From the image, *M* being the midpoint of *BD* looks really intuitive. Proving it, on the other hand, might take a bit of thinking through. If *M* is the midpoint of *BD*, we need to prove that *BM* and *MD* are congruent. The picture tells us that *AM* is congruent to *BM* and *MC* is congruent to *MD*. If *M* is the midpoint of *AC*, that means *AM* and *MC* are congruent and equal in measure. Using the transitive property, we can see that *AM*, *BM*, *MC*, and *MD* are all congruent. In that case, *BM* = *MD*, which means that *M* is the midpoint. Booyah! **Statements** | **Reasons** | 1. *M* is the midpoint of *AC* | Given | 2. *MD* ≅ *MC* | Given in figure | 3. *AM* ≅ *BM* | Given in figure | 4. *AM* ≅ *MC* | Definition of Midpoint (1) | 5. *AM* = *MC* | Definition of Congruence (4) | 6. *AM* = *BM* | Definition of Congruence (3) | 7. *MD* = *MC* | Definition of Congruence (2) | 8. *BM* = *MC* | Transitive Property (5 and 6) | 9. *BM* = *MD* | Transitive Property (7 and 8) | 10. *BM* ≅ *MD* | Definition of Congruence (9) | 11. *M* is the midpoint of *BD* | Definition of Midpoint (10) |
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