If two triangles are equiangular, then the sides that contain the equal angles are proportional, and the sides that correspond are opposite the equal angles. (Angle-Side-Angle. However, because you’re probably not currently working on your Ph.D. in geometry, you shouldn’t sweat this fine point. They are equal to two right angles.

Illustrations of Postulates 1–6 and Theorems 1–3. from your Reading List will also remove any 6.) Let AB be a diameter of a circle, and let the straight line CD be drawn at right angles to AB from its extremity B; then the straight line CD is tangent to the circle. Listed below are six postulates and the theorems that can be proven from these postulates.

(Euclid, I. Postulate 3: Through any two points, there is exactly one line. (Euclid, I. 11. Right triangles are aloof. Let the circles with centers A and D be equal, and let angles BAC, EDF be angles at the center; then, proportionally. The enunciation states the theorem in general terms.

A further condition, Side-Side-Angle, is known as the ambiguous-case. 27 of Plane Geometry.).

The specification follows.

And BD equal to DC, because they are opposite the equal angles BAD, DAC. Definitions, theorems, and postulates are the building blocks of geometry proofs. They are magnitudes of different kinds.

10. Theorem and postulate: Both theorems and postulates are statements of geometrical truth, such as All right angles are congruent or All radii of a circle are congruent. It asserts a condition for triangles to be congruent.

Some Theorems of Plane Geometry. Now here we will learn about the theorems which are covered for Class 10 syllabus. A chord of a circle is a straight line that joins any two points on the circumference. :

(Topic 3:  Trigonometry of right triangles.). The center of a circle lies on the perpendicular bisector of any chord. Let ABC be a triangle in which the angle at B is greater than the angle at C; then the side CA is greater than the side AB. A greater angle of a triangle is opposite a greater side. 11. 18. It is one of four sufficient conditions for triangles to be congruent. Let the straight line DE be tangent to the circle ABC at the point C; let F be the center of the circle, and draw the radius FC; then FC will be perpendicular to DE. Now, angles 1, 2, 3 are together equal to two right angles. When a straight line that stands on another straight line makes angles, either it makes two right angles, or it makes angles that together are equal to two right angles.

The specification restates the theorem with respect to a specific figure. The following theorem shows that for triangles to be similar, it is sufficient that they be equiangular.

Written in if-then form, the theorem All right angles are congruent would read, “If two angles are right angles, then they’re congruent.” Unlike definitions, theorems are generally not reversible. 14. Lines: Intersecting, Perpendicular, Parallel.
If two planes intersect, then their intersection is a line (Postulate 6).

Let any straight line AB stand on the straight line CD making angles ABD, ABC; then either angles ABC, ABD are two right angles, or together they are equal to two right angles. Let the straight line EF meet the two straight lines AB, CD, and let it make the alternate angles AEF, EFD equal; then AB is parallel to CD. This theorem is a partial converse of the previous one. A central angle has its vertex at the center of the circle. With reference to the figure above, if the angles at B and C are equal, then the sides that subtend, or are opposite, those angles will be equal, namely the sides AB, AC. Theorem 1: If a line is drawn parallel to one side of a triangle to intersect the midpoints of the other two sides, then the two sides are divided in the same ratio.

Let ABC be any triangle; then the three angles at A, B, and C will together equal two right angles.

The expression equal "respectively" means each one to each one.

4.) (Euclid, I. right angle will equal the squares drawn on the sides that make the right angle. That is, angles ADB, ADC are right angles, and the straight line BD is equal to the straight line DC.

A tangent is a straight line that touches a circle but does not cut it, however it may be extended. Others are "S. S. S." (Side-Side-Side. In the circle DAB let AB be any chord, and let the straight line CD be its perpendicular bisector; then the center of the circle lies on the straight line CD. To do 19 min read. That fact is the basis for measuring angles, because it is the arc that we actually measure.

16. 12. (Hence to find the center of a circle, draw two chords; draw their perpendicular bisectors; then the center of the circle will be their point of intersection.). 18.) H ERE ARE THE FEW THEOREMS that every student of trigonometry should know.. To begin with, a theorem is a statement that can be proved. Full curriculum of exercises and videos.

Theorem 13. Geometry is a very organized and logical subject. An isosceles triangle has two equal sides. Theorem 2. Leg Acute (LA) and Leg Leg (LL) Theorems. A straight line from the center to the circumference is a called a radius. Theorem 11.

4.) (Euclid, VI. Postulate 4: Through any three noncollinear points, there is exactly one plane. Definition: A definition defines or explains what a term means.

The theorem of Pythagoras. The Pythagorean proof is so simple that we will quickly show it: Through the point A, draw a straight line PQ parallel to BC, forming the Theorem 1. Let angle ABC be inscribed in the semi-circle ABC; that is, let AC be a diameter and let the vertex B lie on the circumference; then angle ABC is a right angle. Theorem 8. For the same reason, the angle at C is equal to angle 3. We shall not prove the theorems here, however. And (keeping the end points fixed) ..... the angle a° is always the same, no matter where it is on the same arc between end points:

A scalene triangle has three unequal sides. The following can be proved directly from Theorem 16: In any circles, the same ratio of arc length to radius determines a unique central angle that the arcs subtend. With very few exceptions, every justification in the reason column is one of these three things. Theorem 5.

Euclid, I.

An inscribed angle a° is half of the central angle 2a° (Called the Angle at the Center Theorem) . Let ABC be an isosceles triangle with the equal sides AB, AC; then the angles at the base, the angles at B and C, are equal. A diameter of a circle is a straight line through the center and terminating in both directions on the circumference.

While the fact of the theorem may be obvious, the proof is quite a different matter, because it requires a satisfactory definition of "have the same ratio." Postulate 5: If two points lie in a plane, then the line joining them lies in that plane.

Let ABC be a right triangle in which angle CAB is a right angle; then the square drawn on BC, opposite the right angle, will equal the two squares together on CA, AB. 5.) For we consider the entire circumference to be an arc, and in degree measure we say that its length is 360°. Therefore the arc that is a sixth of the circumference will subtend a central angle that is a sixth of 360°; it will be 60°. If a straight line that meets two straight lines makes the alternate angles equal, then the two straight lines are parallel.

Previous 13.) Theorem 5. This theorem is known briefly as "S. A. S." (Side-Angle-Side). Theorem 2 is a simple consequence of Theorem 1, and the student should be able to prove it easily. Let ABC, DEB be equiangular triangles with angle ABC equal to angle DEB, angle BCA equal to angle EBD, and finally angle CAB equal to angle BDE; then in those triangles the sides that contain those equal angles are proportional, and the side AB (opposite the angle BCA) corresponds to the side DE (opposite the equal angle EBD), and so on, for each pair of corresponding sides.

(Those from Euclid's First Book are proved here.


A secant is a straight line that cuts a circle. For example, for reason 2 in the first proof in the figure, you choose the version that goes, “If a point is the midpoint of a segment, then it divides the segment into two congruent parts,” because you already know that M is the midpoint of, (because it’s given) and from that given fact you can deduce that. The three angles of any triangle will equal two right angles. Therefore, the three angles A, B, C of the triangle are together equal to angles 1, 2, 3. The enunciation appears in italics. For example, if you reverse this right-angle theorem, you get a false statement: “If two angles are congruent, then they’re right angles.” (If a theorem works in both directions, you’ll get a separate theorem for each version. (Euclid, III. Theorems. and "A. S. 33.)

We shall not prove the theorems here, however. To begin with, a theorem is a statement that can be proved. If this had been a geometry proof instead of a dog proof, the reason column would contain if-then definitions, […] Geometry: Theorems quizzes about important details and events in every section of the book. 17. A postulate is a statement that is assumed true without proof. First, though, here are some basic definitions.

Theorem 7.