And Wednesday night they had an extremely heartening outcome. A 7-1 lead after six innings evaporated, but a two-run, ninth-inning home run by designated hitter Hank Blalock, his second of the game, gave the Rangers a victory. It was the kind of game in which defeat would definitely sting, but victory could be seriously savored.
"That team over there, they don't quit," Rangers manager Ron Washington said, referring to the Angels. "They fight. That's why they've been where they've been the past four or five years. And we're getting there. We don't quit, either."
This was a feisty, defiant manager and that was fine. The Rangers' offense slumped in June and after they were one-hit on Sunday night and lost to the Angels on Monday night, the chorus of local criticism was growing louder and more vociferous.
"You want heart, fire, desire; that was it right there," Washington said to reporters, about his team's performance. "Y'all had us in the grave, buried, with dirt all over us.
"Anaheim came to town and everything we read was that we were gone. We were through. That's what we read. We were supposed to be in Texas. I thought we were in Anaheim. A lot of people jumped off our ship. But you know what? We're going to turn into a submarine and resurface."
A small smile from Washington accompanied that last remark. He had some smiles coming after this series. And he is convinced that his team not only has the talent, but the character to succeed.
The Rangers' improvement this season does not come as a shock to those who have paid close attention.
"From probably the turn of the middle of the season last year until now, I'd be interested in seeing what their record is," Angels manager Mike Scoscia said. "They're a much-improved team. They play great baseball. It's not like it started right now and we're noticing; we noticed it the second half of last season, they played great baseball."
The Rangers started the 2008 season with a miserable 9-18 stretch. Then they had a long run of good baseball, going 51-36 over a period that lasted into August. They were then hit with a wave of pitching injuries and they finished below .500. But there had been some strides taken.
The Rangers built from that point forward this season, holding first place for seven weeks. They had an atypically bad June offensively and they fell out of first place last weekend. But their overall level of pitching and defense has improved since last season. And they showed a nice slice of resilience in coming back to win the last two games of this series.
"Once again, the character inside that clubhouse showed up," Washington said.
The Rangers could also be encouraged by the fact that they won the last two games in distinctly different ways. The Rangers stole five bases Tuesday night, the second time they had done that this season. Prior to that the last time they had done it was in 2001. Their dynamic young shortstop, Elvis Andrus, had four steals, becoming just the fourth player in team history with four or more steals in a game.
This was a clear indication of the kind of versatility the Rangers hope to have on offense. "We do have that capability," Washington said. "And up until we hit a little spell when June rolled in, we were doing that type of stuff. You go through six months of a season and there will be times when things don't go right. And in June, things didn't go right.
"But our guys never dropped their heads. We try to play the game according to the way it is presented to us out there. And I understand we struggled trying to do that, but maybe we're on our way back."
The Rangers left this series just a half-game back of the Angels in the AL West. That's just short of where they want to be, but if this series was any kind of reliable indication, they belong in this neighborhood.