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Home » Triangles » Similar Triangles » How to Prove that Triangles are Similar

How to Prove that Triangles are Similar

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In many of the problems involving similar triangles, you will be asked to prove that the triangles are similar.

The easiest way to do this is to show that all the angles are congruent or have an equal measure.

We can use one of the tools are our disposal to show angles are congruent:
1. If there are vertical angles they are congruent.
2. If there are corresponding angles between parallel lines, they are congruent.
3. If there are congruent triangles, all their angles are congruent.

The tool we choose will depend on the question - are there parallel lines involved, are there similar triangles, etc.

Remember that since the sum of all angles in a triangle is 180°, to show that all three pairs of angles are the same, it is enough to show that two pairs of angles have the same measure, and the third pair will also be the same, and each one of the angles in the third pair will measure 180° minus the sum of the other two. For short, this is often designated as "Angle-Angle", or AA.

Problem

Line segment DE is parallel to the base, BC, of triangle ΔABC. show that ΔABC∼ ΔADE.

Similar triangles in geometry

Strategy

Since we are given two parallel lines, this is the hint to use the fact that corresponding angles between parallel lines are congruent.

DE is parallel to BC, and the two legs of the triangle ΔABC form transversal lines intersecting the parallel lines, so the corresponding angles are congruent.

Proof

(1) ∠BAC≅∠DAE        //Common angle to both triangles

(2) DE || BC                //Given

(3) ∠ABC≅∠ADE        //Corresponding angles of two parallel lines intersected by a transversal

(4) ∠ACB≅∠AED        //Corresponding angles of two parallel lines intersected by a transversal

(5) ΔABC∼ ΔADE      //All three angles are congruent (AAA)

« Arc Length Formula
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About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
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    • Rhombus
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  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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