PSHS-WV will suspend classes on 17 July through 24 July. During these days the school will enforce precautionary measures against Dengue, 3 Level-3 cases of which were recorded in Brgy. Bito-on proper, some 200-m from the school grounds.
All dormer students are encouraged to go home.
Classes will resume, 27-July, Monday.
Wednesday, July 15, 2009
Monday, July 6, 2009
Maths 2: Angle Pairs and Perpendicularity
Definition of Complementary Angles
Two angles are complementary if and only if the sum of their measures is 90.
Definition of Supplementary Angles
Two angles are supplementary if and only if the sum of their measures is 180.
Definition of Linear Pairs
Two angles are linear pairs if and only if they are adjacent angles whose uncommon sides are opposite rays.
Linear Pair Postulate
If two angles form a linear pair then they are supplementary.
Definition of Vertical Angles
Two angles are vertical angles if and only if they are non-adjacent angles formed by two intersecting lines.
Vertical Angle Theorem (VAT)
Vertical angles are always congruent.
Definition of Perpendicular Lines, Line-segments and Rays
Two lines are perpendicular if and only if they intersect to form right angles. Two line-segments or rays are perpendicular if and only if they have a point in common and the lines they determine form right angles.
Definition of Perpendicular Lines to a Plane
A line is perpendicular to a plane at point X if and only if it is perpendicular to every line in the plane that passes through X.
Definition of Perpendicular Planes
Two planes are perpendicular if and only if there is a line lying on one plane which is perpendicular to the other plane.
Theorem
If two lines are perpendicular, then the pairs of adjacent angles they form are congruent.
Corollary
If two lines are perpendicular, then all the four angles formed by the intersection are congruent.
Corollary
If two lines are perpendicular, then all the four angles formed by the intersection are right angles.
Theorem
If two lines intersect to form a pair of congruent adjacent angles then the lines are perpendicular.
Definition of Perpendicular Bisector
A line, ray, line-segment or plane is a perpendicular bisector of a line-segment if and only if the line, ray, line-segment or plane is perpendicular to the segment at its midpoint.
Theorem
If there is any given point on a line in a plane, then there is exactly one line in that plane perpendicular to the given line at the given point.
Corollary
If there is any line-segment in a plane then in that plane there is exactly one line that is a perpendicular bisector of the line-segment.
Theorem
If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Theorem
If there is a point not on a line, then there is exactly one line perpendicular to the given line passing through the given point.
Two angles are complementary if and only if the sum of their measures is 90.
Definition of Supplementary Angles
Two angles are supplementary if and only if the sum of their measures is 180.
Definition of Linear Pairs
Two angles are linear pairs if and only if they are adjacent angles whose uncommon sides are opposite rays.
Linear Pair Postulate
If two angles form a linear pair then they are supplementary.
Definition of Vertical Angles
Two angles are vertical angles if and only if they are non-adjacent angles formed by two intersecting lines.
Vertical Angle Theorem (VAT)
Vertical angles are always congruent.
Definition of Perpendicular Lines, Line-segments and Rays
Two lines are perpendicular if and only if they intersect to form right angles. Two line-segments or rays are perpendicular if and only if they have a point in common and the lines they determine form right angles.
Definition of Perpendicular Lines to a Plane
A line is perpendicular to a plane at point X if and only if it is perpendicular to every line in the plane that passes through X.
Definition of Perpendicular Planes
Two planes are perpendicular if and only if there is a line lying on one plane which is perpendicular to the other plane.
Theorem
If two lines are perpendicular, then the pairs of adjacent angles they form are congruent.
Corollary
If two lines are perpendicular, then all the four angles formed by the intersection are congruent.
Corollary
If two lines are perpendicular, then all the four angles formed by the intersection are right angles.
Theorem
If two lines intersect to form a pair of congruent adjacent angles then the lines are perpendicular.
Definition of Perpendicular Bisector
A line, ray, line-segment or plane is a perpendicular bisector of a line-segment if and only if the line, ray, line-segment or plane is perpendicular to the segment at its midpoint.
Theorem
If there is any given point on a line in a plane, then there is exactly one line in that plane perpendicular to the given line at the given point.
Corollary
If there is any line-segment in a plane then in that plane there is exactly one line that is a perpendicular bisector of the line-segment.
Theorem
If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Theorem
If there is a point not on a line, then there is exactly one line perpendicular to the given line passing through the given point.
Thursday, July 2, 2009
Maths 2: Angles and Angle Measurement
Definition of an Angle
A figure is an angle if and only if it is the union of two non-collinear rays, the sides, with a common endpoint, the vertex.
Definition of Adjacent Angles
Two angles are said to be adjacent if and only if they satisfy the following three conditions:
1. they have a common vertex
2. they have a common side
3. they DO NOT have common interior points
Angle Measure Postulate (AMP)
Given an angle, there is a unique real number BETWEEN 0 and 180 known as its degree measure. (important: this explains why there is no 0 and 180 angles)
Protractor Postulate
In a half-plane with edge, line AB, and any point S between A and B, there is a one-to-one correspondence between the rays that originate at S in that half-plane and the real numbers BETWEEN 0 and 180. The angle measure is taken as the absolute value between the two corresponding real numbers.
Theorem
In a half-plane, through the endpoint of a ray lying in the edge of the half-plane, there is EXACTLY one other ray such that the angle formed by the two rays has a given measure between 0 and 180.
Definition of Angle Congruence
Two angles are said to be congruent if and only if their measures are equal.
Theorem
All right angles are congruent.
Definition of an Angle Bisector
Given three coplanar rays, OA, OT and OB, ray OT is the angle bisector if and only if angle AOT is congruent to angle TOB
A figure is an angle if and only if it is the union of two non-collinear rays, the sides, with a common endpoint, the vertex.
Definition of Adjacent Angles
Two angles are said to be adjacent if and only if they satisfy the following three conditions:
1. they have a common vertex
2. they have a common side
3. they DO NOT have common interior points
Angle Measure Postulate (AMP)
Given an angle, there is a unique real number BETWEEN 0 and 180 known as its degree measure. (important: this explains why there is no 0 and 180 angles)
Protractor Postulate
In a half-plane with edge, line AB, and any point S between A and B, there is a one-to-one correspondence between the rays that originate at S in that half-plane and the real numbers BETWEEN 0 and 180. The angle measure is taken as the absolute value between the two corresponding real numbers.
Theorem
In a half-plane, through the endpoint of a ray lying in the edge of the half-plane, there is EXACTLY one other ray such that the angle formed by the two rays has a given measure between 0 and 180.
Definition of Angle Congruence
Two angles are said to be congruent if and only if their measures are equal.
Theorem
All right angles are congruent.
Definition of an Angle Bisector
Given three coplanar rays, OA, OT and OB, ray OT is the angle bisector if and only if angle AOT is congruent to angle TOB
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