Basic Angle Properties

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Lesson 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.

Vertically opposite angles

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.

Triangle types

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.

Quadrilateral types

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.

Slides

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Practice questions

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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°

Revision notes

## 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).

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°.

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° (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°, 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°. BCD is a straight line. Find angle ACD.

Angle ABC = (180-38)/2 = 71°. Angle ACD = 180 - 71 = 109° (angles on a straight line).

In triangle BCD, ABC is a straight line, BC = BD and angle CBD = 40°. Find angle BDC.

70°. Triangle BCD is isosceles (BC = BD), so the base angles are equal: (180° − 40°) ÷ 2 = 70°.

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°, angle RSQ = 32°. Find angle PQS.

In triangle QRS, angle RQS = 180° − (25° + 32°) = 123°. Since P, Q, R lie on a straight line, angle PQS = 180° − 123° = 57°.

In triangle PRS, Q is on PR, PS = QS, angle QRS = 25°, angle RSQ = 32°. Find angle PSR.

Angle PQS = 57° (see the previous card). Triangle PQS is isosceles (PS = QS), so angle SPQ = angle SQP = 57°, giving angle PSQ = 180° − 57° − 57° = 66°. Angle PSR = angle PSQ + angle QSR = 66° + 32° = 98°.

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°.

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

y = 180 - (25+98) = 57°.

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

Fourth angle = 360 - (97+115+85) = 63°.

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

x = 180 - (60+50) = 70°.

What is the relationship between angles in a rhombus?

Opposite angles are equal.

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