Praising God With the Church in Heaven
15. In the liturgy of the hours the Church exercises the priestly office of its Head and offers to God "without ceasing"  a sacrifice of praise, that is, a tribute of lips acknowledging his name.  This prayer is "the voice of a bride addressing her bridegroom; it is the very prayer that Christ himself, together with his Body, addresses to the Father."  "All who render this service are not only fulfilling a duty of the Church, but also are sharing in the greatest honor of Christ's Bride for by offering these praises to God they are standing before God's throne in the name of the Church, their Mother." 
16. When the Church offers praise to God in the liturgy of the hours, it unites itself with that hymn of praise sung throughout all ages in the halls of heaven;  it also receives a foretaste of the song of praise in heaven, described by John in the Book of Revelation, the song sung continually before the throne of God and of the Lamb. Our close union with the Church in heaven is given effective voice "when we all, from every tribe and tongue and people and nation redeemed by Christ's blood (see Rv 5:9) and gathered together into the one Church, glorify the triune God with one hymn of praise." 
The prophets came almost to a vision of this liturgy of heaven as the victory of a day without night, of a light without darkness: "The sun will no more be your light by day, and the brightness of the moon will not shine upon you, but the Lord will be your everlasting light" (Is 60:19; see Rv 21:23 and 25). "There will be a single day, known to the Lord, not day and night, and at evening there will be light" (Zech 14:7). Already "the end of the ages has come upon us (see I Cor 10:11) and the renewal of the world has been irrevocably established and in a true sense is being anticipated in this world."  By faith we too are taught the meaning of our temporal life, so that we look forward with all creation to the revealing of God's children.  In the liturgy of the hours we proclaim this faith, we express and nourish this hope, we share in some degree the joy of everlasting praise and of that day that knows no setting.
Petition and Intercession
17. But besides the praise of God, the Church in the liturgy of the hours expresses the prayers and desires of all the faithful; indeed, it prays to Christ, and through him to the Father, for the salvation of the whole world.  The Church's voice is not just its own; it is also Christ's voice, since its prayers are offered in Christ's name, that is, "through our Lord Jesus Christ," and so the Church continues to offer the prayer and petition that Christ poured out in the days of his earthly life  and that have therefore a unique effectiveness. The ecclesial community thus exercises a truly maternal function in bringing souls to Christ, not only by charity, good example, and works of penance but also by prayer. 
(Vatican II documents:
SC= Sanctrosactum concilium;
LG= Lumen gentium;
56. See SC art. 83-84.
57. See SC art. 88.
58. See SC art. 94.
59. See PO no. 5.
60. CD no. 30.
61. SC art. 5.
62. See SC art. 83 and 98.
63. SC art. 7.
64. See SC art. 10.
65. SC art. 33.
66. See SC art. 24.
67. See SC art. 33.
68. 1 Thes 5:17.
69. See Heb 13:15.
70. SC art. 84.
71. SC art. 85.
72. See SC art. 83
73. LG no. 50; SC art. 8 and 104.
74. LG no. 48.
75. See Rom 8:19.
76. See SC art. 83.
77. See Heb 5:7.
78. See PO no. 5
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What the Church says about it (from the General Instruction of Liturgy Hours #10-17):
Christ taught us: "You must pray at all times and not lose heart" (Lk 18:1). The Church has been faithful in obeying this instruction; it never ceases to offer prayer and makes this exhortation its own: "Through him (Jesus) let us offer to God an unceasing sacrifice of praise" (Heb 15:15). The Church fulfills this precept not only by celebrating the Eucharist but in other ways also, especially through the liturgy of the hours. By ancient Christian tradition what distinguishes the liturgy of the hours from other liturgical services is that it consecrates to God the whole cycle of the day and the night. 
11. The purpose of the liturgy of the hours is to sanctify the day and the whole range of human activity. Therefore its structure has been revised in such a way as to make each hour once more correspond as nearly as possible to natural time and to take account of the circumstances of life today. 
Hence, "that the day may be truly sanctified and the hours themselves recited with spiritual advantage, it is best that each of them be prayed at a time most closely corresponding to the true time of each canonical hour." 
Liturgy of the Hours and the Eucharist
12. To the different hours of the day the liturgy of the hours extends  the praise and thanksgiving, the memorial of the mysteries of salvation, the petitions and the foretaste of heavenly glory that are present in the eucharistic mystery, "the center and high point in the whole life of the Christian community." 
The liturgy of the hours is in turn an excellent preparation for the celebration of the eucharist itself, for it inspires and deepens in a fitting way the dispositions necessary for the fruitful celebration of the eucharist: faith, hope, love, devotion, and the spirit of self-denial.
Priesthood of Christ in the Liturgy of the Hours
13. In the Holy Spirit Christ carries out through the Church "the task of redeeming humanity and giving perfect glory to God,"  not only when the eucharist is celebrated and the sacraments administered but also in other ways and especially when the liturgy of the hours is celebrated.  There Christ himself is present - in the gathered community, in the proclamation of God's word, "in the prayer and song of the Church." 
Sanctification of God's People
14. Our sanctification is accomplished  and worship is offered to God in the liturgy of the hours in such a way that an exchange or dialogue is set up between God and us, in which "God is speaking to his people ... and his people are responding to him by both song and prayer." 
Those taking part in the liturgy of the hours have access to holiness of the richest kind through the lifegiving word of God, which in this liturgy receives great emphasis. Thus its readings are drawn from sacred Scripture, God's words in the psalms are sung in his presence, and the intercessions, prayers, and hymns are inspired by Scripture and steeped in its spirit. 
Hence, not only when those things are read "that are written for our instruction" (Rom 15:4), but also when the Church prays or sings, faith is deepened for those who take part and their minds are lifted up to God, in order to offer him their worship as intelligent beings and to receive his grace more plentifully. 
Continued next week, references will be in Part II.