If a quadrilateral is a parallelogram, then its opposite angles are congruent. If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. If a quadrilateral is a parallelogram, then its opposite sides are congruent. 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. 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. 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. Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. 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. P erpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the end points of the segment. Base Angles Converse Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent. Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite of them are 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. 3rd Angles Theorem: If two angles of 1 triangle are congruent to two angles of another, then the 3rd angle is also congruent. 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. Triangle Sum Theorem: The sum of the measures of the interior angles is 180 degrees. 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. 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. 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. Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of consecutive interior angles are supplementary.

Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate exterior angles are congruent. Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate interior angles are congruent. Vertical angles Theorem: Vertical angles are congruent. (5) m∠5 + m∠4 = 180° //using (3) and (4), and performing algebraic substitution, replacing m∠1 with the equivalent m∠5Īnd we can repeat this proof for the second pair of interior angles.You ought to be very familiar with and, if applicable, prove the following theorems.Ĭongruent Complements / Supplements Theorem: If 2 angles are complementary or supplementary to the same angle (or to congruent angles) then they are congruent. (3) m∠1 = m∠5 //definition of congruent angles (2) ∠1 ≅ ∠5 //from the axiom of parallel lines – corresponding angles Here's how you prove the Consecutive Interior Angles Theorem: So let’s proceed to the proof, using what we already know about angles that are next to each other and which form a straight line. So we will try to use that here, too, since here we also need to prove that the sum of two angles is 180°. So how do we go about this? We already know that the two angles that are next to each other and which form a straight line are “ Supplementary angles” and their sum is 180°. The Consecutive Interior Angles Theorem states that the two interior angles formed by a transversal line intersecting two parallel lines are supplementary (i.e: they sum up to 180°).

Supplementary angle theorems