Reflection and Congruence
Geometry · Reflection and Congruence
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 5 Reflection and Congruence
Lesson objectives
By the end of this topic, you should be able to:
- State the properties of reflection as a transformation.
- Use the properties of reflection in construction and identification of images.
- Determine the images of objects under reflection in given lines.
- State and apply the conditions for congruence of triangles.
Reflection and Congruence
A reflection flips a figure across a line, producing a mirror image.
a) Properties of reflection
The mirror line is the line of reflection.
Each point and its image are the same distance from the mirror line, on opposite sides.
The line joining a point to its image is perpendicular to the mirror line. The mirror bisects that joining line.
Any point on the mirror line stays where it is. Such points are invariant.
Reflection preserves length and angle, so the image is the same size and shape as the object.
But it reverses orientation. A clockwise lettering becomes anticlockwise. That is what distinguishes reflection from rotation.
b) Reflection in the Cartesian plane
Reflecting in the x-axis changes the sign of y.
Reflecting in the y-axis changes the sign of x.
Reflecting in y = x swaps the coordinates.
Reflecting in y = −x swaps them and changes both signs.
Learn these four. They cover almost every examination question.
c) Constructing a reflection
From each object point, drop a perpendicular to the mirror line.
Measure the distance to the line, then continue the same distance beyond it.
Mark the image point there.
Join the image points in the same order as the object.
d) Finding the mirror line
Given an object and its image, join each point to its image.
The mirror line is the perpendicular bisector of any of those joining lines.
Construct it with compasses, and check with a second pair of points.
e) Congruence
Two figures are congruent if they have the same shape and the same size.
Reflection, rotation and translation all produce congruent images. They change position or orientation, never size.
f) Conditions for congruent triangles
| Condition | Meaning |
|---|---|
| SSS | three sides equal |
| SAS | two sides and the included angle |
| ASA | two angles and the included side |
| AAS | two angles and a non-included side |
| RHS | right angle, hypotenuse and one side |
SAS requires the angle to lie between the two sides. An angle elsewhere does not prove congruence.
SSA is not a valid condition, except in the special right-angled case RHS. Two triangles can share two sides and a non-included angle and still differ.
AAA proves similarity, not congruence. Equal angles fix the shape but not the size.
g) Where this is used
Symmetry in design and architecture. Manufacturing identical parts. Proving geometric results. Pattern making in textiles.
Words to know
- Transformation -- a rule that changes the position, size or orientation of a figure.
- Mirror line -- the line across which a reflection takes place.
- Invariant point -- a point that does not move under a transformation.
- Congruent -- having exactly the same size and shape.
- Included angle -- the angle lying between two named sides.
:::checkpoint Check yourself
- Find the image of (4, −3) after reflection in the x-axis.
- Find the image of (2, 5) after reflection in the line y = x.
- Which property does reflection reverse that rotation does not?
- Why is AAA not a condition for congruence? :::
Bridge to practice
The exercises begin with the properties of reflection and invariant points, move through coordinate reflections and the meaning of congruence, and finish with the four tests and their use in proof. For every congruence claim, name the test and check that any angle used in SAS truly lies between the two sides.