Angles and Geometry

An obtuse angle is more than 90°. Two of them together must be more than $90^\circ + 90^\circ = 180^\circ$, so they can never sum to exactly 180°. (Supplementary pairs are one acute and one obtuse, or two right angles.)

The angle opens past 90° but is well short of 180°, so it is obtuse. The arc is drawn on the inside of the angle, not the reflex side.

Complementary angles add to 90°, so the complement of 35° is $90^\circ-35^\circ=55^\circ$. (Supplementary angles, by contrast, add to 180°.)

Supplementary angles add to 180°, so the supplement of 110° is $180^\circ-110^\circ=70^\circ$.

Vertically opposite angles are always equal, so the angle opposite the 65° angle is also $65^\circ$. (The angle beside it would be $115^\circ$, since they are supplementary.)

Angles on a straight line add to 180°, so the other angle is $180^\circ-130^\circ=50^\circ$.

Angles at a point add to 360°, so the third angle is $360^\circ-110^\circ-90^\circ=160^\circ$. Use 360° for a full turn, not 180°.

The angles of a triangle add to 180°, so the third angle is $180^\circ-65^\circ-80^\circ=35^\circ$.

The three angles add to 180°, so the two equal base angles together are $180^\circ-40^\circ=140^\circ$. Each base angle is $\dfrac{140^\circ}{2}=70^\circ$.

The three angles add to 180°. The right angle at C is 90° and $\angle BAC = 35^\circ$, so $\angle ABC = 180^\circ-90^\circ-35^\circ = 55^\circ$. (The two acute angles in a right triangle are complementary.)

Equal sides force equal angles, so all three angles are equal. Since they sum to $180^\circ$, each is $\dfrac{180^\circ}{3} = 60^\circ$. (A triangle cannot be equilateral and also have a right angle or an obtuse angle.)

A triangle is classified by its largest angle. Here 100° is greater than 90°, so it is an obtuse-angled triangle. Equilateral and scalene are side-based names, not angle-based.

The angles of a quadrilateral add to 360°, so the fourth angle is $360^\circ-80^\circ-95^\circ-110^\circ=75^\circ$. (A triangle adds to 180°; a quadrilateral adds to 360°.)

A rectangle only needs opposite sides equal, not all four. A rectangle with all four sides equal is a square, so "all four sides equal" is not a property of every rectangle.

A rhombus is defined by having all four sides equal. Its angles are not generally 90° and its diagonals are not generally equal, as those would make it a square.

A parallelogram has both pairs of opposite sides parallel and equal in length. The other statements describe more specific shapes (rhombus, rectangle or square).

All four sides equal makes it a rhombus, and all four angles 90° makes it a rectangle. The shape that is both at once is a square.

An equilateral triangle has 3 lines of symmetry, one from each vertex through the midpoint of the opposite side. (An isosceles triangle that is not equilateral has 1; a scalene triangle has 0.)

A square looks identical after a turn of 90°, 180°, 270° or 360° about its centre, which is four positions in a full turn. So the order of rotational symmetry is 4. (90° is the angle of rotation, not the order.)

The two parts make 9 units in total and add to 180°, so one unit is $\dfrac{180^\circ}{9}=20^\circ$. The larger angle has 7 units: $7\times 20^\circ=140^\circ$. (The smaller is $2\times 20^\circ=40^\circ$.)

Let $\angle B = x$. Then $\angle A = 2x$. The angles sum to 180°: $2x + x + 30^\circ = 180^\circ$, so $3x = 150^\circ$ and $x = 50^\circ$. Therefore $\angle A = 2\times 50^\circ = 100^\circ$.

By the exterior angle theorem, an exterior angle equals the sum of the two non-adjacent interior angles: $50^\circ+60^\circ=110^\circ$. (Check: the third interior angle is $180^\circ-50^\circ-60^\circ=70^\circ$, and $70^\circ+110^\circ=180^\circ$.)

The exterior angle at A and $\angle BAC$ sit on a straight line, so $\angle BAC = 180^\circ-120^\circ = 60^\circ$. The triangle angles add to 180°: $60^\circ+35^\circ+\angle BCA = 180^\circ$, giving $\angle BCA = 85^\circ$. (Or use the exterior angle theorem: $120^\circ = 35^\circ + \angle BCA$.)

A rectangle has equal diagonals but not perpendicular; a rhombus has perpendicular diagonals but not equal. A square has both, so it is the only one. Parallelograms and trapeziums have neither in general.

Since $AB = AC$, the angles opposite them are equal, so $\angle B = \angle C$. Let $\angle A = x$, then $\angle B = \angle C = x + 30^\circ$. The sum is 180°: $x + (x+30^\circ) + (x+30^\circ) = 180^\circ$, so $3x + 60^\circ = 180^\circ$ and $x = 40^\circ$. (Check: A = 40°, B = C = 70°.)