1. angle A B C side angle A0A0 C0C0 B0B0 side angle Angle-Side-Angle Congruence by basic rigid motions A geometric realization of a proof in H. Wu’s “Teaching Geometry According to the Common Core Standards”

2. A B C0C0 B0B0 A0A0 C0C0 A0A0 A composition of basic rigid motions maps  ABC onto A0B0C0 A0B0C0 and proves ASA: a translation,a rotation, and a reflection. C But let’s go through it carefully, step by step….

3. If two triangles have two pairs of equal angles and the common side of the angles in one triangle is equal to the corresponding side in the other triangle, then the triangles are congruent. ASA criterion for congruence:

4. Given two triangles,  ABC and  A 0 B 0 C 0. Assume two pairs of equal corresponding angles We want to prove the triangles are congruent. A0A0 C0C0 B0B0 B C A with the sides between them equal.

5. angle In other words, A B C A0A0 C0C0 B0B0  B =  B 0,with  A =  A 0, and |AB| = |A 0 B 0 |, we must find a composition of basic rigid motions that maps  ABC exactly onto A0B0C0.A0B0C0. side angle given  ABC and A0B0C0,A0B0C0,

6. A B C C0C0 B0B0 A0A0

7. Step 2: Bring sides AB and A 0 B 0 together. A B C C0C0 B0B0 A0A0

8. C0C0 Step 3: Bring vertices C and C 0 together. A B C A0A0 B0B0 We want to get the third vertex C to coincide with C0 C0 so we reflect the image of  ABC across A0B0.A0B0.

9. Can we be sure this reflection takes the image of C exactly to C 0 and not to some other place? A B C A0A0 C0C0 B0B0 ?

10. Yes, we can. Basic rigid motions preserve degrees of angles. Therefore the reflected angle  CAB =  C0A0B0C0A0B0 and the reflected angle  CBA =  C0B0A0.C0B0A0. A B C A0A0 C0C0 B0B0

11. Therefore the reflected ray BC coincides with ray B 0 C 0 and the reflected ray AC coincides with ray A 0 C 0. A B C The intersection of the reflected rays BC and AC (i.e., the reflected C) coincides with the intersection of ray B 0 C 0 and A0C0 A0C0 (i.e., C 0 ). A0A0 C0C0 B0B0

12. A B C0C0 B0B0 A0A0 C0C0 A0A0 So, a composition of basic rigid motions— maps  ABC onto A0B0C0 A0B0C0 and proves the ASA criterion for congruence. a translation,a rotation,and a reflection— C