For clarity: The question is well written as follows;
If m∠10 = 77°, m∠7 = 47° and m∠16 = 139°, find the measure of each missing angle.
The following should be noted:
i. Corresponding angles: Two angles are said to be corresponding if they have the same position with respect to some given parallel lines and a transversal to the parallel lines. For example, 1 and 10 are corresponding angles and are also equal.
ii. Alternate angles: These are angles that are on opposite sides of a transversal. For example, 8 and 10, 5 and 15 are all pairs of alternate angles and are also equal.
iii. Vertical or Opposite angles: These are two pairs of opposite angles that are formed when two lines intersect. For example, 1 and 8, 2 and 9, 15 and 17, 16 and 18 are opposite angles and are also equal.
Therefore, we can solve for the missing angles as follows;
(a) m∠1
1 and 10 are corresponding angles. Therefore, since m∠10 = 77°, m∠1 = 77°
(b) m∠2
1 and 2 are angles on a straight line, therefore the sum of the angles should be 180°. i.e
m∠1 + m∠2 = 180°
Solve for m∠2
m∠2 = 180° - m∠1
Substitute for m∠1 = 77°
m∠2 = 180° - 77°
m∠2 = 103°
(c) m∠3
m∠15 + m∠16 = 180° --------------[Sum of angles on a straight line is 180°]
m∠16 = 139°
m∠15 + 139° = 180°
m∠15 = 180° - 139°
m∠15 = 41°
Since;
m∠3 = m∠15 -----------------------[corresponding angles are equal]
m∠3 = 41°
(d) m∠4
m∠4 = m∠16 -----------------------[corresponding angles are equal]
m∠4 = 139°
(e) m∠5
m∠5 = m∠3 -----------------------[opposite angles are equal]
m∠5 = 41°
(f) m∠6
m∠6 + m∠7 = m∠4 ----------------------[opposite angles are equal]
m∠6 + 47° = 139°
m∠6 = 139° - 47°
m∠6 = 92°
(g) m∠8
m∠8 = m∠1 -------------------------[opposite angles are equal]
m∠8 = 77°
(h) m∠9
m∠9 = m∠2 ------------------------[opposite angles are equal]
m∠9 = 103°
(i) m∠11
m∠11 + m∠8 + m∠7 = 180° ----------------[sum of angles in a triangle is 180°]
m∠11 + 77° + 47° = 180°
m∠11 + 124° = 180°
m∠11 = 180° - 124°
m∠11 = 56°
(j) m∠12
m∠12 = m∠7 -------------------------[alternate angles are equal]
m∠12 = 47°
(k) m∠13
m∠13 = m∠10 -------------------------[opposite angles are equal]
m∠13 = 77°
(l) m∠14
m∠14 = m∠9 -------------------------[corresponding angles are equal]
m∠14 = 103°
(m) m∠15
m∠15 = m∠3 -------------------------[corresponding angles are equal]
m∠15 = 41°
(n) m∠17
m∠17 = m∠15 -------------------------[opposite angles are equal]
m∠17 = 41°
(o) m∠18
m∠18 = m∠16 -------------------------[opposite angles are equal]
m∠18 = 139°
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