# Angle Sum of a Triangle

Examples, solutions, worksheets, videos, and lessons to help Grade 8 students know the Angle Sum Theorem for triangles; the sum of the interior angles of a triangle is always 180°.

Students present informal arguments to draw conclusions about the angle sum of a triangle.

### New York State Common Core Math Grade 8, Module 2, Lesson 13

Concept Development
• The Angle Sum Theorem for triangles states that the sum of the interior angles of a triangle is always 180° (∠ sum of △).
• It does not matter what kind of triangle (i.e., acute, obtuse, right) when you add the measure of the three angles, you always get a sum of 180°.

We want to prove that the angle sum of any triangle is . To do so, we will use some facts that we already know about geometry:
• A straight angle is 180° in measure.
• Corresponding angles of parallel lines are equal in measure.
• Alternate interior angles of parallel lines are equal in measure.

Exploratory Challenge 1
Let triangle ABC be given. On the ray from B to C, take a point D so that C is between B and D. Through point C, draw a line parallel to AB as shown. Extend the parallel lines AB and CD. Line AC is the transversal that intersects the parallel lines.

a. Name the three interior angles of triangle ABC.
b. Name the straight angle.
c. What kinds of angles are ∠ABC and ∠ECD? What does that mean about their measures?
d. What kinds of angles are ∠BAC and ∠ECA? What does that mean about their measures?
e. We know that ∠BCD = ∠BCA + ∠ECA +∠ECD = 180°. Use substitution to show that the three interior angles of the triangle have a sum of 180°.

Exploratory Challenge 2
The figure below shows parallel lines L1 and L2. Let m and n be transversals that intersect L1 at points B and C respectively, and L2 at point F, as shown. Let A be a point on L1 to the left of B, D be a point on L1 to the right of C, G be a point on L2 to the left of F and E be a point on L2 to the right of F.
a. Name the triangle in the figure.
b. Name a straight angle that will be useful in proving that the sum of the interior angles of the triangle is 180°.

Lesson Summary:

• All triangles have a sum of interior angles equal to 180°
• The proof that a triangle has a sum of interior angles equal to 180° is dependent upon the knowledge of straight angles and angles relationships of parallel lines cut by a transversal.

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