Our, One's Own, My
adjective
Here we have I followed by the particle が, giving us a somewhat old-fashioned and stylized way of say our, one's own, or my. One explanation for this is that in ancient Japanese, が was used as a possessive particle, much like の is in modern Japanese.
わが
You haven't learned this reading for 我 so here's a mnemonic to help you: Let us present you with our walrus (わ). Well it's my walrus really, but I'm happy to share.
我が家の今日のディナーはエビフライです。
Today's dinner at our house is deep-fried shrimp.
だれだって、結局我が子が一番可愛いんだよ。
In the end, everyone loves their own child the most.
課長、あんたの受け売りをそのまま、我が物顔で言ってたよ。
The section chief was saying exactly what you said, as if it were his own words.