Congruence, Similarity and Circle Theorems

Congruence Tests #

SSS, SAS, ASA and RHS.

Three matching facts, in one of four patterns, prove two triangles are identical

Congruent means identical: same shape and same size, however it is turned.

Three matching sides leave a triangle no room to flex: side-side-side, SSS.

Two sides with the angle between them fix the third side: side-angle-side, SAS.

Two angles fix the shape and the side between them fixes the size: angle-side-angle, ASA.

A right angle with the hypotenuse and one side fixes one triangle: RHS.

Similarity Conditions #

Equal angles are enough, whatever the size.

Equal angles make two shapes similar even when their sizes differ

Similar means same shape, any size — and two equal angles are enough to force it.

Every pair of matching sides shares the same scale factor.

The Midpoint Theorem #

Half the base, and never meeting it.

The line joining two midpoints runs parallel to the third side, at half its length

Mark the midpoints of two sides and join them. The join runs parallel to the base.

Count it on the grid: the midline spans 2 dots across; the base spans 4 — exactly half.

Why: the top triangle is the whole at half scale — equal angles, sides in ratio 1 : 2.

Ratio of Areas #

Scale the lengths by k, the area by k squared.

Scaling every length by k scales the area by $k^2$k², not by k

A 2 by 2 square covers 4 unit squares.

Triple the sides and it covers 36 — that is $3^2=9$3² = 9 times as much.

Area grows by the square of the scale factor, never by the factor.

Ratio of Volumes #

Three dimensions, so the factor cubes.

Scaling every length by k scales the volume by $k^3$k³, not by k

A cube of side 2 holds 8 unit cubes.

Double the side and it holds 64 — that is $2^3=8$2³ = 8 times, not twice.

One dimension for lengths, two for area, three for volume.

Angle in a Semicircle #

An angle on the diameter is always a right angle.

An angle drawn on the diameter from anywhere on the edge is always a right angle

Draw a line straight through the middle. That is the diameter.

Now join both ends of it, A and B, to any other point P on the edge.

Drag P along the edge: the corner there stays a right angle, every time.

Chords, Arcs, Sectors and Segments #

Four pieces, told apart by what closes them.

A chord, an arc, a sector and a segment are four different pieces of one circle

A chord is a straight line joining two points on the edge.

The arc is the curved edge running between those very same two points.

Join both points to the center instead, and the slice between is a sector.

Cut along the chord and the piece beyond it is a segment, with no center in it.

Arcs and Sectors #

A fraction of the whole circle.

An arc or sector is just a fraction of the whole circle

A sector is a slice of the circle. This one turns 90° of the full 360°.

So it is $\frac{90}{360}$90/360 of the circle, a quarter: a quarter of the edge and of the area.

Any sector works the same way: its fraction of the circle is its angle divided by 360.

The Area of a Segment #

The sector, less the triangle inside it.

A segment is what is left when the triangle is taken out of its sector

A sector is the slice between two radii, opened out by the angle at the center.

Join the two ends with a chord. The sector is now a triangle plus the segment.

Both pieces have a formula, and the angle at the center is all they need.

A quarter circle of radius 10: 78.5 minus 50, so the segment is 28.5.

Angle at the Center #

Always twice the angle at the circumference.

The angle at the center is always twice the angle at the circumference

A and B are fixed. P sits anywhere else on the edge.

Drag P around the edge. Its angle never changes, and neither does the center one.

The center angle is exactly double: 140 at the middle, 70 at the edge.

Angles in the Same Segment #

Same arc, same side, same angle.

Angles standing on the same arc from the same side are equal

P and Q both look at the same chord AB from the same side.

Drag P anywhere on that arc. It always reads the same as Q.

It follows from the center rule: both are half of the same center angle.

Cyclic Quadrilaterals #

Opposite angles add to 180 degrees.

Opposite angles of a quadrilateral drawn in a circle add to 180 degrees

All four corners sit on the circle. That is a cyclic quadrilateral.

A and C are opposite. Whatever one is, the other makes up 180.

A looks across at the arc through B, C and D. C looks at the rest of the edge.

The two arcs make the whole 360°, and each angle is half its own: so A + C = 180.

Tangent Properties #

At a right angle to the radius, and equal in pairs.

A tangent meets the radius at a right angle and two tangents from a point are equal

A tangent touches the circle once, at A — the radius to A meets it at a right angle.

From an outside point P, two tangents touch at A and B, each at 90° to its radius.

Both right triangles share the side to the center and have equal radii, so PA = PB.

The Alternate Segment Theorem #

The tangent-chord angle crosses the circle.

The angle between a tangent and a chord equals the inscribed angle in the alternate segment

The tangent-chord angle at P and the angle at R are both marked a: they are equal.

Try a diameter chord: both angles come out at 90°. Tilting the chord keeps them equal.

Know either one of the pair and the other follows: the two angles are always equal.

Chords and the Center #

The center's perpendicular halves the chord.

The perpendicular from the center meets any chord at its exact midpoint

Drop a perpendicular from the center O onto the chord. It lands at the exact middle.

The two triangles from O share a side, a right angle and a radius — congruent by RHS.

Matching triangles mean matching halves — the two ticked parts are equal.

Equal chords sit equally far from the center: same halves, same radius, same drop.

Circle Theorems with Algebra #

The theorem writes the equation.

A circle theorem links the two expressions in one equation; then solve for x

The angle at the center is double the angle at the rim: 3x + 40 = 2(3x − 10), so x = 20.

Opposite angles of a cyclic quadrilateral add to 180°: 5x + 10 = 180, so x = 34.