Theorems
Alternate Interior Angles Theorem- If two parallel angles are cut by a transversal, then the pair of alternate interior angles are congruent.
Since Line AB is parallel to Line CD, angle 4 and angle 6 are congruent. Also angle 3 and 5 are congruent.
Alternate Exterior Angles Theorem- If two parallel angles are cut by a transversal, then the pair of alternate exterior angles are congruent.
Since AB is parallel to line CD, angles 1 and 7 are congruent according to the alternate exterior angles theorem. Also angles 2 and 8 are congruent because they are alternate exterior angles.
Same-side Interior Angles Theorem- If two parallel angles are cut by a transversal, then the pair of same-side interior angles are supplementary.
Since line AB is parallel to line CD, angles 4 and 5 are supplementary according to the same-side interior angles theorem. Also angles 3 and 6 are supplementary.
Linear Pair Theorem -If two angles form a linear pair, then they are supplementary.
Since A and B are a linear pair they are supplementary or equal 180 degrees.
Congruent Supplements Theorem- If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent.
Angles 1 and 3 are supplementary to angle 2 so angles 1 and 3 are congruent. Angles 4 and 2 are supplementary to angle 3 and angle 1 so angles 4 and 2 are congruent.
Right Angle Congruence Theorem- All right angles are congruent.
Right angles H, E, and B are congruent because of the right angles congruence theorem.
Congruent Complements Theorem- If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent.
Angles ABC and EBD are complementary to angle CBE so angle ABC are congruent to angle EBD.
Common Segments Theorem -Given collinear points A, B, C, and D arranged on a segment such that A and D are the endpoints, B is between A and C, and C is between B and D. If , AB=CD then AC equals BD.
Since AB=CD then AC=BD.
Vertical Angles Theorem- Vertical angles are congruent.
Angles BPC and DPA are congruent according to the vertical angles theorem.
Theorem 2-7-3- If two congruent angles are supplementary, then each angle is a right angle.
Angles DBA and CBA are right because they are congruent supplementary angles.
Converse of the Alternate Interior Angles Theorem- If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel.
Since angles 3 and 5 are alternate interior angles, the lines AB and CD are parallel according to the Converse of the Alternate Interior Angles Theorem.
Converse of the Alternate Exterior Angles Theorem- If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel.
Since angles 2 and 8 are alternate exterior angles, the lines AB and CD are parallel according to the Converse of the Alternate Exterior Angles Theorem.
Converse of the Same-Side Interior Angles- Theorem If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel.
Since angles 4 and 5 are same-side interior angles, the lines AB and CD are parallel according to the Converse of the Same-Side Interior Angles Theorem.
Parallel Lines Theorem- In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.
Lines AB and CD are parallel because they have the same slope.
Perpendicular Lines Theorem-In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is −1. Vertical and horizontal lines are perpendicular.
Lines k and t are perpendicular because their slopes equal -1 when multiplied.
Theorem 12-4-1- A composition of two isometries is an isometry.
This shape went through two isometries so it is an isometry.
Theorem 12-4-2-The composition of two reflections across two parallel lines is equivalent to a translation. The translation vector is perpendicular to the lines. The length of the translation vector is twice the distance between the lines. The composition of two reflections across two intersecting lines is equivalent to a rotation. The center of rotation is the intersection of the lines. The angle of rotation is twice the measure of the angle formed by the lines.
This translation is an example of two reflectionsThis is a translation because it went through two reflections.
Theorem 12-4-3-Any translation or rotation is equivalent to a composition of two reflection.
This translation is an example of two reflections.
Postulates
Corresponding Angles Postulate- If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Angles 1 and 5 are congruent because they are corresponding. Also lines 2 and 6 are corresponding.
Converse of the Corresponding Angles Postulate- If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel.
Since angles 1 and 5 are corresponding, lines S and T are parallel.
Parallel Postulate- Through a point P not on line l, there is exactly one line parallel to l .
Through the point C there is one line parallel to line AB.