sides opposite to equal angles are also equal. Therefore, if two angles in a triangle are equal, the sides opposite those angles must be equal, making it isosceles.

What happens when height, median, and angle bisector coincide in triangles?

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When any two of these three lines (height, median, angle bisector) coincide in a triangle, it proves the triangle is isosceles. In isosceles triangles, all three of these special lines from the vertex angle to the base are actually the same line.

Can you identify isosceles triangles without measuring sides?

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Yes, you can identify isosceles triangles without measuring sides by using angle measurements or geometric properties. Check for equal angles, or verify if special lines like height, median, or angle bisector coincide - these methods don't require side measurements.

What's the difference between isosceles triangle identification and other triangle types?

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Isosceles triangles have exactly two equal sides and two equal base angles, while equilateral triangles have all sides equal and scalene triangles have no equal sides. The identification methods for isosceles triangles specifically look for these 'two equal' properties.

How do you solve isosceles triangle problems step by step?

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Follow these steps: 1) Identify given information about angles, sides, or special lines, 2) Check which identification condition applies, 3) Apply the appropriate rule (equal angles, coinciding lines, etc.), 4) Use isosceles properties to find unknown values, 5) Verify your answer makes geometric sense.

What are common mistakes when identifying isosceles triangles?

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Common mistakes include: assuming a triangle is isosceles without proof, confusing isosceles with equilateral triangles, not recognizing when special lines coincide, and forgetting that equal angles indicate equal opposite sides. Always verify using one of the four identification methods.

More Identification of an Isosceles Triangle Questions

Click on any question to see the complete solution with step-by-step explanations

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Continue Your Math Journey

Master Identification of an Isosceles Triangle first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

AreaThe sides or edges of a triangleTriangle HeightThe Sum of the Interior Angles of a TriangleExterior angles of a triangleTypes of TrianglesObtuse TriangleEquilateral triangleScalene triangleAcute triangleIsosceles triangleThe Area of a TriangleArea of a right triangleArea of Isosceles TrianglesArea of a Scalene TriangleArea of Equilateral TrianglesAreas of Polygons for 7th GradeRight TriangleMedian in a triangleCenter of a Triangle - The Centroid - The Intersection Point of MediansHow do we calculate the area of complex shapes?How to calculate the area of a triangle using trigonometry?All terms in triangle calculationPerimeterTrianglePerimeter of a triangleHow do we calculate the perimeter of polygons?

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