lesson
Properties constrain the possibilities
1 · Learn
Triangle angles total 180°. In an isosceles triangle, the angles opposite the equal sides are equal. A quadrilateral's interior angles total 360°, but its specific properties decide what else is equal or parallel.
A rectangle has four right angles and opposite equal sides; a rhombus has four equal sides but need not have right angles. A square has both sets of properties and belongs to both families.
Why are the base angles of an isosceles triangle equal? For AB = AC, join A to the midpoint D of BC. Triangles ABD and ACD have AB = AC, BD = CD and the common side AD, so they are congruent by SSS. Their corresponding angles at B and C are therefore equal. A quadrilateral's angle total can be derived by splitting it along an internal diagonal into two triangles: 180° + 180° = 360°.
2 · Worked example
An isosceles triangle has vertex angle 40°. Its equal base angles total 140°, so each is 70°.
3 · Your turn
An isosceles triangle has vertex angle 50°. Find the equal base angles. Must every rhombus be a square?
Check your answer
65° each. No; a rhombus need not have four right angles.
4 · Apply your learning
On equal-scale axes draw rectangle (0,0),(4,0),(4,2),(0,2) and rhombus (0,2),(3,0),(0,-2),(-3,0), joining each list in order. Compare their equal sides, parallel sides and right angles. Explain why neither is a square.
Adult guidance and safety
Require a reasoned method and an independent check. Use accurate number lines, geometric drawings and data displays where the task needs them. These text models require varied practice and diagnostic assessment before curriculum approval.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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