Not necessarily. I can be exasperated with the red tape in government, and not be angry. I know that governmental processes are filled with checks and security. I do get exasperated and frustrated, but not angry.
I get angry with myself when I fail to meet a goal. I am disappointed, but not exasperated with myself.
Some people get exasperated then they try to explain something to someone else that they think in simple. A few do get angry when they are exasperated, but most either give up or find someone else to handle the situation.
Teaching can be exasperating when the instructor knows facts and/or skills and the student fails to grasp what is being taught or thinks they have a better way that the instructor knows will endanger the safety of others.
Interpersonal relationships can be exasperating, but the underlying friendship should mend the relationship after any anger cools.