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Geometry Acceptable Shortened Forms Theorems Conditional Form Shortened Form If two angles are right angles, then they are congruent. right ∠s ⇒ ≅ If two angles are straight angles, then they are congruent. st. ∠s ⇒ ≅ If two angles form a linear pair, then they are supplementary. linear pair ⇒ supp If angles are supplementary to the same angle, then they are congruent. supps of same ∠ ⇒ ≅ If angles are supplementary to congruent angles, then they are congruent. supps of ≅ ∠s ⇒ ≅ If angles are complementary to the same angle, then they are congruent. comps of same ∠ ⇒ ≅ If angles are complementary to congruent angles, then they are congruent. comps of ≅ ∠s ⇒ ≅ If the same segment/angle is added to congruent segments/angles, then the sums are congruent. Addition Prop. of ≅ If congruent segments/angles are added to congruent segments/angles, then the sums are congruent. Addition Prop. of ≅ **Similarly, you may use Subtraction Prop of ≅ Subtraction Prop. of ≅ If segments/angles are congruent, then their like multiples are congruent. Like multiples ≅ If segments/angles are congruent, then their like divisions are congruent. Like divisions ≅ If two angles are vertical angles, then they are congruent. (No “def. of vertical angles” step required) Vertical ∠s ⇒ ≅ If angles/segments are congruent to the same or congruent angles/segments, then they are congruent. Transitive Prop of ≅ If angles/segments are congruent, then one may replace the other. Substitution Prop. of ≅ If two angles are supplementary and congruent, then they are right angles. supp + ≅ ⇒ rt ∠s Definitions Shortened Form A right angle is an angle whose measure is 90° rt. ∠ ⇒ 90° Congruent segments/angles have equal measures. ≅ ⇒ = meas. A midpoint divides a segment into two congruent segments. mdpt. ⇒ 2 ≅ segs A segment/angle bisector divides a segment/angle into two congruent segments/angles. bis ⇒ 2 ≅ segs/∠s Complementary angles are two angles whose sum is a right angle (90°). comp ⇒ rt ∠ Supplementary angles are two angles whose sum is a straight angle (180°). supp ⇒ st ∠ Perpendicular lines, rays, segments meet to form ⊥ ⇒ rt ∠s An altitude of a triangle is a line segment drawn from any vertex of the triangle to the opposite side, extended if necessary, and perpendicular to that side. (An altitude of a triangle forms right [90°] angles with one of the sides.) alt ⇒ ⊥ or alt ⇒ rt ∠ A median of a triangle is a line segment drawn from any vertex of the triangle to the midpoint of the opposite side. (A median of a triangle divides into two congruent segments, or bisects the side to which it is drawn.) med ⇒ mdpt or med ⇒ 2 ≅ segs A scalene triangle is a triangle in which no two sides are congruent. scalene Δ ⇒ no sides ≅ An isosceles triangle is a triangle in which at least two sides are congruent. isos Δ ⇒ legs ≅ An equilateral triangle is a triangle in which all sides are congruent. equilateral Δ ⇒ all sides ≅ An acute triangle is a triangle in which all angles are acute. acute Δ ⇒ all ∠s acute An obtuse triangle is a triangle in which one angle is obtuse. obtuse Δ ⇒ obtuse ∠ A right triangle is a triangle in which one angle is right. congruent. right Δ ⇒ rt ∠ or alt ⇒ 90° ∠ or med ⇒ bisect