Unit 1, Lesson 16
Parallel Lines and the Angles in a Triangle
Grade 8
16.1
Warm-up
Here is triangle . Line is parallel to line .
Point B lies on line D E. Triangle B A C is drawn. Angle A B C is labeled b degrees. Angle B A C is labeled a degrees. Angle A C B is labelled c degrees.
16.2
Activity
Here is triangle .
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Rotate triangle around the midpoint of side . Label the new vertex .
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Rotate triangle around the midpoint of side . Label the new vertex .
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Look at angles , , and . Without measuring, write what you think is the sum of the measures of these angles. Explain or show your reasoning.
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Is the measure of angle equal to the measure of any angle in triangle ? If so, which one? Explain your reasoning.
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Is the measure of angle equal to the measure of any angle in triangle ? If so, which one? Explain your reasoning.
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What is the sum of the measures of angles , , and ? Explain your reasoning.
16.3
Activity
Here is triangle . Line is parallel to line .
Point B lies on line D E. Triangle B A C is drawn. Angle A B C is labeled b degrees. Angle B A C is labeled a degrees. Angle A C B is labelled c degrees.
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What is the sum of the measures of angle , angle , and angle ?
- Use your answer to explain why .
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Explain why your argument will work for any triangle: that is, explain why the sum of the angle measures in any triangle is .
16.4
Activity
This diagram shows a square that has been made by images of triangle under rigid transformations.
Quadrilateral B D F H. Angles B, D, F, and H are marked as right angles. Point C lies on B D. Point E lies on D F. Point G lies on F H. Point A lies on B H. Segments A C, C E, E G, and G A are drawn.
Given that angle measures , find as many other angle measures as you can.
Student Lesson Summary
Using parallel lines and rotations, we can understand why the angles in a triangle always add to . Here is triangle . Line is parallel to and contains .
Point B lies on segment D E. Triangle B A C is drawn. Angle D B A is labeled x degrees. Angle A B C is labeled y degrees. Angle E B C is labeled z degrees. Angle B A C is labeled x degrees. Angle B C A is labeled z degrees.
A rotation of triangle around the midpoint of interchanges angles and so they have the same measure (in the picture these angles are marked as ).
A rotation of triangle around the midpoint of interchanges angles and so they have the same measure (in the picture, these angles are marked as ).
Also, is a straight line because rotations take lines to parallel lines.
So the three angles with vertex make a line and they add up to (). But are the measures of the three angles in triangle so the sum of the angles in a triangle is always !
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