Miter boxes, table saws and radial arm saws all depend on the user's quick mental math to find the supplementary angle to the desired angle. Supplementary Angles ExamplesĪ common place to find supplementary angles is in carpentry. This property stems directly from the Same Side Interior Angles Theorem, because any side of a parallelogram can be thought of as a transversal of two parallel sides. Whatever angle you choose, that angle and the angle next to it (in either direction) will sum to 180 °. Since the converse of the theorem tells us the interior angles will be supplementary if the lines are parallel, and we see that 145 ° - 35 ° = 180 °, then the lines must be parallel.Ĭonsecutive Angles in a Parallelogram are Supplementary - One property of parallelograms is that their consecutive angles (angles next to each other, sharing a side) are supplementary. Here are two lines and a transversal, with the measures for two same side interior angles shown: This is an especially useful theorem for proving lines are parallel. The converse theorem tells us that if a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel. The converse of the Same Side Interior Angles Theorem is also true. Same Side Interior Angles Theorem – If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.Ī transversal through two lines creates eight angles, four of which can be paired off as same side interior angles. Since either ∠ C or ∠ A can complete the equation, then ∠ C = ∠ A. We know two true statements from the theorem: Two theorems involve parallel lines.Ĭongruent Supplements Theorem - If two angles - we'll call them ∠ C and ∠ A - are both supplementary to a third angle (we'll call it ∠ T), then ∠ C and ∠ A are congruent. Supplementary angles are seen in three geometry theorems. The third set has three angles that sum to 180 ° three angles cannot be supplementary. Only those pairs are supplementary angles. Notice the only sets that sum to 180 ° are the first, fifth, sixth and eighth pairs. Identify the ones that are supplementary: Here are eight sets of angles in degrees. The two angles must either both be right angles, or one must be an acute angle and the other an obtuse angle.Only two angles can sum to 180 ° - three or more angles may sum to 180 ° or π radians, but they are not considered supplementary.Supplementary angles have two properties: Supplementary angles can also have no common sides or common vertex: Supplementary angles can also share a common vertex but not share a common side: Supplementary angles sharing a common side will form a straight line: Supplementary angles are easy to see if they are paired together, sharing a common side. Supplementary angles sum to exactly 180 ° or exactly π radians. Straight angles - measuring exactly 180 ° or exactly π radiansĬomplementary angles sum to exactly 90 ° or exactly π 2 radians.Right angles - measuring exactly 90 ° or exactly π 2 radians.Obtuse angles - measuring greater than 90 ° or greater than π 2 radians.Acute angles - measuring less than 90 ° or less than π 2 radians.Two types of angle pairs are complementary angles and supplementary angles. (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.Angles and angle pairs are everywhere in geometry. (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°).
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