1. Parallel Line Segments and the Midpoint Theorem Slideshow 35, Mathematics Mr. Richard Sasaki, Room 307

2. ObjectivesObjectives Understand the influence of parallel line segments on triangle structures Understand the influence of parallel line segments on triangle structures Be able to calculate missing line segments for structures with parallel line segments Be able to calculate missing line segments for structures with parallel line segments Prove triangles are similar with parallel line segments Prove triangles are similar with parallel line segments Use the midpoint theorem Use the midpoint theorem

3. Parallel Line Segments Have a look at the diagram below. They’re parallel. As they are parallel, the triangles must be. similar Why would a pair of parallel line segments imply similarity? We’d know the angles either side are equal, so the triangles are similar by test AA.

4. Parallel Line Segments If a pair of parallel line segments exist and by test AA, similarity exists, we can assume all edges are in the same proportions (SSS). Let’s make an equation with ratios for the given line segments. Also… The opposite also applies too. ① ② Let’s use ②.

5. Answers – Easy (Q1 – 2)

6. Answers – Easy (Question 3)

7. Answer - Hard

8. The Midpoint Theorem What can we say about a line segment built from two midpoints of line segments on a triangle? Mid-segment

9. The Midpoint Theorem A segment that connects the midpoints of any two edges is parallel to and half the length of the third edge. Also, we can use the midpoint theorem with trapeziums and parallelograms by splitting them into triangles.

10. Answers – Part 1 The area of the triangle is a third of the area of the trapezium.

11. Answers – Part 2, Question 1

12. Answers – Part 2 Two kites and two parallelograms Both pairs of shapes are congruent Scale Factor: ½ and the area of the original is a quarter of the size The two kites are similar to the main one