Angles, triangles and polygons

Angle relationships on parallel lines, triangle angle sum and exterior angle, isosceles triangles, polygon angle sums and quadrilateral properties, with reasons.

Key facts

Parallel lines: alternate angles equal, corresponding angles equal, co-interior angles add to $180^\circ$.
Triangle: angles add to $180^\circ$; an exterior angle equals the sum of the two opposite interior angles.

Polygon angle sum: $(n - 2) \times 180^\circ$.
Quadrilaterals add to $360^\circ$; parallelogram opposite angles are equal; rhombus diagonals cross at right angles.

Example 1●●

In an isosceles triangle the apex angle is $40^\circ$. Find each base angle.

$(180 - 40) \div 2 = 70^\circ$ (base angles of an isosceles triangle are equal)

Example 2●●

Co-interior angles on parallel lines are $115^\circ$ and $x^\circ$. Find $x$.

$x = 180 - 115 = 65$ (co-interior angles, parallel lines)

Flashcards

Co-interior angles

Add to $180^\circ$ (parallel lines)

Exterior angle of a triangle

Sum of the two opposite interior angles

Rhombus diagonals

Bisect each other at right angles

Practice

Isosceles triangle, apex angle $40^\circ$. Each base angle?
Co-interior angles are $115^\circ$ and $x^\circ$. Find $x$.
A triangle has interior angles $70^\circ$ and $45^\circ$. Find the exterior angle at the third vertex.
Each interior angle of a regular octagon?
A quadrilateral has angles $90^\circ, 100^\circ, 85^\circ$ and $x^\circ$. Find $x$.
Which quadrilateral always has diagonals that meet at right angles?
Alternate angles on parallel lines: one is $72^\circ$. The other?
A parallelogram has one angle of $70^\circ$. Find the angle next to it.