Matthew 1:20κατ᾽
But after he had pondered these things, an angel of the Lord appeared to him in a dream and said, “Joseph, son of David, do not be afraid to embrace Mary as your wife, for the One conceived in her is from the Holy Spirit.
Luke 1:9κατὰ
he was chosen by lot, according to the custom of the priesthood, to enter the temple of the Lord and burn incense.
Acts 3:13κατὰ
The God of Abraham, Isaac, and Jacob, the God of our fathers, has glorified His servant Jesus. You handed Him over and rejected Him before Pilate, even though he had decided to release Him.
Romans 1:3κατὰ
regarding His Son, who was a descendant of David according to the flesh,
2 Corinthians 1:8καθ᾽
We do not want you to be unaware, brothers, of the hardships we encountered in the province of Asia. We were under a burden far beyond our ability to endure, so that we despaired even of life.
Hebrews 1:10κατ᾽
And: “In the beginning, O Lord, You laid the foundations of the earth, and the heavens are the work of Your hands.

Abbott-Smith

κατά (bef. a vowel κατ᾽, καθ᾽; on the frequently neglect of elision, see Tdf., Pr., 95; WH, App., 146a),
prep. with genitive, accusative,
down, downwards.

I. C. genitive (WM, §47, k; Bl., §42, 2).

1. C. genitive of thing(s), in local sense;

(a) down, down from: Mat.8:32, Mrk.5:13, Luk.8:33, 1Co.11:4;

(b) throughout (late usage; Bl, l.with): κ. ὅλης κ.τ.λ., Luk.4:14 23:5, Act.9:31 10:37;

(with) in a peculiar adjectival phrase: ἡ κ. βάθους, deep or extreme poverty, 2Co.8:2.

2. C. genitive of person(s), usually in hostile sense;

(a) against (in cl. only after verbs of speaking, witnessing, etc.): opposite to ὑπέρ, Mrk.9:40; μετά, Mat.12:30; after ἐπιθυμεῖν, Gal.5:17; λαλεῖν, Act.6:13; διδάσκειν, Act.21:28; ψεύδεσθαι, Jas.3:14; after verbs of accusing, etc., Mat.5:23, Luk.23:14, Rom.8:33, al.; verbs of fighting, prevailing, etc., Mat.10:35, Act.14:2, 1Co.4:6, al.;

(b) of swearing, by: όμνυμι κ. (BL, §34, 1), He 6:13,16, cf. Mat.26:63.

II. C. accusative (WM, §49d; BL, §42, 2).

1. Of motion or direction;

(a) through, throughout: Luk.8:39 9:6 10:4, Act.8:1, 36 al.;

(b) to, towards, over against

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Sources: Abbott-Smith’s Manual Greek Lexicon of the New Testament (1922, public domain) and glosses from STEPBible TBESG (CC BY 4.0); STEPBible TAGNT (Greek New Testament, CC BY 4.0); the Berean Standard Bible (public domain). Reformatted by Word Study, wording unchanged. All sources and licenses.