الرياضيات
Isosceles Triangle: Two Equal Sides, Two Equal Angles
An isosceles triangle has two equal sides and two equal base angles. See how one given angle produces the other two, in two lines of working.
Isosceles triangle
An isosceles triangle has two sides of equal length, and the two angles opposite those equal sides are also equal to each other.
- أين يقابله الطلاب
- Grade 3 mathematics, when triangles are first sorted by side length, and again everywhere from GCSE circle theorems to isosceles trapezia.
الإجابة باختصار
An isosceles triangle has two sides of equal length, and the two angles opposite those sides are equal to each other. It has exactly one line of symmetry, running from the apex to the midpoint of the third side. One given angle is therefore enough to find the other two.
مثال
(180° − 40°) ÷ 2 = 70°
The apex angle of an isosceles triangle is 40 degrees. Subtracting it from 180 leaves 140 degrees to be shared, and because the two base angles are equal that share is 70 degrees each. Reverse the question and the arithmetic reverses with it: a base angle of 70 degrees means 140 degrees are used at the base, so the apex is 40.
Three facts that always travel together
Two equal sides, two equal angles, one line of symmetry. These are not three separate things to memorise — the equal angles exist because folding the triangle along the line of symmetry lands one equal side exactly on the other, carrying its angle with it.
The vocabulary is worth getting right, because questions use it. The two matching sides are the legs, the odd one out is the base, the angle between the legs is the apex angle, and the two angles sitting on the base are the base angles.
The base does not have to be at the bottom. An isosceles triangle printed lying on its side, or upside down, is the same shape with the same rules, and the marks on the sides rather than the orientation say which angles are equal.
Which angle were you given?
Everything on this topic depends on one question: was the angle you were handed the apex or a base angle? The two cases need different arithmetic, and answering the wrong one is the way marks disappear here.
Given the apex, subtract it from 180 and halve what is left. Given a base angle, double it and subtract from 180. A diagram usually settles which is which — the equal sides are marked with matching dashes, and the apex is the corner where those dashes meet.
The two cases at the edges
An equilateral triangle satisfies the isosceles definition, since having three equal sides certainly means having two. Most syllabuses name it separately because 60 degrees everywhere is a stronger fact, but no rule is broken by calling it isosceles.
A right-angled isosceles triangle has the right angle at the apex, which forces both base angles to be 45 degrees. It appears constantly in exact trigonometry, because it is where the values for sin, cos and tan of 45 degrees come from.
One more place it turns up is inside every circle. Any triangle drawn from the centre to two points on the circumference has two sides that are radii, so it is isosceles automatically — and that observation is the first line of the proof of most circle theorems.
أسئلة شائعة
Can an isosceles triangle have a right angle?
Yes, and there is only one shape it can take. The right angle must be the apex, because two right angles at the base would already use up all 180 degrees. That leaves 90 degrees to be split evenly, so both base angles are 45. A right angle at the base is impossible.
Is an equilateral triangle also isosceles?
By the definition, yes — three equal sides include two equal sides. Textbooks usually list the two types separately because equilateral carries the far more useful fact that every angle is 60 degrees. If an exam asks you to name a triangle with three equal sides, write equilateral.
How many lines of symmetry does an isosceles triangle have?
Exactly one, from the apex down to the midpoint of the base, meeting the base at a right angle. It has no rotational symmetry beyond a full turn. An equilateral triangle has three lines of symmetry, which is one of the clearest ways to tell the two apart in a shape-sorting question.
How do you know two sides are equal if the diagram is not to scale?
By the dash marks, never by appearance. A single dash on two sides means those two are equal in length, and matching arcs at two corners mean those angles are equal. Exam diagrams are frequently drawn out of proportion on purpose, so measuring with a ruler is the one method guaranteed to mislead you.
آخر تحديث
معرفة الكلمة ليست كاستخدامها
يستطيع المعلّم أن يرى الطالب وهو يستعمله في سؤال، فيتبيّن أين يتوقف الفهم بالضبط. الحصة الأولى مجانية.