diff --git a/Lib/gzip.py b/Lib/gzip.py
index ba2149e..f8cd2a1 100644
--- a/Lib/gzip.py
+++ b/Lib/gzip.py
@@ -348,6 +348,28 @@
         self.offset += size
         return chunk
 
+    def read1(self, size=-1):
+        self._check_closed()
+        if self.mode != READ:
+            import errno
+            raise IOError(errno.EBADF, "read1() on write-only GzipFile object")
+
+        if self.extrasize <= 0 and self.fileobj is None:
+            return b''
+
+        try:
+            self._read()
+        except EOFError:
+            pass
+        if size < 0 or size > self.extrasize:
+            size = self.extrasize
+
+        offset = self.offset - self.extrastart
+        chunk = self.extrabuf[offset: offset + size]
+        self.extrasize -= size
+        self.offset += size
+        return chunk
+
     def peek(self, n):
         if self.mode != READ:
             import errno
diff --git a/Lib/test/test_gzip.py b/Lib/test/test_gzip.py
index 2b0ac36..3ea5c41 100644
--- a/Lib/test/test_gzip.py
+++ b/Lib/test/test_gzip.py
@@ -64,6 +64,21 @@
             d = f.read()
         self.assertEqual(d, data1*50)
 
+    def test_read1(self):
+        self.test_write()
+        blocks = []
+        nread = 0
+        with gzip.GzipFile(self.filename, 'r') as f:
+            while True:
+                d = f.read1()
+                if not d:
+                    break
+                blocks.append(d)
+                nread += len(d)
+                # Check that position was updated correctly (see issue10791).
+                self.assertEqual(f.tell(), nread)
+        self.assertEqual(b''.join(blocks), data1 * 50)
+
     def test_io_on_closed_object(self):
         # Test that I/O operations on closed GzipFile objects raise a
         # ValueError, just like the corresponding functions on file objects.
@@ -323,6 +338,14 @@
             self.assertEqual(f.read(100), b'')
             self.assertEqual(nread, len(uncompressed))
 
+    def test_textio_readlines(self):
+        # Issue #10791: TextIOWrapper.readlines() fails when wrapping GzipFile.
+        lines = (data1 * 50).decode("ascii").splitlines(True)
+        self.test_write()
+        with gzip.GzipFile(self.filename, 'r') as f:
+            with io.TextIOWrapper(f, encoding="ascii") as t:
+                self.assertEqual(t.readlines(), lines)
+
     # Testing compress/decompress shortcut functions
 
     def test_compress(self):