Hebrews 1:4

"So he became as much superior to the angels as the name he has inherited is superior to theirs." NIV translation

With the resurrection of Jesus Christ and His going to sit at the right hand of the Father, it was demonstrated that He, in fact, is the Son of God.

Hebrews 1:5

"For to which of the angels did God ever say, 'You are my Son; today I have become your Father'? Or again, 'I will be his Father, and he will be my Son'?" NIV translation

Paul quotes from Psalm 2:7 and it is talking about the fact that God declared that Jesus was His Son. He also quotes from 2 Samuel 7:14 where God promises King David that a king would come from his family line whom God would call His Son. That was fulfilled through the family tree of Jesus. 

Paul uses these quotes to demonstrate that Jesus is far superior to the angels just as the Father is superior to them. It seems that this fact is even more relevant today as many people have elevated angels up to the point where they actually worship them instead of God. We must remember that both the Father and the Son are far superior to the angels and the only ones that deserve our worship and praise.

Hebrews 1:6

"And again, when God brings his firstborn into the world, he says, 'Let all God's angels worship him.' NIV translation

We see the third example of where God said that Jesus was superior to the angels.

Hebrews 1:7 - 9

"In speaking of the angels he says, 'He makes his angels winds, his servants flames of fire.' But about the Son he says, 'Your throne, O God, will last for ever and ever, and righteousness will be the scepter of your kingdom. You have loved righteousness and hated wickedness; therefore God, your God, has set you above your companions by anointing you with the oil of joy.'" NIV translation

Paul continues to explain how Jesus is superior to the angels by quoting from the book of Psalms. As he is talking to Jewish believers, he speaks of the anointing "with the oil of joy". They would have understood anointing because of the anointing oil that was used in the temple.