Question

Given MPRK is a parallelogram, how would you prove, in two-column format, that the measure of angle MTK = measure of angle PMT + measure of angle RKT?

Answer 1

I assume, point `T` is any point inside of parallelogram.
See below the proof.

Explanation:

Draw a line through point `T` parallel to sides `MP` and `KR` of a parallelogram `MPRK`.
Let's assume that it intersects side `KM` at point `X`.

Statement:
`m/_PMT = m/_MTX`
Reason:
Angles `/_PMT` and `/_MTX` are alternate interior angles with parallel lines `MP` and `TX` and transversal `MT`.

Statement:
`m/_KTX = m/_RKT`
Reason:
Angles `/_KTX` and `/_RKT` are alternate interior angles with parallel lines `KR` and `TX` and transversal `KT`. Hence, they have the same measure.

Statement:
`m/_PMT + m/_RKT = m/_MTX + m/_KTX`
Reason:
Above equalities of corresponding measures.

Statement:
`m/_PMT + m/_RKT = m/_MTK`
Reason:
`m/_MTX + m/_KTX = m/_MTK`

End of proof.