Angle properties of triangles and quadrilaterals.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Angle properties of triangles and quadrilaterals

Angles and trigonometry

Lesson 1 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Before We Start

Angles and trigonometry

Lesson 1 of 7

Answer each one, then check.

1

How many degrees are there in a full turn?

2

What is the name of an angle bigger than 90° but smaller than 180°?

3

Solve 3x + 30 = 180.

4

How many lines of symmetry does a rhombus have?

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Before We Start - Answers

Angles and trigonometry

Lesson 1 of 7

1

How many degrees are there in a full turn?

360°

2

What is the name of an angle bigger than 90° but smaller than 180°?

Obtuse

3

Solve 3x + 30 = 180.

x = 50

4

How many lines of symmetry does a rhombus have?

2

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Learning Objectives

Angles and trigonometry

Lesson 1 of 7

1

Use the angle facts for lines, points and triangles.

2

Use the angle sum of a quadrilateral and the properties of special quadrilaterals.

3

Find angles made by parallel lines: alternate, corresponding and co-interior.

4

Give a reason for every step of an angle calculation.

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The Basic Angle Facts

Angles and trigonometry

Lesson 1 of 7

Every angle problem is built from a handful of facts. Learn the exact wording - it is what earns the reason marks.

Angles on a straight line

Angles on a straight line add up to 180°.

Angles around a point

Angles around a point add up to 360°.

Vertically opposite angles

Where two straight lines cross, the opposite angles are equal.

Angles in a triangle

The angles in a triangle add up to 180°.

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Special Triangles

Angles and trigonometry

Lesson 1 of 7

The sides tell you about the angles.

Equilateral

Three equal sides and three equal angles of 60°.

Isosceles

Two equal sides, and the two base angles opposite them are equal.

Right-angled

One angle is 90°, so the other two add up to 90°.

Exterior angle of a triangle

The exterior angle equals the sum of the two interior opposite angles.

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An Isosceles Triangle

Angles and trigonometry

Lesson 1 of 7

In triangle ABC, AB = AC and angle BAC = 40°. Work out angle ABC.

1

The triangle is isosceles, so the base angles are equal

angle ABC = angle ACB

2

Angles in a triangle add up to 180°

180 − 40 = 140

3

Share between the two equal base angles

140 ÷ 2 = 70

ANSWER

angle ABC = 70°

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Quadrilaterals

Angles and trigonometry

Lesson 1 of 7

A quadrilateral splits into two triangles, so its angles add up to 2 × 180°= 360°.

Square and rectangle

Four right angles. Diagonals are equal and bisect each other.

Parallelogram

Opposite sides parallel and equal; opposite angles equal.

Rhombus

A parallelogram with four equal sides; the diagonals cross at right angles.

Kite and trapezium

A kite has one pair of equal opposite angles; a trapezium has one pair of parallel sides.

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Angles in Algebra

Angles and trigonometry

Lesson 1 of 7

The angles of a quadrilateral are x, 2x, 3x and 4x. Work out the size of the largest angle.

1

Angles in a quadrilateral add up to 360°

x + 2x + 3x + 4x = 360

2

Collect like terms

10x = 360

3

Solve

x = 36

4

The largest angle is 4x

4 × 36 = 144

ANSWER

144°

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PART TWO

Parallel Lines

A line crossing two parallel lines makes equal angles in three patterns.

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Alternate, Corresponding and Co-interior

Angles and trigonometry

Lesson 1 of 7

Z for alternate, F for corresponding, C for co-interior.

Alternate angles are equal (a Z shape). Corresponding angles are equal (an F shape). Co-interior angles add up to 180° (a C shape). In the exam, write the proper name - "Z angles" does not earn the mark.

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The Parallel Line Facts

Angles and trigonometry

Lesson 1 of 7

Angle pair

Shape

Fact

Alternate

Z

Alternate angles are equal.

Corresponding

F

Corresponding angles are equal.

Co-interior (allied)

C

Co-interior angles add up to 180°.

Vertically opposite

X

Vertically opposite angles are equal.

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Parallel Lines with Reasons

Angles and trigonometry

Lesson 1 of 7

Two parallel lines are crossed by a straight line. One of the angles is 72°. Angle y is co-interior to it. Angle z is on a straight line with y. Work out y and z.

1

Co-interior angles add up to 180°

y = 180 − 72 = 108

2

Angles on a straight line add up to 180°

z = 180 − 108 = 72

3

Check: z is alternate (or corresponding) to the 72° angle, so equal to it

z = 72

ANSWER

y = 108°, z = 72°

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Reasons That Earn Marks

Angles and trigonometry

Lesson 1 of 7

LOSES THE MARK

• "Z angles"

• "Because it's a straight line"

• "Triangle = 180"

• "They're the same"

EARNS THE MARK

• "Alternate angles are equal"

• "Angles on a straight line add up to 180°"

• "Angles in a triangle add up to 180°"

• "Base angles of an isosceles triangle are equal"

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Key Terms

Angles and trigonometry

Lesson 1 of 7

Vertically opposite

The angles opposite each other where two lines cross; they are equal.

Isosceles

A triangle with two equal sides and two equal angles.

Parallel

Lines that are always the same distance apart and never meet.

Transversal

A line that crosses two or more other lines.

Alternate angles

Angles on opposite sides of a transversal, between parallel lines; they are equal.

Co-interior angles

Angles on the same side of a transversal, between parallel lines; they add up to 180°.

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Angles and trigonometry

Lesson 1 of 7

YOUR TASK

Angle Chase

10 minutes

Draw two parallel lines and two different transversals that cross each other between the parallel lines, making a triangle. Mark one angle at each crossing. Swap with a partner, who must find every other angle and give a reason for each.

1

Mark the parallel lines with arrows.

2

Find one angle at a time.

3

Write the reason next to each angle.

WHAT A GOOD ANSWER SHOWS

Students use alternate and corresponding angles to find the triangle's angles, then check that they add up to 180° - which is in fact a proof that the angles in a triangle add up to 180°.

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Can I...?

Angles and trigonometry

Lesson 1 of 7

Use angles on a line and around a point.

Use vertically opposite angles.

Use angles in a triangle, including isosceles triangles.

Use angles in a quadrilateral.

Recall the properties of special quadrilaterals.

Find alternate, corresponding and co-interior angles.

Form and solve an equation from angle facts.

Give a correct reason for each step.

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Summary & Exam Focus

Line 180°, point 360°, triangle 180°, quadrilateral 360°.

Vertically opposite angles are equal.

Alternate and corresponding angles are equal; co-interior angles add up to 180°.

Isosceles: the base angles are equal.

EXAM FOCUS

AB and CD are parallel lines. Angle APQ = 65°. Work out the size of angle x. Give a reason for each stage of your working. (3 marks)

"Give reasons" questions have marks for the reasons as well as the angles. Write each reason as a full sentence using the proper words: alternate, corresponding, co-interior, vertically opposite.

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