Geometric Twins — Class 7 Mathematics
Ganita Prakash Part 2 · Chapter 1
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Chapter summary
This chapter is about congruence — figures that are exact copies of each other, having the same shape and size, so that one can be superimposed exactly on the other (after rotating or flipping if needed). The focus is on congruence of triangles and the minimum measurements needed to guarantee it. You discover four sufficient conditions for general triangles — SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side) and AAS (two angles and a non-included side) — and one special condition for right triangles, RHS (right angle, hypotenuse and one side). Importantly, SSA (two sides and a non-included angle) does not guarantee congruence, since two different triangles can fit those measurements. The correct correspondence of vertices is written carefully, e.g.
Congruent figures and superimpositionSSS and SAS conditionsASA and AAS conditionsRHS condition and the SSA cautionAngles of isosceles and equilateral triangles
Key ideas & terms
- Congruent figures
- Figures of the same shape and size that can be superimposed exactly on one another.
- SSS condition
- Two triangles are congruent if their three pairs of sides are equal.
- SAS condition
- Congruence from two sides and the included angle being equal.
- ASA condition
- Congruence from two angles and the included side being equal.
- AAS condition
- Congruence from two angles and a non-included side being equal.
- RHS condition
- For right triangles: congruence from the right angle, hypotenuse and one side.
- Hypotenuse
- The side opposite the right angle in a right-angled triangle.
Extra questions & answers
What does it mean for two figures to be congruent?
They have the same shape and size and can be superimposed exactly on each other.
State the SSS condition for congruence.
If the three sides of one triangle equal the three sides of another, the triangles are congruent.
State the SAS condition.
If two sides and the included angle are equal, the triangles are congruent.
State the ASA condition.
If two angles and the included side are equal, the triangles are congruent.
Does SSA guarantee congruence?
No — two non-congruent triangles can satisfy two sides and a non-included angle.
What is the RHS condition used for?
It guarantees congruence of right triangles from the right angle, hypotenuse and one side.
What is the hypotenuse?
The side opposite the right angle in a right-angled triangle.
If
In an isosceles triangle, what is true of the base angles?
The angles opposite the equal sides are equal.
What is each angle of an equilateral triangle?
Each angle is