Basic Angle Properties

Learn it by playing

Answer these questions to earn energy, then fish and explore. No account needed.

For teachers: ready-to-use lesson slides, revision notes, diagrams for Basic Angle Properties (Maths [CIE], Core) — use them in your lesson, or run the topic as a live class game.

Notes

Angles around Points & Lines

**Angles around a point** add up to **360°**.
**Angles on a straight line** add up to **180°**.
**Vertically opposite angles** are equal when two lines intersect.
Use known angle properties to find missing angles, even if they are not the target angle.

Angles in Triangles

The **three interior angles** of any triangle sum to **180°**.
In an **isosceles triangle**, the two angles opposite the equal sides are equal.
In an **equilateral triangle**, all three angles are **60°**.
A **right-angled triangle** has one **90°** angle.

Angles in Quadrilaterals

The **four interior angles** of any quadrilateral sum to **360°**.
In a **square** or **rectangle**, all angles are **90°**.
In a **parallelogram** or **rhombus**, opposite angles are equal.
In a **kite**, one pair of opposite angles are equal.

Problem-Solving Tips

Mark all missing angles you can find on the diagram using angle properties.
Use vertically opposite angles and straight line angles to find triangle angles.
For isosceles triangles, identify the equal angles from the equal sides.
Combine triangle and quadrilateral angle sums with straight line facts.

Vertically Opposite Angles

Triangle Angle Sum

Isosceles Triangle

Quadrilateral Angle Sum

Practice questions

Free preview — 8 of 40 questions. Sign up to see them all.

1.What is the sum of angles around a point?
Easy
- A180°
- B360°
- C90°
- D270°
2.Angles that form a straight line add up to
Easy
- A90°
- B180°
- C360°
- D270°
3.Vertically opposite angles are
Easy
- Aequal
- Bsupplementary
- Ccomplementary
- Ddifferent
4.The three interior angles of any triangle add up to
Easy
- A90°
- B180°
- C360°
- D270°
5.In an equilateral triangle, each interior angle is
Easy
- A45°
- B60°
- C90°
- D30°
6.A right-angled triangle has one angle equal to
Easy
- A45°
- B60°
- C90°
- D180°
7.The sum of interior angles of a quadrilateral is
Easy
- A180°
- B270°
- C360°
- D90°
8.In a rectangle, each interior angle is
Easy
- A45°
- B60°
- C90°
- D180°

Summary

## Basic Angle Properties ### Angles around Points & Lines - **Angles around a point** add up to **360°**. - **Angles on a straight line** add up to **180°**. - **Vertically opposite angles** are equal (when two lines intersect). ### Angles in Triangles - **Interior angles of a triangle** add up to **180°**. - **Isosceles triangle**: two sides equal → two base angles equal. - **Equilateral triangle**: all angles = 60°. - **Right-angled triangle**: one angle = 90°. ### Angles in Quadrilaterals - **Interior angles of a quadrilateral** add up to **360°**. - Square/rectangle: all angles = 90°. - Parallelogram/rhombus: opposite angles equal. - Kite: one pair of opposite angles equal. ### Tips - Mark all missing angles you can find on the diagram. - Use known angles to find others step by step. ### Example (from source) Find x and y where three lines intersect: vertically opposite gives x = 25°; angles on a straight line: 25 + y + 98 = 180 → y = 57°. ### Common Question Types - Find missing angle in triangle/quadrilateral. - Use isosceles property (base angles equal). - Combine straight line and triangle properties. - Explain why a triangle sum is wrong (e.g., 55+85+50=190 ≠ 180).

Key terms

Angles around a point: Angles that meet at a point sum to 360°.
Angles on a straight line: Angles that form a straight line sum to 180°.
Vertically opposite angles: When two lines intersect, the angles opposite each other are equal.
Interior angles of a triangle: The three angles inside any triangle sum to 180°.
Isosceles triangle: A triangle with two equal sides; the angles opposite those sides are equal.
Equilateral triangle: A triangle with all sides equal; all angles are 60°.
Right-angled triangle: A triangle with one angle of 90°.
Interior angles of a quadrilateral: The four angles inside any quadrilateral sum to 360°.

Slides

Basic Angle Properties

Angles around a point add up to 360°.
Angles on a straight line add up to 180°.
Vertically opposite angles are equal.

What do angles around a point add up to?

90°
180°
270°
360°

Angles in Triangles

The three interior angles of any triangle add up to 180°.
In an isosceles triangle, two angles are equal (base angles).
In an equilateral triangle, all angles are 60°.
A right-angled triangle has one 90° angle.

The interior angles of a triangle add up $to ____$ degrees.

Angles in Quadrilaterals

The four interior angles of any quadrilateral add up to 360°.
In a square or rectangle, all angles are 90°.
In a parallelogram or rhombus, opposite angles are equal.
In a kite, one pair of opposite angles are equal.

What is the sum of interior angles in a quadrilateral?

180°
270°
360°
540°

Using Angle Properties

Find all missing angles you can, even if not the target – they may help.
Use vertically opposite angles, straight lines, and triangle/quadrilateral sums.
Mark angles on the diagram as you go.

Which of these are true?

Vertically opposite angles are equal.
Angles on a straight line add up to 360°.
The sum of angles in a triangle is 180°.

Worked Example: Angles around a Point

Three straight lines intersect at a point.
Given: one angle is 98°, another is 25° (vertically opposite).
Find x and y.

Vertically opposite angles $are ____$ . So $x = 25^{\circ}$ .

Worked Example: Triangle and Straight Line

In a triangle, one angle is 60° (vertically opposite), another is 50° (straight line).
The third angle x is found using triangle sum: $x + 60 + 50 = 180$ .
So $x = 70^{\circ}$ .

In a triangle, two angles are 60° and 50°. What is the third angle?

70°
80°
90°
110°

Worked Example: Quadrilateral

Irregular quadrilateral with three angles: 97°, 115°, 85°.
Sum of given angles $= 297^{\circ}$ .
Fourth angle $= 360 - 297 = 63^{\circ}$ .
Then use straight line to find $y: y + 63 = 180$ , so $y = 117^{\circ}$ .

Which statements are correct?

The sum of angles in a quadrilateral is 360°.
In an isosceles triangle, all angles are equal.
Angles on a straight line add up to 180°.

Exam Tip: Find All Missing Angles

Always find and mark every missing angle you can.
Even if not the target, they may lead to the answer.
Use the properties: straight line, vertically opposite, triangle sum, quadrilateral sum.

An isosceles triangle has one angle of 32°. The other two angles are equal. What is each equal angle?

74°
148°
64°
32°

Summary

Angles around a point: 360°.
Angles on a straight line: 180°.
Vertically opposite angles: equal.
Triangle interior sum: 180°.
Quadrilateral interior sum: 360°.

Flashcards

What do angles around a point add up to?

Angles around a point add up to 360°.

What do angles on a straight line add up to?

Angles on a straight line add up to 180°.

What are vertically opposite angles?

Vertically opposite angles are equal. They occur when two lines intersect.

What do the interior angles of a triangle add up to?

The interior angles of a triangle add up to 180°.

In an isosceles triangle, which angles are equal?

The two angles opposite the equal sides are equal.

What are the angles in an equilateral triangle?

Each angle in an equilateral triangle is 60°.

What is the sum of interior angles in a quadrilateral?

The sum of interior angles in a quadrilateral is 360°.

In a parallelogram, what is true about opposite angles?

Opposite angles in a parallelogram are equal.

In a kite, which angles are equal?

One pair of opposite angles are equal in a kite.

What is the angle in a right-angled triangle that is 90°?

One angle in a right-angled triangle is 90°.

If a triangle has angles 55°, 85°, and 50°, why is it incorrect?

The sum is 190°, but the interior angles of a triangle must add up to 180°.

In an isosceles triangle, one angle is 96°. Find the other two angles.

The other two angles are 42° and $42^{\circ} (since 180 - 96 = 84$ , divided by 2).

Kim says if one angle of an isosceles triangle is 48°, another must be 66°. Why is he wrong?

The 48° could be the vertex angle, making base angles $(180-48)/2 = 66^{\circ}$ , or it could be a base angle, making the other base angle also 48° and vertex 84°.

In triangle $ABC, AB = AC$ and angle $BAC = 38^{\circ}$ . BCD is a straight line. Find angle ACD.

Angle $ABC = (180-38)/2 = 71^{\circ}$ . Angle $ACD = 180 - 71 = 109^{\circ} (angles$ on a straight line).

In the diagram, ABC is a straight line and BCD is an isosceles triangle. Find x.

Use angles on a straight line and isosceles triangle properties. (Specific value depends on diagram, but method: find missing angles step by step.)

What is the value of x in a trapezium where three angles are given?

Sum of angles in a quadrilateral is 360°, so $x = 360 - (sum$ of given angles).

In triangle PRS, Q is on $PR, PS = QS$ , angle $QRS = 25^{\circ}$ , angle $RSQ = 32^{\circ}$ . Find angle PQS.

Angle $PQS = 180 - (25+32) = 123^{\circ} (angles$ in triangle QRS).

In triangle PRS, Q is on $PR, PS = QS$ , angle $QRS = 25^{\circ}$ , angle $RSQ = 32^{\circ}$ . Find angle PSR.

Angle $PQS = 123^{\circ}$ , so angle $PQS = 180-123 = 57^{\circ} (angles$ on straight line). In triangle $PQS, PS=QS$ so base angles equal: $(180-57)/2 = 61.5^{\circ}$ . Angle $PSR = 61.5+32 = 93.5^{\circ}$ .

What is the sum of angles in any quadrilateral?

360°.

If a quadrilateral is a square or rectangle, what are its angles?

All angles are 90°.

What is the value of x if angles around a point are 25°, 98°, and x?

$x = 360 - (25+98) = 237^{\circ}$ .

What is the value of y if angles on a straight line are 25°, y, and 98°?

$y = 180 - (25+98) = 57^{\circ}$ .

In an irregular quadrilateral, three angles are 97°, 115°, and 85°. Find the fourth angle.

Fourth angle $= 360 - (97+115+85) = 63^{\circ}$ .

If a triangle has angles x, 60°, and 50°, find x.

$x = 180 - (60+50) = 70^{\circ}$ .

What is the relationship between angles in a rhombus?

Opposite angles are equal.

Unlock all 40 questions, slides, flashcards & more

Create a free account to see every question, the slides, flashcards and revision notes for this topic.

Past papers

Past-paper practice for this topic is coming soon.

🗂️ Coming soon