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Vital Geometry Theorems
You ought to be very familiar with and, if applicable, prove the following theorems.
Congruent Complements / Supplements Theorem: If 2 angles are complementary or supplementary to the same angle (or to congruent angles) then they are congruent.
Vertical angles Theorem: Vertical angles are congruent.
Parallel Line Theorems:
- Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate interior angles are congruent.
- Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate exterior angles are congruent.
- Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of consecutive interior angles are supplementary.
- Alternate Interior Angles Converse Theorem: If two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.
- Alternate Exterior Angles Converse Theorem: If two lines are cut by a transversal such that the alternate exterior angles are congruent, then the lines are parallel.
- Consecutive Interior Angles Converse Theorem: If two lines are cut by a transversal such that the consecutive interior angles are supplementary, then the lines are parallel.
Triangle Theorems:
- Triangle Sum Theorem: The sum of the measures of the interior angles is 180 degrees.
- Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two nonadjacent interior angles.
- 3rd Angles Theorem: If two angles of 1 triangle are congruent to two angles of another, then the 3rd angle is also congruent.
- Angle-Angle-Side (AAS) Congruence Theorem: If two angles and an non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
- Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite of them are congruent.
- Base Angles Converse Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent.
- Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the end points of the segment.
- Perpendicular Bisector Theorem Converse: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.
- Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
- Angle Bisector Converse: If a point is in the interior of an angle, equidistant from its sides, then it lies on the bisector of the angle.
- Mid Segment Theorem: The segments connecting the midpoints of two sides of a triangle is parallel to the third side and half the third side.
- Triangle Inequality: If one side of a triangle is longer than another side, then the angle opposite the longest side is larger than the angle opposite of the shorter side.
Parallelogram Theorems:
- If a quadrilateral is a parallelogram, then its opposite sides are congruent.
- If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
- If a quadrilateral is a parallelogram, then its opposite angles are congruent.
- If a quadrilateral is a parallelogram, then its diagonals bisect each other.
- If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
- If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
- If an angle of a quadrilateral is supplementary to both its consecutive angles, then the quadrilateral is a parallelogram.
- If a parallelogram is a rhombus, then its diagonals are perpendicular.
- If the diagonals of a parallelogram are perpendicular, then its a rhombus.
Similarity Theorems:
- SSS Similarity Theorem: If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.
- SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the length of the sides including these angles are proportional, then the triangles are similar.
- Triangle Proportionality Theorem: If a line parallel to one side of a triangle, intersects the other two sides, then it divides 2 sides proportionally.
- Equal Products Theorem: If 2 secants intersect outside a circle, the product of the lengths of one of the secants and its external segment equals the product of the lengths of the other secant and its external segment.
- Corresponding angle bisectors of 2 triangles have the same ratio as a pair of corresponding sides.
- Corresponding medians of 2 similar triangles have the same ratio as a pair of corresponding sides.
- If 2 chords intersect within a circle, the product of the lengths of the segments of 1 chord equals the product of the lengths of the segments of the other.
- If a tangent and a secant intersect outside a circle, then the length of the tangent is a geometric mean of the length of the secant and its external segment.
- If 2 secants intersect outside a circle, then the product of the lengths of one of the secants and its external segment equals the length of the other secant and its external segment.
- When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean of the lengths of the segments of the hypotenuse.
- When the altitude is drawn to the hypotenuse of a right triangle, the length of each leg is the geometric mean of the adjacent segment on the hypotenuse and the length of the hypotenuse.
The Pythagorean Theorem
Circle Theorems:
- If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.
- If two segments from the same exterior point are tangents to a circle, then they are congruent.
- If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
- If two inscribed angles of a circle intersect the same arc, then the angles are congruent.
- The angle in a semicircle is a right angle (if one side of an inscribed triangle is the diameter of a circle, then the triangle is a right triangle).
- If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
- Chords equidistant from the centre of a circle are congruent.
- In a circle, congruent chords are equidistance from the centre.
- In a circle, a diameter that is perpendicular to a chord bisects the chords and its arcs.
- In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord.
- In a circle, the perpendicular bisector of a chord contains the centre of the circle.
- The opposite angles of a quadrilateral inscribed in a circle are supplementary.
- The measure of an angle formed by a tangents and a chord is half the measure of the intercepted arc.
- The measure of an angle formed by 2 lines that intersect inside a circle is half the sum of the measures of its intercepted arcs.
- The measure of an angle formed by 2 lines that intersect outside a circle is half the difference of the measures of its intercepted arcs.
Polygon Angle Theorems:
* The sum of the interior angles of an n-sided polygon is (n-2)180 degrees.
* The sum of the exterior angles of an n-sided polygon is 360 degrees.
You ought to be very familiar with and, if applicable, prove the following theorems.
Congruent Complements / Supplements Theorem: If 2 angles are complementary or supplementary to the same angle (or to congruent angles) then they are congruent.
Vertical angles Theorem: Vertical angles are congruent.
Parallel Line Theorems:
- Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate interior angles are congruent.
- Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate exterior angles are congruent.
- Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of consecutive interior angles are supplementary.
- Alternate Interior Angles Converse Theorem: If two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.
- Alternate Exterior Angles Converse Theorem: If two lines are cut by a transversal such that the alternate exterior angles are congruent, then the lines are parallel.
- Consecutive Interior Angles Converse Theorem: If two lines are cut by a transversal such that the consecutive interior angles are supplementary, then the lines are parallel.
Triangle Theorems:
- Triangle Sum Theorem: The sum of the measures of the interior angles is 180 degrees.
- Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two nonadjacent interior angles.
- 3rd Angles Theorem: If two angles of 1 triangle are congruent to two angles of another, then the 3rd angle is also congruent.
- Angle-Angle-Side (AAS) Congruence Theorem: If two angles and an non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
- Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite of them are congruent.
- Base Angles Converse Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent.
- Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the end points of the segment.
- Perpendicular Bisector Theorem Converse: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.
- Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
- Angle Bisector Converse: If a point is in the interior of an angle, equidistant from its sides, then it lies on the bisector of the angle.
- Mid Segment Theorem: The segments connecting the midpoints of two sides of a triangle is parallel to the third side and half the third side.
- Triangle Inequality: If one side of a triangle is longer than another side, then the angle opposite the longest side is larger than the angle opposite of the shorter side.
Parallelogram Theorems:
- If a quadrilateral is a parallelogram, then its opposite sides are congruent.
- If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
- If a quadrilateral is a parallelogram, then its opposite angles are congruent.
- If a quadrilateral is a parallelogram, then its diagonals bisect each other.
- If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
- If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
- If an angle of a quadrilateral is supplementary to both its consecutive angles, then the quadrilateral is a parallelogram.
- If a parallelogram is a rhombus, then its diagonals are perpendicular.
- If the diagonals of a parallelogram are perpendicular, then its a rhombus.
Similarity Theorems:
- SSS Similarity Theorem: If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.
- SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the length of the sides including these angles are proportional, then the triangles are similar.
- Triangle Proportionality Theorem: If a line parallel to one side of a triangle, intersects the other two sides, then it divides 2 sides proportionally.
- Equal Products Theorem: If 2 secants intersect outside a circle, the product of the lengths of one of the secants and its external segment equals the product of the lengths of the other secant and its external segment.
- Corresponding angle bisectors of 2 triangles have the same ratio as a pair of corresponding sides.
- Corresponding medians of 2 similar triangles have the same ratio as a pair of corresponding sides.
- If 2 chords intersect within a circle, the product of the lengths of the segments of 1 chord equals the product of the lengths of the segments of the other.
- If a tangent and a secant intersect outside a circle, then the length of the tangent is a geometric mean of the length of the secant and its external segment.
- If 2 secants intersect outside a circle, then the product of the lengths of one of the secants and its external segment equals the length of the other secant and its external segment.
- When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean of the lengths of the segments of the hypotenuse.
- When the altitude is drawn to the hypotenuse of a right triangle, the length of each leg is the geometric mean of the adjacent segment on the hypotenuse and the length of the hypotenuse.
The Pythagorean Theorem
Circle Theorems:
- If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.
- If two segments from the same exterior point are tangents to a circle, then they are congruent.
- If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
- If two inscribed angles of a circle intersect the same arc, then the angles are congruent.
- The angle in a semicircle is a right angle (if one side of an inscribed triangle is the diameter of a circle, then the triangle is a right triangle).
- If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
- Chords equidistant from the centre of a circle are congruent.
- In a circle, congruent chords are equidistance from the centre.
- In a circle, a diameter that is perpendicular to a chord bisects the chords and its arcs.
- In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord.
- In a circle, the perpendicular bisector of a chord contains the centre of the circle.
- The opposite angles of a quadrilateral inscribed in a circle are supplementary.
- The measure of an angle formed by a tangents and a chord is half the measure of the intercepted arc.
- The measure of an angle formed by 2 lines that intersect inside a circle is half the sum of the measures of its intercepted arcs.
- The measure of an angle formed by 2 lines that intersect outside a circle is half the difference of the measures of its intercepted arcs.
Polygon Angle Theorems:
* The sum of the interior angles of an n-sided polygon is (n-2)180 degrees.
* The sum of the exterior angles of an n-sided polygon is 360 degrees.