Reaction Paper
Angles: Right Angles
All right angles are congruent. Straight Angles
All straight angles are congruent. Congruent Supplements
Supplements of the same angle, or congruent angles, are congruent.
Congruent Complements
Complements of the same angle, or congruent angles, are congruent.
Linear Pair
If two angles form a linear pair, they are supplementary. Vertical Angles
Vertical angles are congruent. Triangle Sum
The sum of the interior angles of a triangle is 180º. Exterior Angle
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
The measure of an exterior angle of a triangle is greater than either non-adjacent interior angle.
Base Angle Theorem
(Isosceles Triangle)
If two sides of a triangle are congruent, the angles opposite these sides are congruent.
Base Angle Converse
(Isosceles Triangle)
If two angles of a triangle are congruent, the sides opposite these angles are congruent.
Angle Postulates
Angle Addition Postulate
If a point lies on the interior of an angle, that angle is the sum of two smaller angles with legs that go through the given point.
Consider the figure below in which point T lies on the interior of ?QRS. By this postulate, we have that ?QRS = ?QRT + ?TRS. We have actually applied this postulate when we practiced finding the complements and supplements of angles in the previous section.
Corresponding Angles Postulate
If a transversal intersects two parallel lines, the pairs of corresponding angles are congruent.
Converse also true: If a transversal intersects two lines and the corresponding angles are congruent, then the lines are parallel.
The figure above yields four pairs of corresponding angles.
Parallel Postulate
Given a line and a point not on that line, there exists a unique line through the point parallel to the given line.
The parallel postulate is what sets Euclidean geometry apart from non-Euclidean geometry.
All right angles are congruent. Straight Angles
All straight angles are congruent. Congruent Supplements
Supplements of the same angle, or congruent angles, are congruent.
Congruent Complements
Complements of the same angle, or congruent angles, are congruent.
Linear Pair
If two angles form a linear pair, they are supplementary. Vertical Angles
Vertical angles are congruent. Triangle Sum
The sum of the interior angles of a triangle is 180º. Exterior Angle
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
The measure of an exterior angle of a triangle is greater than either non-adjacent interior angle.
Base Angle Theorem
(Isosceles Triangle)
If two sides of a triangle are congruent, the angles opposite these sides are congruent.
Base Angle Converse
(Isosceles Triangle)
If two angles of a triangle are congruent, the sides opposite these angles are congruent.
Angle Postulates
Angle Addition Postulate
If a point lies on the interior of an angle, that angle is the sum of two smaller angles with legs that go through the given point.
Consider the figure below in which point T lies on the interior of ?QRS. By this postulate, we have that ?QRS = ?QRT + ?TRS. We have actually applied this postulate when we practiced finding the complements and supplements of angles in the previous section.
Corresponding Angles Postulate
If a transversal intersects two parallel lines, the pairs of corresponding angles are congruent.
Converse also true: If a transversal intersects two lines and the corresponding angles are congruent, then the lines are parallel.
The figure above yields four pairs of corresponding angles.
Parallel Postulate
Given a line and a point not on that line, there exists a unique line through the point parallel to the given line.
The parallel postulate is what sets Euclidean geometry apart from non-Euclidean geometry.