EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Angle properties of triangles and quadrilaterals
Angles and trigonometry · Lesson 1 of 7
Before We Start
Answer each one, then check.
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
Learning Objectives
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.
The Basic Angle Facts
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°.
Special Triangles
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.
An Isosceles Triangle
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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°
Quadrilaterals
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.
Angles in Algebra
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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°
PART TWO
Parallel Lines
A line crossing two parallel lines makes equal angles in three patterns.
Alternate, Corresponding and Co-interior
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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. |
Z for alternate, F for corresponding, C for co-interior. |
The Parallel Line Facts
|
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. |
Parallel Lines with Reasons
|
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°
Reasons That Earn Marks
|
LOSES THE MARK |
EARNS THE MARK |
|
▸ "Z angles" ▸ "Because it's a straight line" ▸ "Triangle = 180" ▸ "They're the same" |
▸ "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" |
Key Terms
|
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°. |
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. |
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°.
Can I...?
☐ 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.
Summary
✓ 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.
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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. |