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To bisect a given finite straight line. Result: A given straight line is bisected, creating two equal segments.

If in a triangle, two angles are equal to one another, then the sides opposite to the equal angles are also equal to one another. Result: A triangle's sides opposite the equal angles are proven to be congruent.

To construct an equilateral triangle on a given finite straight line. Result: An equilateral triangle is constructed.

To draw a straight line at right angles to a given straight line from a given point on it. Result: A perpendicular line is drawn from a given point on a line.

To place at a given point (as an extremity) a straight line equal to a given straight line. Result: A straight line is constructed at a point equal in length to the given line.

Given two unequal straight lines, to cut off from the greater a straight line equal to the less. Result: A segment equal to the shorter line is cut from the longer line.

The angles at the base of an isosceles triangle are equal to each other. Result: Isosceles triangle base angles are proven to be congruent.

Given two straight lines constructed from a point and at a straight angle to one another, there is no straight line that can be constructed from the point that will be at an angle smaller than a right angle to the existing lines. Result: It is shown that a straight line constructed at a point forms equal angles with the two given lines.

To bisect a given angle. Result: A given angle is bisected.

If a straight line stands on another straight line, then it makes either two right angles or angles whose sum equals two right angles. Result: It is proven that the angles are either right angles or their sum is equal to that of two right angles.

To construct a triangle out of three given straight lines which are such that two of them taken together in any manner are greater than the third. Result: A triangle is constructed from the three given lines, provided they can form a triangle.

If two triangles have two sides and the angle contained by the sides of the one equal to two sides and the contained angle of the other, the triangles will also have their bases or third sides equal. Result: Triangles are proven congruent based on the Side-Angle-Side (SAS) postulate.