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Home » Triangles » Sum of Angles in a Triangle

Sum of Angles in a Triangle

Last updated: Jan 4, 2020 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

One of the basic properties of triangles is that the sum of the measure of angles, in every triangle, is 180°. We will now prove this, using what we know about parallel lines and the angles formed by a transversal line.

Drawing showing the sum of angles in a triangle.

Take the triangle ABC, formed by the line segments AB, AC and BC. Let’s draw a parallel line to AB, that goes through point C (the dashed line, above).

How do we know we can do this? Remember the axiom we stated in the previous section: “for every straight line and every point not on that line, there is one straight line that passes through that point, and never intersects the first line.”

Here we have a straight line (AB), and a point (C ) not on that line, so we know from the axiom that we can draw a line through C which will be parallel to AB.

So we now have two parallel lines (AB and the dashed line through C). And those two parallel lines are intersected by 2 transversal lines – AC and BC, which form the triangle.

Strategy

The key to this proof is that we want to show that the sum of the angles in a triangle is 180°. And we already know that a straight line's angle measures 180°. So we look for straight lines that include the angles inside the triangle.

Proof

(1)   m∠1 + m∠2 + m∠3= 180°                      // straight line measures 180° 

(2)   m∠1 = m∠4                         //Corresponding angles of parallel lines, with AC as transversal

(3)   m∠2 = m∠5                        //Alternating interior angles of parallel lines, with BC as transversal

(4)   m∠4 + m∠5+ m∠3= 180°   //using (1),(2) & (3) and performing algebraic substitution. We replace  m∠2 with the equivalent m∠5 and m∠1with the equivalent m∠4

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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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About

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
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  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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