Question bank — Triangles — scalene, isosceles, equilateral; acute, right, obtuse

True — isosceles means at least two equal sides, and "all three equal" satisfies "at least two". Equilateral is the special, most symmetric case of isosceles.

False — isosceles only guarantees two equal sides; the third can be any length allowed by the Triangle Inequality, e.g. sides $5,5,6$, which is not equilateral.

True — the $45°\text{-}45°\text{-}90°$ triangle has two equal legs and one $90°$ angle, so it wears the "right" label from the angle box and "isosceles" from the side box.

False — an equilateral triangle forces every angle to $60°$ (three equal angles summing to $180°$), so it can never contain a $90°$ angle.

True — all three angles are exactly $60°$, and $60°<90°$, so every angle is acute by definition.

True — "scalene" only says the sides differ; the angles are free to be all-small (acute), one right, or one large (obtuse). Side-class and angle-class are independent.

False — obtuse needs an angle above $90°$, but equilateral fixes all angles at $60°$; the two conditions cannot coexist.

True — a larger side always faces a larger angle (see the stretch figure above), so distinct sides mean distinct opposite angles; equal angles would demand equal opposite sides.

True — the third angle is $180°-60°-60°=60°$, so all angles are $60°$, and equal angles force equal sides (Congruence Criteria (SSS, SAS, ASA) via equal-angles ⇒ equal-sides).

False — from the Pythagorean Theorem, $c^2=a^2+b^2>a^2$ and $>b^2$, so $c$ beats both legs; the hypotenuse is always the longest side.

Error — $2+3=5<7$ violates the Triangle Inequality $a+b>c$; the two short sides cannot reach across the long one, so no triangle exists at all, thin or otherwise.

Error — it's isos (equal) + skelos (leg), meaning two equal legs, not all three; "all three equal" is the separate word equilateral.

Error — $100°+95°=195°>180°$ already, leaving a negative third angle. A triangle can hold at most one obtuse angle.

Error — the hypotenuse is the side opposite the $90°$ angle, not whichever side happens to be horizontal; misnaming it wrecks every subsequent calculation.

Error — acute (angle box) and isosceles (side box) are independent labels; an equilateral triangle is acute and isosceles simultaneously, so "acute" never rules out equal sides.

Error — those two angles alone sum to $185°>180°$, impossible. A single triangle holds at most one non-acute angle.

Error — a single line of symmetry only proves two matching sides, i.e. isosceles; equilateral needs three such mirror lines (all three vertices interchangeable), a stronger condition the eye can't confirm from one axis.

Because stretching a side forces the corner across from it to open wider (see the stretch figure above), while its own end-corners are unaffected; so the ordering of sides and of opposite angles rise and fall together.

Walking the perimeter you make three turns totalling one full rotation ($360°$); each exterior turn is $180°$ minus its interior angle, so $3(180°)-360°=180°$ of interior angle remains (see the walk figure above). Related: Interior Angles of Polygons.

A right triangle spends exactly $90°$ on one angle; an obtuse triangle already spends more than $90°$ on one angle, and having both would exceed $180°$ before the third angle is counted.

The two equal sides and shared altitude give matching hypotenuses and a common right-angle side, so by Congruence Criteria (SSS, SAS, ASA) the halves are congruent — which is why the altitude also bisects the base.

Full three-fold symmetry means each vertex sees an identical picture, so every "centre line" from a vertex plays all four roles at once — a symmetry no less-equal triangle has.

No — the three points are collinear, so the figure has zero area and no genuine corners; the definition demands three non-collinear vertices.

No — with $a=4,b=4,c=8$ we get $a+b=8$, not greater than $c=8$; the two sides lie flat along the third, collapsing to a line, so the Triangle Inequality fails at its boundary.

The right triangle — as the largest angle grows from below $90°$ up through $90°$ and beyond, the triangle passes acute → right → obtuse, with the right triangle marking $c^2=a^2+b^2$ exactly.

If $c^2<a^2+b^2$ it's acute, if $c^2=a^2+b^2$ it's right (Pythagorean Theorem), and if $c^2>a^2+b^2$ it's obtuse — the longest side's square tells you where the biggest angle sits relative to $90°$.

No — a $0°$ apex folds the two equal sides onto each other, giving base angles of $90°$ each and zero area, so it's a collapsed segment, not a triangle.