Concept Development• The Angle Sum Theorem for triangles states that the sum of the interior angles of a triangle is always 180° (∠ sum of △).
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°.
c. Write your proof below.
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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