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# Proof of the triangle proportionality theorem.

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Provide reasons for the proof of the triangle proportionality theorem.

Given: Line segment KL Prove: KM/JK = LN/JL

Statement Reason

1. Line segment KL box line segment MN

1. Given

2.

2.

3. box JKL two boxes JMN

3.

4. JM/JK = JN/JL

4.

5. JM = JK + KM and JN = JL + LN

5.

6. JK + KM/ JK = JL + LN/JL

6.

7. JK/JK + KM/JK = JL/JL + LN/JL

7. Rewrite as a sum.

8. 1 + KM/JK = 1 + LN/JL

8. Simplify.

9. KM/JK = LN/JL

9.

Nov 26, 2018

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1. Line segment KL is parallel to segment MN

1. Given

2. Angle JKL = Angle JMN ; Angle JLK = Angle JLK = Angle JNM

2. Parallel lines cut by a transversal make corresponding angles equal

3. Triangle JKL is similar to Triangle JMN

3. AA congruency

4. JM / JK =  JN /JL

4. Correspnding sides of similar triangles are proportional to each other

5.  JM = JK + KM and JN = JL + LN

6. JK + KM/ JK = JL + LN/JL

6. Substitution of (5) into (4)

7. JK/JK + KM/JK = JL/JL + LN/JL

7. Rewrite as a sum.

8. 1 + KM/JK = 1 + LN/JL

8. Simplify.

9. KM/JK = LN/JL

9. Equality Property of Subtraction

Nov 26, 2018