The values of x and y are;
15. x = 21, y = 8
16. x = 29, y = 9
17. x = 18, y = 11
The details of the above solution are;
15. Linear pair angles are supplementary (have a sum of 180°), therefore, the sum of the linear pair angles (10·x - 61)° and (x + 10)° is 180°
(10·x - 61)° + (x + 10)° = 180°
11·x - 51 = 180
11·x = 180 + 51
x = (180 + 51)/11
(180 + 51)/11 = 21
x = 21
The vertical angle theorem that the vertical angles are congruent
The measure of congruent angles are the same, therefore;
The vertical angles (10·x - 61)° and (18·y + 5)° are congruent
(10·x - 61)° = (18·y + 5)°
10 × 21 - 61 = 18·y + 5
10 × 21 - 61 = 149
149 = 18·y + 5
18·y + 5 = 149
18·y = 149 - 5
y = (149 - 5)/18
(149 - 5)/18 = 8
y = 8
16. The linear pair angles (5·x - 17)° and (3·x - 11)° are supplementary, therefore;
(5·x - 17) + (3·x - 11) = 180°
(8·x - 28) = 180
x = (180 + 28)/8
(180 + 28)/8 = 26
x = 26
The angles, (3·x - 11)°, and the right angle plus the (2·y + 5)° angle form a line pair, therefore;
(3·x - 11)° + 90° + (2·y + 5)° = 180°
(3 × 26 - 11)° + 90° + (2·y + 5)° = 180°
67 + 90 + (2·y + 5)° = 180°
2·y = 180 - (67 + 90) - 5
y = (180 - (67 + 90) - 5)/2
y = 9
17. m∠MNQ = (8·x + 12)°, m∠PNQ = 78°, and m∠RNM = (3·y - 9)°
m∠MNQ = m∠PNQ + m∠PNM
∠PNQ ≅ ∠PNM (Definition of bisected angle, ∠MNQ)
m∠PNQ = m∠PNM
m∠MNQ = m∠PNQ + m∠PNQ
m∠MNQ = 2 × m∠PNQ
m∠PNQ = 78°
m∠MNQ = 2 × 78°
2 × 78° = 156°
m∠MNQ = 156°
m∠MNQ = (8·x + 12)°
(8·x + 12)° = 156°
x = (156 - 12)/8
(156 - 12)/8
x = 18
∠MNQ and ∠RNM are linear pair angles, therefore;
m∠MNQ + m∠RNM = 180°
m∠RNM = (3·y - 9)°
Therefore;
(8·x + 12)° + (3·y - 9)° = 180°
(8 × 18 + 12)° + (3·y - 9)° = 180°
156° + (3·y - 9)° = 180°
(3·y - 9)° = 180° - 156°
y = ((180° - 156°) + 9)/3
((180° - 156°) + 9)/3 = 11
y = 11