1. Warm-up
Can you conclude that the triangles are congruent? Explain.
1. 2.
3. Can you conclude that the triangles are congruent? Explain.
a.  AZK and DRS   
b.  SDR and JTN   
c.  ZKA and NJT
What additional information do you need to prove congruence by the HL Theorem?
4. LMX LOX AMD CNB
Yes; use the congruent hypotenuses and leg BC to prove ABC DCB
SAS
c.  Since AZK  DRS and SDR  JTN, by the Transitive Property of , ZKA  NJT.
a. Two pairs of sides are congruent. The included angles are congruent. Thus, the two triangles are congruent by SAS.
b. aas
LM LO
AM CN or
MD NB
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2. 5.8 Triangles and Coordinate Proof
Geometry
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5.8 coordinate proof
5.8

3. Goals Place geometric figures in the coordinate plane.
Write a coordinate proof.
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4. Placing Figures in the Coordinate Plane
You have used coordinate geometry to find the midpoint of a line segment and to find the distance between two points. Coordinate geometry can also be used to prove conjectures.
A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step of a coordinate proof is to position the given figure in the plane. You can use any position, but some strategies can make the steps of the proof simpler.
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5. September 14, 2018
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6. Place a 2 unit by 6 unit rectangle in the coordinate plane.
Example Problem
Place a 2 unit by 6 unit rectangle in the coordinate plane.
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7. Once the figure is placed in the coordinate plane, you can use slope, the coordinates of the vertices, the Distance Formula, or the Midpoint Formula to prove statements about the figure.
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8. If a coordinate proof requires calculations with fractions, choose coordinates that make the calculations simpler.
For example, use multiples of 2 when you are to find coordinates of a midpoint. Once you have assigned the coordinates of the vertices, the procedure for the proof is the same, except that your calculations will involve variables.
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9. Since the triangles are congruent,
Then B is the midpoint of segment AC by definition. Using the midpoint formula yields
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10. To do this, assign variables as the coordinates of the vertices.
A coordinate proof can also be used to prove that a certain relationship is always true.
To do this, assign variables as the coordinates of the vertices.
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11. Find the coordinates of C.
B(b, c) C
(a+b, c) c c b b
A(0, 0)
D(a, 0)
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12. Prove ΔOPQ
ΔQRO
Q(h, k+n)
P(0, n)
R(h, k)
O(0, 0)
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13. SAS OP is same length as RQ OP = n – 0 = n RQ = k + n – k = n
OP and QR are both vertical which implies they have same slope, parallel
Congruent: Why?
SAS
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