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McGraw Hill Integrated II, 2012
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McGraw Hill Integrated II, 2012 View details
5. Parts of Similar Triangles
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Exercise 26 Page 588

Apply the Triangle Midsegment Theorem twice and use the Parallelogram Opposite Angles Theorem.

Statements
Reasons
1.
∠ H a right angle, L, K, and M are midpoints
1.
Given
2.
KL and KM are midsegments of △ GHJ
2.
Definition of midsegment
3.
KL∥ JH and KM∥ GH
3.
Triangle Midsegment Theorem
4.
KLHM is a parallelogram
4.
Definition of parallelogram
5.
∠ H ≅ ∠ LKM
5.
Parallelogram Opposite Angles Theorem
6.
∠ LKM is a right angle
6.
Definition of congruent angles
Practice makes perfect

Let's consider the right triangle △ GHJ where K, L, and M are the midpoints of each sides.

Notice that both KL and KM are midsegments of △ GHJ. We can apply the Triangle Midsegment Theorem.
Triangle Midsegment Theorem
△ GHJ and KL △ GHJ and KM
KL ∥ JH KM ∥ GH

From the above we get that KLHM is a parallelogram.

Then, by the Parallelogram Opposite Angles Theorem we conclude that ∠ H ≅ ∠ LKM. Since ∠ H is a right angle we get that ∠ LKM is a right angle as well, and this is what we wanted to prove.

Two-Column Proof

Given: & ∠ H a right angle & L, K, and M are midpoints Prove: & ∠ LKM is a right angle We will summarize the proof we did above in the following two-column table.

Statements
Reasons
1.
∠ H a right angle, L, K, and M are midpoints
1.
Given
2.
KL and KM are midsegments of △ GHJ
2.
Definition of midsegment
3.
KL∥ JH and KM∥ GH
3.
Triangle Midsegment Theorem
4.
KLHM is a parallelogram
4.
Definition of parallelogram
5.
∠ H ≅ ∠ LKM
5.
Parallelogram Opposite Angles Theorem
6.
∠ LKM is a right angle
6.
Definition of congruent angles
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arrow_right Exercises p.586-590
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Exercises 2 (Page 586)
Exercises 3 (Page 586)
Exercises 4 (Page 587)
Exercises 5 (Page 587)
Exercises 6 (Page 587)
Exercises 7 (Page 587)
Exercises 8 (Page 587)
Exercises 9 (Page 587)
Exercises 10 (Page 587)
Exercises 11 (Page 587)
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Exercises 15 (Page 588)
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Exercises 17 (Page 588)
Exercises 18 (Page 588) engineering
Exercises 19 (Page 588) engineering
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Exercises 25 (Page 588) engineering
Exercises 26 (Page 588)
Exercises 27 (Page 589)
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Exercises 35 (Page 589) engineering
Exercises 36 (Page 590)
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Exercises 51 (Page 590)