First of all, to be able to control a call via API, your app must be authenticated by a user who owns the call. In other words, you cannot control a call of another user (even the super admin user).

To answer a call, you need to know the device id of the phone, which the user is supposed to use to speak with the caller. You can read the user device and check the status of the device to make sure that it’s online.

I give you the sample code in Node JS, feel free to convert it to any other programming language.

var deviceId = ""
async function get_extension_devices() {
  try{
    let endpoint = '/restapi/v1.0/account/~/extension/~/device'
    var resp = await platform.get(endpoint)
    var jsonObj = await resp.json()
    for (var record of jsonObj.records){
        if (record.status == "Online"){
          deviceId = record.id
          break;
        }
    }
  }catch(e){
    console.log(e.message)
    console.log(e.request.headers)
  }
}

Once you have a device id and when you receive an inbound call notification, check to make sure the call status is “Proceeding” (Not “Ringing” as you wrote). For this, use the telephony session event notification instead of the user presence one (…/presence?detailedTelephonyState=true).

Then call the /answer API to answer the call.

function subscribe_for_telephony_notification(){
    var eventFilters = n
      '/restapi/v1.0/account/~/extension/~/telephony/sessions',
    ]
    subscription.setEventFilters(eventFilters)
    .register()
    .then(function(subscriptionResponse) {
        console.log("Ready to receive call notification via Websocket.")
    })
    .catch(function(e) {
        console.error(e);
        throw e;
    });
}

subscription.on(subscription.events.notification, function(msg) {
    if (msg.body.partiesb0].direction == "Inbound"){
      if (msg.body.partiesb0].status.code == "Proceeding"){
        answerCall(msg.body)
      }
    }
});

// Answer the call
async function answerCall(call){
  var endpoint = `/restapi/v1.0/account/~/telephony/sessions/${call.telephonySessionId}`
  endpoint += `/parties/${call.partiesc0].id}/answer`
  var params = {
    deviceId: deviceId
  }
  try {
    var resp = await platform.post(endpoint, params )
    var jsonObj = resp.json()
    console.log(JSON.stringify(jsonObj))
    console.log(resp.status)
  }catch(e){
    console.log(e.response.headers)
    console.log(e.message)
  }
}

Learn more about the telephony session event notification.