Euclidean Geometry
Euclidean geometry in Bagrut is proof-based. Questions ask you to justify angle equalities, prove triangles congruent or similar, apply circle theorems, and calculate lengths or areas from a diagram described in words. Every statement needs a reason.
Core rules
- Angles in a triangle sum to 180°; an exterior angle equals the sum of the two remote interior angles.
- Isosceles triangle: equal sides imply equal base angles, and the converse holds.
- Congruence: SSS, SAS, ASA, or AAS. Similarity: AA, SSS, or SAS with proportional sides.
- In a circle, the angle at the centre is twice the inscribed angle on the same arc.
- A cyclic quadrilateral has opposite angles summing to 180°.
- A tangent is perpendicular to the radius at the point of contact; the tangent-chord angle equals the angle in the alternate segment.
- Thales: an angle subtended by a diameter is a right angle.
Worked example
In cyclic quadrilateral ABCD, angle A=70° and angle B=95°. Find angles C and D. Opposite angles in a cyclic quadrilateral are supplementary, so C=180°−70°=110° and D=180°−95°=85°. The check is that all four angles sum to 360°: 70+95+110+85=360. The reason for each step is the cyclic quadrilateral theorem, not merely the numerical subtraction.
Traps
Assuming a quadrilateral is cyclic without proof is a common gap. Matching vertices in the wrong order in a congruence or similarity statement invalidates the correspondence. Do not use a tangent-chord angle without identifying the alternate segment. A diagram is not a proof; name the theorem.