Angles (Copy)
1.
At a point O, three rays OA, OB, and OC meet. Angle AOB = 120°, angle BOC = 100°.
Find angle AOC. Give a reason.
2.
Angles on a straight line PQ are split by a ray PR. Angle QPR = 75°.
Find angle RPS. Give a reason.
3.
Two straight lines intersect at point X. One angle is 68°.
Find each of the other three angles. State your reasons.
4.
In ΔABC, angle A = 50°, angle B = 65°.
Find angle C.
5.
A quadrilateral PQRS has angles P = 95°, Q = 80°, R = 120°.
Find angle S.
6.
In ΔDEF, exterior angle at E is 128°. Interior angle at F is 46°.
Find angle D.
7.
AB ∥ CD. A transversal PQ intersects them.
∠APQ = 72°.
Find ∠PQC. State the geometrical reason.
8.
AB ∥ CD. A transversal EF cuts them.
∠AEF = 60°.
Find ∠EFD. Explain.
9.
AB ∥ CD, cut by transversal XY.
∠AXY = 105°.
Find ∠XYD. State the reason.
10.
AB ∥ CD, cut by transversal PQ.
∠BPQ = 77°.
Find ∠PQD.
11.
In ΔXYZ, XY ∥ PQ and Z is joined to P.
If ∠X = 70°, ∠Y = 50°, find ∠PZQ.
12.
A triangle has angles in the ratio 2:3:4.
Find the three angles.
13.
A quadrilateral has angles in the ratio 3:4:5:6.
Find each angle.
14.
A regular hexagon is drawn.
(a) Find each interior angle.
(b) Find each exterior angle.
15.
A regular octagon is drawn.
(a) Find its interior angle.
(b) State the sum of all interior angles.
16.
The exterior angle of a regular polygon is 24°.
Find the number of sides of the polygon.
17.
The interior angle of a regular polygon is 150°.
Find the number of sides.
18.
In ΔPQR, ∠P = 90°, ∠Q = 62°.
Find ∠R.
19.
Angles at a point are divided into four equal angles.
Find each angle.
20.
Angles on a straight line are in the ratio 2:3.
Find each angle.
21.
Two parallel lines are cut by a transversal. One angle is 68°.
Find each of the other seven angles.
22.
A parallelogram ABCD has ∠A = 70°.
Find ∠B, ∠C, and ∠D.
23.
A rhombus PQRS has ∠P = 72°.
Find ∠Q, ∠R, and ∠S.
24.
An isosceles triangle has base angles of 58°.
Find the vertex angle.
25.
A triangle has sides in ratio 5:5:8.
Find all angles.
26.
A regular decagon is drawn.
(a) Find one interior angle.
(b) Find the sum of exterior angles.
27.
A regular polygon has each interior angle 135°.
(a) Find the number of sides.
(b) Find its order of rotational symmetry.
28.
In ΔXYZ, angle bisectors of ∠X, ∠Y, and ∠Z meet at O.
Prove that the angles at O sum to 180°.
29.
A triangle has angles 2x, 3x, 4x.
Find the angles.
30.
The exterior angle of a regular polygon is 15°.
Find the number of sides and each interior angle.
1.
∠AOC = 360° − (120° + 100°) = 140°.
Reason: angles at a point sum to 360°.
2.
∠RPS = 180° − 75° = 105°.
Reason: angles on a straight line sum to 180°.
3.
If one angle = 68°, opposite angle = 68°. Adjacent angles = 180° − 68° = 112° each.
Reason: vertically opposite angles equal; linear pair = 180°.
4.
∠C = 180° − (50° + 65°) = 65°.
Reason: angles in a triangle sum to 180°.
5.
∠S = 360° − (95° + 80° + 120°) = 65°.
Reason: angles of a quadrilateral sum to 360°.
6.
Exterior angle at E = 128° = ∠D + ∠F.
So ∠D = 128° − 46° = 82°.
7.
∠PQC = 72°.
Reason: corresponding angles between parallel lines are equal.
8.
∠EFD = 60°.
Reason: alternate angles between parallel lines are equal.
9.
∠XYD = 105°.
Reason: alternate angles are equal.
10.
∠PQD = 77°.
Reason: corresponding angles are equal.
11.
∠PZQ = ∠X + ∠Y = 70° + 50° = 120°.
Reason: exterior angle = sum of two opposite interior angles.
12.
Angles ratio 2:3:4 → total 9 parts.
180 ÷ 9 = 20°.
Angles = 40°, 60°, 80°.
13.
Sum = 360°, ratio = 3+4+5+6 = 18 parts.
360 ÷ 18 = 20°.
Angles = 60°, 80°, 100°, 120°.
14.
(a) Interior = (n−2)×180 ÷ n = (4×180)/6 = 120°.
(b) Exterior = 60°.
15.
(a) Interior = (n−2)×180 ÷ n = (6×180)/8 = 135°.
(b) Sum = (8−2)×180 = 1080°.
16.
Exterior angle = 24°.
n = 360 ÷ 24 = 15 sides.
17.
Interior = 150°.
Exterior = 30°.
n = 360 ÷ 30 = 12 sides.
18.
∠R = 180° − (90° + 62°) = 28°.
19.
360 ÷ 4 = 90°.
Each angle = 90°.
20.
Angles on line ratio 2:3 → total 180°.
2x+3x=180 → 5x=180 → x=36°.
Angles = 72°, 108°.
21.
If one angle = 68°:
Opposite = 68°.
Adjacent = 112°.
All 8 angles alternate between 68° and 112°.
22.
Parallelogram: ∠A = 70°, ∠C = 70° (opposite).
∠B = 110°, ∠D = 110° (opposite).
Reason: opposite angles equal; adjacent angles supplementary.
23.
Rhombus: ∠P = 72°. Opposite ∠R = 72°.
∠Q = ∠S = 108°.
Reason: adjacent angles supplementary.
24.
Base angles = 58° each.
Vertex = 180 − (58+58) = 64°.
25.
Sides 5:5:8 → isosceles.
Use cosine rule: cosθ = (5²+5²−8²)/(2×5×5) = (25+25−64)/50 = −14/50 = −0.28.
∠ opposite 8 = cos⁻¹(−0.28) ≈ 106°.
Other two angles ≈ 37° each.
26.
(a) Interior = (10−2)×180 ÷10 = 144°.
(b) Exterior sum always = 360°.
27.
Interior = 135°. Exterior = 45°.
n = 360 ÷ 45 = 8 sides.
Order of rotational symmetry = 8.
28.
Angle bisectors always meet at in-centre.
Proof: ∠XOY + ∠YOZ + ∠ZOX = 180°.
Reason: angles round a point on a straight line property.
29.
2x+3x+4x=180 → 9x=180 → x=20.
Angles = 40°, 60°, 80°.
30.
Exterior = 15°.
n = 360 ÷ 15 = 24 sides.
Interior = 180−15 = 165°.