Congruence, Similarity and Circle Theorems
Stage 10 of 23 Strand 4 of 9 16 lessons
16 illustrated lessons, each teaching the why before the how.
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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 fix a triangle completely: 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, the hypotenuse and one other side fix the triangle: right angle-hypotenuse-side, RHS.
Now you
In two triangles, all three angles match, and no side is known. Are the triangles congruent?
Which congruence test is right angle, hypotenuse and one side?
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Similarity Conditions #
Equal angles are enough, whatever the size.
Equal angles make two triangles similar even when their sizes differ
Similar means the same shape at any size, and for triangles two equal angles are enough to prove it.
Every pair of matching sides shares the same scale factor.
Now you
Two similar triangles have scale factor 2. Which length matches a side of 5?
Two similar triangles have scale factor 5. Which length matches a side of 3?
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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 is 2 units long and the base is 4, exactly half.
The reason: the top triangle is similar to the whole triangle at half scale, with sides in the ratio 1 : 2.
Now you
The midline is 9 cm long. How long is the side it is parallel to?
The third side is 12 cm. How long is the midline joining the midpoints of the other two sides?
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Ratio of Areas #
Scale the lengths by k, the area by k squared.
Scaling every length by k scales the area by , not by k
A 2 by 2 square covers 4 unit squares.
Triple the sides and it covers 36 — that is times as much.
Area grows by the square of the scale factor, never by the factor.
Now you
Two similar shapes have scale factor 2. What does an area of 7 become?
Two similar shapes have scale factor 4. What does an area of 5 become?
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Ratio of Volumes #
Three dimensions, so the factor cubes.
Scaling every length by k scales the volume by , not by k
A cube of side 2 holds 8 unit cubes.
Double the side and it holds 64 — that is times, not twice.
One dimension for lengths, two for area, three for volume.
Now you
Two similar solids have scale factor 2. What does a volume of 9 become?
Two similar solids have scale factor 2. What does a volume of 5 become?
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Angle in a Semicircle #
An angle on the diameter is always a right angle.
An angle standing on a diameter, from any point on the circle, is a right angle
A straight line through the center, from one side of the circle to the other, is a diameter.
Now join both ends of it, A and B, to any other point P on the circle.
Drag P around the circle: the angle at P stays a right angle, every time.
Now you
A triangle stands on a diameter, and one angle is 55°. What is the third angle?
A triangle stands on a diameter, and one angle is 35°. What is the third angle?
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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 circle.
An arc is the part of the circle between those same two points.
Join both points to the center, and the region between the two radii and the arc is a sector.
The chord cuts off a region between itself and the arc: that is a segment.
Now you
What is the marked part of this circle called?
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Arcs and Sectors #
A fraction of the whole circle.
An arc or a sector is a fraction of the whole circle
A sector is the part of a circle between two radii. This one has an angle of 90° out of the full 360°.
So it is of the circle, a quarter: a quarter of the circumference and of the area.
Any sector works the same way: its fraction of the circle is its angle divided by 360.
Now you
A sector has angle 180° and radius 6. What is its area?
A sector has angle 90° and radius 8. How long is its arc?
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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 region between two radii and an arc, set 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.
For a quarter circle of radius 10, the segment is 78.5 − 50 = 28.5.
Now you
The radius is 8 cm and the angle at the center is 90°. What is the area of the segment, to 1 decimal place?
A sector of area 28.3 cm² holds a triangle of area 18 cm². What is the area of the segment?
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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, and P is any other point on the circle.
Drag P around the circle: the angle at P never changes, and neither does the angle at the center.
The angle at the center is exactly double: 140° at the center, 70° at the circumference.
Now you
The angle at the center is 54°. What is the angle at the circumference?
The angle at the circumference is 26°. What is the angle at the center?
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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 stand on the same chord AB, on the same side of it.
Drag P anywhere on that arc: the angle at P always equals the angle at Q.
It follows from the angle at the center: both are half of the same angle at the center.
Now you
P and Q stand on the same arc. The angle at P is 69°. What is the angle at Q?
P and Q stand on the same arc. The angle at P is 52°. What is the angle at Q?
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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 angles: whatever A is, C makes the total 180°.
Angle A stands on the arc through B, C and D, and angle C stands on the rest of the circle.
The two arcs make the whole 360°, and each angle is half its own: so A + C = 180.
Now you
A cyclic quadrilateral has one angle of 137°. What is the opposite angle?
A cyclic quadrilateral has one angle of 125°. What is the opposite angle?
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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.
Now you
One tangent from an outside point is 14. How long is the other?
One tangent from an outside point is 7. How long is the other?
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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.
If the chord is a diameter, both angles are 90°, and as the chord turns the two angles stay equal.
Know either one of the pair and the other follows: the two angles are always equal.
Now you
The angle between a tangent and a chord equals the angle…?
The tangent-chord angle at P is 55°. What is the angle at R, in the alternate segment?
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Chords and the Center #
The center's perpendicular halves the chord.
The perpendicular from the center meets any chord at its exact midpoint
Draw the perpendicular from the center O to the chord: it meets the chord at its midpoint.
The two triangles from O share a side, a right angle and a radius — congruent by RHS.
Congruent triangles have equal sides, so the two ticked halves are equal.
Equal chords are the same distance from the center: the same half-length and radius give the same perpendicular distance.
Now you
Which pair of triangles proves the two halves equal?
How far from the center do two equal chords of one circle sit?
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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.
Now you
Opposite angles of a cyclic quadrilateral are 3x − 20 and 2x + 30. Find x.
The angle at the center is x + 80 and the angle at the rim is 3x − 10. Find x.
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