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parallel lines
two lines that lie in the dame plane and do not intersect each other no matter how far they are extended in both directions.
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Theorem 71
Tow perpendiculars to the same line cannot intersect no matter how far they are extended.
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test for parallel lines
- when 2 lines are intersected by a third line and it turns out that:
- 1. some corresponding angles are congruent, or
- 2. some alternate angles are congruent, or
- 3. the sum of some same side interior or same side exterior angles is 2d, then these two lines are parallel
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the parallel postulate
through a given point, one cannot draw two different lines parallel to the same line
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Angles formed by intersection of parallel lines by a transversal.
- Therom converse to 73.
- If two parallel lines are intersected by a line then:
- 1. Corresponding angles are congruent
- 2. Alternate angles are congruent
- 3. Sum of same-side interior angles is 2d, and sum of same-side exterior angles is 2d.
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Theorem 79.
Angles with respectively parallel sides.
If the sides of one angle are respectively parallel to the sides of another angle, then such angles are either congruent or add up to 2d.
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Theorem 80.
Angles with respectively perpendicular lines.
If the sides of one angle are respectively perpendicular to the sides of another one, then such angles are either congruent or add up to 2d.
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Theorem 81.
Angle sum of a polygon.
The sum of angles of a triangle is 2d.
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Theorem 81 corollaries:
- 1. Any exterior angle of a triangle is congruent to the sum of the interior angles not supplementary to it.
- 2. If two angles of one triangle are congruent respectively to two angles of another, then the remaining angles are congruent as well.
- 3. The sum of the two acute angles of a right triangle is congruent to one right angle 90 degrees.
- 4. In an isosceles right triangle, each acute angle is 1/2d, 45 degrees.
- 5. In an equilateral triangle, each angle is 2/3d, 60 degrees.
- 6. If any right triangle, one of the acute angles is 30 degrees, then the leg opposite to it is congruent to half of the hypotenuse.
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Theorem 82.
The sum of angles of a convex polygon having n sides is congruent to two right angles repeated n-2 times. 180(n-2)
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Theorem 83.
If at each vertex of a convex polygon we extend one of the sides of this angle, then the sum of the exterior angles thus formed is congruent to 4d, 360 degrees.
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Parallelogram:
A quadrilateral whose opposite side are pair-wise parallel.
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Theorem 85: Properties of sides and angles.
In any parallelogram opposite sides are congruent, opposite angles are congruent, and the sum of angles adjacent to one side is 2d, 180 degrees.
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Theorem 85 corollary 1.
If one of the angles of a parallelogram is right, then the other three are also right.
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Theorem 85 corollary 2.
- If two lines are parallel, then all points of each of them are the same distance away from the other line; in short parallel lines are everywhere the same distance apart.
- * All altitudes between the same bases of a parallelogram are congruent to each other.
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Theorem 86: Two tests for Parallelograms.
- If in a convex quadrilateral,
- 1. Opposite sides are congruent to each other, or
- 2. Two opposite sides are congruent and parallel,
- then this quadrilateral is a parallelogram
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Theorem 87: Diagonals and their property.
- 1. If a quadrilateral is a parallelogram, then its diagonals bisect each other
- 2. Vice-versa, in a quadrilateral, if the diagonals bisect each other, then this quadrilateral is a parallelogram.
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Center of Symmetry:
In a parallelogram, the intersection point of the diagonals.
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Rectangle:
- The parallelogram, all of whose angles are right
- 1. Diagonals are congruent
- 2. Has two axes of symmetry
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Rhombus:
- A parallelogram, all of whose sides are congruent,
- 1. Diagonals are perpendicular and bisect the angles of the rhombus
- 2. Each diagonal is its axes of symmetry
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Square:
- A parallelogram, all of whose sides are congruent and all of whose angles are right,
- 1. Square has four axes of symmetry
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Theorem 93: Based on properties of parallelograms.
If on one side of an angels we mark segments congruent to each other, and through their endpoints we draw parallel lines until their intersections with the other side of the angle, then the segments cut out on the side will be congruent to each other.
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Theorem 94 corollary:
The line passing through the midpoint of one side of a triangle and parallel to another side, bisects the third side.
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Theorem 95: The mid-line theorem.
The line segment connecting the mid-points of two sides of a triangle is parallel to the third side and congruent to a half of it.
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Trapezoid:
Quadrilateral which has two opposite sides parallel, and the other two opposite sides non-parallel (parallel sides are called bases and non-parallel sides are called lateral sides).
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Theorem 97: Mid-line of a trapezoid.
Line segment connecting the mid-points of the lateral sides of a trapezoid.
Mid-line of a trapezoid is parallel to the bases and congruent to their semi-sum.
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Converse Theorem:
If Q then P.
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Broken Line:
Straight segments not lying on the same line, if the endpoint of the first segment is the beginning of the second one, etc.
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Polygon:
A figure formed by non-self intersecting closed broken line.
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Altitude:
A perpendicular dropped from the vertex to the base.
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Medium:
The segment connecting the vertex of a triangle with the mid-point of the base.
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Congruent tests for triangles:
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Theorem 42:
An exterior angle of a triangle is greater than each interior angle not supplementary to it.
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Theorem 42 corollary:
If in a triangle, one angle is right or obtuse, then the other two angles are acute.
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Theorem 44: Relationships between sides and angles of a triangle.
- In any triangle,
- 1. The angles opposite to congruent sides are congruent
- 2. The angle opposite to a greater side is greater
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Theorem 48:
In a triangle, each side is smaller than the sum of the other two sides.
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Theorem 49:
The line segment connecting any two points is smaller than any broken line connecting these points.
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Theorem 50:
- If two sides of one triangle are congruent respectively to two sides of another triangle, then
- 1. The greater angle contained by these sides is opposite to the greater side
- 2. Vice-versa, the greater of the non-congruent sides is opposite to the greater angle
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Direct Theorem:
If P then Q.
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Converse Theorem:
If Q then P.
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Inverse Theorem:
If not Q then not P.
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Contrapositive Theorem:
If not Q then not P.
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Theorem 52
- If the perpendicular and some slants are drawn to a line from the same point outside this line, then:
- 1. if the feet of the slants are the same distance away from the foot of the perpendicular then such slants are congruent;
- 2. if the feet of two slants are not the same distance away from the foot of the perpendicular then the slant whose foot is farther away from the foot of the perpendicular is greater.
- ~Converse~
- If some slants and the perpendicular are drawn to a line from the same point outside this line, then;
- 1. If two slants are congruent then their feet are the same distance away from the foot of the perpendicular
- 2. If two slants are not congruent then the foot of the greater one is farther away from the foot of the perpendicular
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Congruence tests for right triangles
- right triangles are congruent
- 1. if the legs of one of them are congruent respectively to the legs of the other
- 2. if a leg and the acute angle adjacent to it in one triangle are congruent respectively to a leg and the acute angle adjacent to it in the other triangle
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Theorem 55: two tests requiring special proofs
- two right triangles are congruent:
- 1. if the hypotenuse and an acute angle of one triangle are congruent to respectively the hypotenuse and an acute angle of the other
- 2. If the hypotenuse and a leg of one triangle are congruent respectively to the hypotenuse and a leg of the other.
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