Worked example · Easy
Missing angle around a point
Three angles around a point are \(124^\circ\), \(76^\circ\) and \(x\).
\(x=360-124-76=\boxed{160^\circ}\), angles at a point total \(360^\circ\).
Chapter 37 · Completely rewritten
Angles
Source-matched teaching notes, visual explanations, graduated worked examples and independent examination-style practice.
What you will learn
Use the key facts confidently, select an appropriate method, communicate each working step, and check the reasonableness of the final result.
- Angles formed by straight lines
- Angles formed within parallel lines
- The angles in a quadrilateral
- The angles in a polygon
1
Subtopic 1
Angles formed by straight lines
Mark the vertex and the exact angle being requested. A full turn at a point is \(360^\circ\); adjacent angles on a straight line total \(180^\circ\); vertically opposite angles formed by crossing lines are equal.
Three first-response facts
- At a point: \(\sum\theta=360^\circ\).
- On a straight line: adjacent angles sum to \(180^\circ\).
- Intersecting lines: vertically opposite angles are equal.
Explain the rule in words next to your equation. Never trust the visual scale of an exam diagram, and avoid confusing neighbouring angles with opposite ones.
Worked examples: easy → medium → exam-style
Study the reason for each step; in an examination the method and relevant geometric explanation can earn marks even if a later calculation goes wrong.
Worked example · Easy
A straight line
Two adjacent angles on a line are \(3x\) and \(2x\).
\(3x+2x=180\); \(5x=180\); \(\boxed{x=36^\circ}\).
Worked example · Medium
Vertically opposite algebra
Opposite angles at an intersection are \((5x+8)^\circ\) and \((3x+44)^\circ\). Find \(x\).
Opposite angles are equal: \(5x+8=3x+44\); \(2x=36\); \(\boxed{x=18}\).
Worked example · Medium
Ratio on a straight line
Adjacent angles on a straight line are in the ratio \(4:5\). Find both.
Nine equal parts make \(180^\circ\). One part \(=20^\circ\); angles \(=\boxed{80^\circ,100^\circ}\).
Worked example · Exam-style
Four algebraic angles
Four angles around a point are \(x,2x,(x+30),(3x-20)\) degrees. Find \(x\) and the largest angle.
\(x+2x+x+30+3x-20=360\), so \(7x+10=360\), \(x=50\). Largest \(3(50)-20=\boxed{130^\circ}\).
2
Subtopic 2
Angles formed within parallel lines
A transversal crosses two parallel lines. Corresponding angles (matching corners) are equal; alternate interior angles (opposite sides inside the parallels) are equal; co-interior angles (same side inside) add to \(180^\circ\).
Angles in a triangle
The three interior angles add to \(180^\circ\). An exterior angle equals the sum of the two opposite interior angles. Isosceles triangles have equal base angles; equilateral triangles have three angles of \(60^\circ\); right-angled triangles contain \(90^\circ\).
Do not assume corresponding or alternate angles are equal unless the lines really are marked parallel. In proofs, state the correct pair of angles and the parallel-line reason.
Worked examples: easy → medium → exam-style
Study the reason for each step; in an examination the method and relevant geometric explanation can earn marks even if a later calculation goes wrong.
Worked example · Easy
Corresponding angles
Two parallel lines are crossed by a transversal; an angle of \(68^\circ\) corresponds to \(x\).
Corresponding angles are equal: \(\boxed{x=68^\circ}\).
Worked example · Easy
Co-interior angles
Co-interior angles are \(112^\circ\) and \(x\).
\(x=180-112=\boxed{68^\circ}\).
Worked example · Medium
Isosceles triangle
An isosceles triangle has vertex angle \(42^\circ\). Find each base angle.
Equal base angles each \((180-42)/2=\boxed{69^\circ}\).
Worked example · Medium
Exterior angle of a triangle
Opposite interior angles are \(46^\circ\) and \(58^\circ\). Find the exterior angle.
Exterior angle \(=46+58=\boxed{104^\circ}\).
Worked example · Exam-style
Parallel line and triangle
Triangle \(ABC\) has \(\angle A=53^\circ\), \(\angle B=72^\circ\). A line through C parallel to AB creates a corresponding angle equal to \(\angle A\). Find that angle and \(\angle C\).
Corresponding angle \(=\boxed{53^\circ}\). Triangle sum gives \(\angle C=180-53-72=\boxed{55^\circ}\).
3
Subtopic 3
The angles in a quadrilateral
The four interior angles of any quadrilateral add to \(360^\circ\). In a parallelogram, opposite angles are equal and adjacent angles add to \(180^\circ\). A rectangle has four right angles; an isosceles trapezium has equal base angles at each parallel side.
Multi-step quadrilateral problems
Identify right angles, parallel lines, equal sides or equal angles from given markings. Form an equation using \(360^\circ\) or one of the special-shape facts; then calculate, and give a reason.
Worked examples: easy → medium → exam-style
Study the reason for each step; in an examination the method and relevant geometric explanation can earn marks even if a later calculation goes wrong.
Worked example · Easy
Missing quadrilateral angle
Three interior angles are \(90^\circ,108^\circ,77^\circ\). Find the fourth.
\(x=360-90-108-77=\boxed{85^\circ}\).
Worked example · Easy
Parallelogram
One angle of a parallelogram is \(74^\circ\). Find its adjacent angle.
Adjacent parallelogram angles are supplementary: \(180-74=\boxed{106^\circ}\).
Worked example · Medium
Four algebraic angles
Quadrilateral angles are \(x,2x,(x+20),(2x+40)\). Find \(x\).
\(6x+60=360\), so \(\boxed{x=50}\).
Worked example · Medium
Isosceles trapezium
An isosceles trapezium has one angle of \(116^\circ\). Find the other three.
The matching angle on that base is \(116^\circ\). Each adjacent angle on a leg is \(180-116=64^\circ\). Other three: \(\boxed{116^\circ,64^\circ,64^\circ}\).
Worked example · Exam-style
Diagonal dividing a quadrilateral
A quadrilateral is divided by a diagonal. One triangle has angles \(37^\circ,65^\circ,x\); the other has \(82^\circ,45^\circ,y\). Find \(x+y\).
\(x=180-37-65=78^\circ\); \(y=180-82-45=53^\circ\). Thus \(\boxed{x+y=131^\circ}\).
4
Subtopic 4
The angles in a polygon
For any simple polygon with \(n\) sides, draw diagonals from one vertex to divide it into \(n-2\) triangles. The sum of its interior angles is therefore \((n-2)180^\circ\).
Regular polygons
For a regular \(n\)-gon, each exterior angle is \(\frac{360^\circ}{n}\); each interior angle is \(180^\circ-\frac{360^\circ}{n}\). For a convex polygon the exterior angles, one at each vertex and measured consistently, total \(360^\circ\). At a vertex, an interior angle and the adjacent exterior angle add to \(180^\circ\).
Use the total exterior-angle fact to find the number of sides of a regular polygon. An irregular polygon still has the same total interior angle sum, but its individual angles need not match.
Worked examples: easy → medium → exam-style
Study the reason for each step; in an examination the method and relevant geometric explanation can earn marks even if a later calculation goes wrong.
Worked example · Easy
Interior-angle sum
Find the total interior angle sum of a heptagon.
\((7-2)180=\boxed{900^\circ}\).
Worked example · Easy
Exterior angle of regular polygon
Find the exterior angle of a regular 12-gon.
\(360/12=\boxed{30^\circ}\).
Worked example · Medium
Find interior angle
Find the interior angle of a regular octagon.
Exterior \(360/8=45^\circ\), interior \(180-45=\boxed{135^\circ}\).
Worked example · Medium
Number of sides
A regular polygon has exterior angle \(18^\circ\). Find its number of sides.
\(n=360/18=\boxed{20}\).
Worked example · Exam-style
Unknown interior angle
Five angles of an irregular hexagon are \(95^\circ,112^\circ,124^\circ,137^\circ,108^\circ\). Find the sixth.
Total for six sides \(=720^\circ\). Known sum \(=576^\circ\). Missing \(=720-576=\boxed{144^\circ}\).
★
Independent practice
Exercises with step-by-step Show Answer solutions
Questions in each subtopic progress from confidence-building to developed methods and original exam-style problems. The problems are original practice, not copied past-paper questions. Answers stay closed until selected.
Angles formed by straight lines
Work through the basics first; save the examination-style questions for last.
1Question 1Basic
Two adjacent angles on a line are \(115^\circ\) and \(x\). Find \(x\).
Show Answer
Step-by-step solution: \(x=180-115=\boxed{65^\circ}\) (straight line).
2Question 2Basic
An angle vertically opposite \(73^\circ\) is \(x\). Find \(x\).
Show Answer
Step-by-step solution: Vertically opposite angles are equal: \(\boxed{x=73^\circ}\).
3Question 3Basic
Angles \(90^\circ,110^\circ\) and \(x\) meet at a point. Find \(x\).
Show Answer
Step-by-step solution: \(x=360-90-110=\boxed{160^\circ}\).
4Question 4Developing
Adjacent angles are \((2x+10)^\circ\) and \((x+20)^\circ\) on a straight line. Find \(x\).
Show Answer
Step-by-step solution: \(2x+10+x+20=180\), hence \(3x=150\), \(\boxed{x=50}\).
5Question 5Developing
Vertically opposite angles are \((4x-15)^\circ\) and \((2x+29)^\circ\). Find \(x\).
Show Answer
Step-by-step solution: \(4x-15=2x+29\); \(2x=44\); \(\boxed{x=22}\).
6Question 6Developing
Angles at a point are \(2x,3x,4x\) degrees. Find the smallest.
Show Answer
Step-by-step solution: \(9x=360\), \(x=40\); smallest \(2x=\boxed{80^\circ}\).
7Question 7Exam-style
Two adjacent angles are in ratio \(7:8\) on a straight line. Find the smaller.
Show Answer
Step-by-step solution: \(15\) parts give \(180^\circ\), each \(12^\circ\). Smaller \(7\times12=\boxed{84^\circ}\).
8Question 8Exam-style
Four angles around a point are \(3x,(2x+30),(x+15),(2x-5)\). Find \(x\).
Show Answer
Step-by-step solution: \(8x+40=360\), so \(\boxed{x=40}\).
9Question 9Exam-style
One angle at a crossing is \(138^\circ\). Find the other three.
Show Answer
Step-by-step solution: Opposite angle \(=138^\circ\). Adjacent each \(180-138=42^\circ\). Other three: \(\boxed{138^\circ,42^\circ,42^\circ}\).
Angles formed within parallel lines
Work through the basics first; save the examination-style questions for last.
10Question 10Basic
An alternate interior angle to \(82^\circ\) is \(x\). Find \(x\).
Show Answer
Step-by-step solution: Alternate angles between parallel lines are equal: \(\boxed{82^\circ}\).
11Question 11Basic
Find an angle co-interior with \(131^\circ\).
Show Answer
Step-by-step solution: \(180-131=\boxed{49^\circ}\).
12Question 12Basic
A triangle has angles \(65^\circ,45^\circ,x\). Find \(x\).
Show Answer
Step-by-step solution: \(x=180-65-45=\boxed{70^\circ}\).
13Question 13Developing
Corresponding angles are \((3x+10)^\circ\) and \((5x-30)^\circ\). Find \(x\).
Show Answer
Step-by-step solution: \(3x+10=5x-30\); \(2x=40\); \(\boxed{x=20}\).
14Question 14Developing
Co-interior angles are \((4x+12)^\circ\) and \((2x+18)^\circ\). Find \(x\).
Show Answer
Step-by-step solution: \(6x+30=180\), so \(\boxed{x=25}\).
15Question 15Developing
An equilateral triangle has one angle labelled \((2x+14)^\circ\). Find \(x\).
Show Answer
Step-by-step solution: Each angle is \(60^\circ\); \(2x+14=60\), so \(\boxed{x=23}\).
16Question 16Exam-style
An isosceles triangle has equal base angles \((3x-9)^\circ\) and a vertex angle \((2x+18)^\circ\). Find \(x\).
Show Answer
Step-by-step solution: \(2(3x-9)+2x+18=180\); \(8x=180\); \(\boxed{x=22.5}\).
17Question 17Exam-style
At a triangle vertex, the exterior angle is \(127^\circ\) and one opposite interior angle is \(68^\circ\). Find the other opposite angle.
Show Answer
Step-by-step solution: Exterior angle \(=\) sum of remote interiors, so \(127-68=\boxed{59^\circ}\).
18Question 18Exam-style
An exterior angle of a triangle is \((5x+8)^\circ\); its opposite interiors are \((2x+20)^\circ\) and \((x+42)^\circ\). Find \(x\).
Show Answer
Step-by-step solution: \(5x+8=2x+20+x+42\); \(2x=54\), hence \(\boxed{x=27}\).
The angles in a quadrilateral
Work through the basics first; save the examination-style questions for last.
19Question 19Basic
Interior angles are \(100^\circ,85^\circ,95^\circ,x\). Find \(x\).
Show Answer
Step-by-step solution: \(360-100-85-95=\boxed{80^\circ}\).
20Question 20Basic
One parallelogram angle is \(125^\circ\). Find its opposite angle.
Show Answer
Step-by-step solution: Opposite angles in a parallelogram are equal: \(\boxed{125^\circ}\).
21Question 21Basic
What is the sum of the interior angles of any quadrilateral?
Show Answer
Step-by-step solution: Divide into two triangles: \(2\times180=\boxed{360^\circ}\).
22Question 22Developing
Quadrilateral angles are \(x,2x,3x,4x\). Find \(x\).
Show Answer
Step-by-step solution: \(10x=360\), so \(\boxed{x=36^\circ}\).
23Question 23Developing
Three quadrilateral angles are \((2x+10)^\circ\), \((x+30)^\circ\) and \(100^\circ\); the fourth is \(80^\circ\). Find \(x\).
Show Answer
Step-by-step solution: \(3x+220=360\), so \(3x=140\), \(\boxed{x=46\tfrac23}\).
24Question 24Developing
Adjacent angles of a parallelogram are \((3x-15)^\circ\) and \((2x+20)^\circ\). Find \(x\).
Show Answer
Step-by-step solution: \(3x-15+2x+20=180\); \(5x=175\), \(\boxed{x=35}\).
25Question 25Exam-style
An isosceles trapezium has an upper angle \(73^\circ\). Find the two lower angles.
Show Answer
Step-by-step solution: Base angles at the upper base both \(73^\circ\). Along each leg, \(180-73=\boxed{107^\circ}\).
26Question 26Exam-style
A quadrilateral has two right angles, one angle \((4x-10)^\circ\), and the last \((2x+40)^\circ\). Find \(x\).
Show Answer
Step-by-step solution: \(90+90+4x-10+2x+40=360\); \(6x+210=360\); \(\boxed{x=25}\).
27Question 27Exam-style
A quadrilateral interior-angle ratio is \(2:3:4:6\). Find the largest.
Show Answer
Step-by-step solution: Fifteen parts sum to \(360^\circ\), so one part \(24^\circ\). Largest \(6\times24=\boxed{144^\circ}\).
The angles in a polygon
Work through the basics first; save the examination-style questions for last.
28Question 28Basic
Find the interior angle sum of a pentagon.
Show Answer
Step-by-step solution: \((5-2)180=\boxed{540^\circ}\).
29Question 29Basic
Find each exterior angle of a regular hexagon.
Show Answer
Step-by-step solution: \(360/6=\boxed{60^\circ}\).
30Question 30Basic
Find each interior angle of a regular hexagon.
Show Answer
Step-by-step solution: \(180-360/6=\boxed{120^\circ}\).
31Question 31Developing
A regular polygon has 15 sides. Find each interior angle.
Show Answer
Step-by-step solution: Exterior \(360/15=24^\circ\), interior \(180-24=\boxed{156^\circ}\).
32Question 32Developing
A regular polygon has exterior angle \(45^\circ\). Find its number of sides.
Show Answer
Step-by-step solution: \(n=360/45=\boxed{8}\).
33Question 33Developing
The interior sum of a polygon is \(1440^\circ\). Find the number of sides.
Show Answer
Step-by-step solution: \((n-2)180=1440\) gives \(n-2=8\), so \(\boxed{n=10}\).
34Question 34Exam-style
A regular polygon has interior angle \(165^\circ\). How many sides?
Show Answer
Step-by-step solution: Exterior \(180-165=15^\circ\); \(n=360/15=\boxed{24}\).
35Question 35Exam-style
A pentagon has four angles \(102^\circ,119^\circ,108^\circ,126^\circ\). Find the fifth.
Show Answer
Step-by-step solution: Total \(540^\circ\). Known sum \(455^\circ\), so fifth \(=\boxed{85^\circ}\).
36Question 36Exam-style
An irregular octagon has seven known interior angles totalling \(930^\circ\). Find the eighth.
Show Answer
Step-by-step solution: Total \((8-2)180=1080^\circ\), remaining \(1080-930=\boxed{150^\circ}\).
37Question 37Exam-style
A convex polygon has exterior angles \(30^\circ,45^\circ,65^\circ,80^\circ,70^\circ\) and one unknown. Find the unknown.
Show Answer
Step-by-step solution: Total exterior angles \(=360^\circ\). Known sum \(290^\circ\); unknown \(=\boxed{70^\circ}\).
High-grade revision checklist
- Can I explain and correctly apply: Angles formed by straight lines?
- Can I explain and correctly apply: Angles formed within parallel lines?
- Can I explain and correctly apply: The angles in a quadrilateral?
- Can I explain and correctly apply: The angles in a polygon?
- Have I shown reasons, units, and sufficient steps rather than just giving final numbers?
- Have I checked which quantity or angle the question actually asks for?