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 5. 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 September 14, 2018 5.8

5.8 Triangles and Coordinate Proof Geometry September 14, 2018 5.8 coordinate proof 5.8

Goals Place geometric figures in the coordinate plane. Write a coordinate proof. September 14, 2018 5.8

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. September 14, 2018 5.8

September 14, 2018 5.8

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. September 14, 2018

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. September 14, 2018 5.8

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. September 14, 2018 5.8

Since the triangles are congruent, Then B is the midpoint of segment AC by definition. Using the midpoint formula yields September 14, 2018

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. September 14, 2018 5.8

Find the coordinates of C. B(b, c) C (a+b, c) c c b b A(0, 0) D(a, 0) September 14, 2018

Prove ΔOPQ ΔQRO Q(h, k+n) P(0, n) R(h, k) O(0, 0) September 14, 2018

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 September 14, 2018 5.8