Question

solve the problems. write the complete proof in your paper homework and for online (only) complete the probing statement (if any) that is a part of your proof or related to it

Answer 1

m∠A ≅ m∠C by reason Base angles theorem

△ AMO≅△ CPO by reason ASA / SAA

Answer 2

`m \angle A = m \angle C` by reason
`\overline{AB} \cong \overline{BC}` and
`m \angle B = m \angle M = m \angle P`.

`\triangle AMO \cong \triangle CPO` SAS Theorem

Explanation:

We proceed to demonstrate the statement by Geometric means:

1)
`\overline{AB} \cong \overline{BC}`,
`\overline {AM} \cong \overline {PC}`,
`m\angle AMO = m\angle CPO` Given.

2)
`(AM)/(AB) = (PC)/(BC)` Proportionality.

3)
`(AM)/(AM + MB) = (PC)/(BP + PC)` Definition of line segments.

4)
`(1)/(1+(MB)/(AM) ) = (1)/((BP)/(PC)+1)` Algebra.

5)
`(BP)/(PC) + 1 = 1 +(MB)/(AM)` Algebra.

6)
`(BP)/(PC) = (MB)/(AM)` Algebra.

7)
`BP = BM` By 1)

8)
`m \angle B = m \angle M = m \angle P` By 1), 7)

9)
`\triangle AMO \sim \triangle ABC`,
`\triangle CPO \sim \triangle ABC` By 1), 7), 8). Defintion of simmilarity.

10)
`(AM)/(MO) = (AB)/(BC)`,
`(PO)/(PC) = (AB)/(BC)` Definition of proportionality.

11)
`(AM)/(MO) = (PO)/(PC)` Algebra.

12)
`AM^(2) = PO\cdot MO` Algebra.

13)
`PO = MO` By 12) and Algebra.

14)
`\overline{PO} \cong \overline{MO}` By 13).

15)
`\triangle AMO \cong \triangle CPO` SAS Theorem/Result.