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PROPERTIES AND ATTRIBUTES OF TRIANGLES By: Mariana Beltranena 9-5 Perpendicular Bisector The perpendicular bisector of a segment is aligned perpendicular to other segment and to its midpoint. Perpendicular Bisector Theorem… If a point is in the perpendicular bisector of a segment, then it is equidistant from both endpoints. … and its Converse If a point is equidistant from both endpoints of a segment then it is the perpendicular bisector of a segment. Angle Bisector Is a line that divides an angle into two congruent angles. Angle Bisector Theorem… If a point is on the angle bisector of an angle then the perpendicular distance to each side of the angle is the same. … and its Converse If the perpendicular distance from a point to both sides of an angle is the same, then the point is on the angle bisector of that angle. Concurrency of Perpendicular Bisectors Concurrent means the coincidence on a point. Where three or more lines intersect at one point. The concurrency of bisectors is when the three perpendicular bisectors of a triangle are on a point called circumcenter. This point can be inside or outside of the triangle. Also around the triangle it could be drawn a circle touching all of the corners. Circumcenter Therorem The circumcenter is equidistant from the 3 vertices of the triangle. Concurrency of Angle Bisectors The three angle bisectors of a triangle are concurrent on a point called incenter, which is always in the triangle. Incenter Theorem Is always in and is where the angle bisector meets. The perpendicular distance from the incenter to the three sides of the triangle is the same. Medians The median of a side of a triangle is the line from the midpoint of that side to the vertex opposite to it. Centroid: the point of concurrency of the medians of a triangle. The centroid is always inside of the triangle. Centroid Theorem The distance from the vertex to the centroid is 2/3 of the distance from the vertex to the opposite side midpoint. Concurrency of altitudes of a triangle theorem The three altitudes of a triangle are concurrent in a point called orthocenter which has no special properties. The altitude of a triangle is the line from the vertex to its opposite side or the prolongation of that side if the triangle is obtuse. Midsegments The midsegment of a triangle is the line that joins the midpoints of two of its sides. Midsegment Theorem The midsegment of a triangle is parallel to the other side and measures half the measure of that side. Midsegment examples the relationship between the longer and shorter sides of a triangle and their opposite angles In the same triangle or in congruent triangles with no congruent angles the side opposite to the biggest angle is the biggest and the side opposite to the smallest angle is the smallest side. Triangle Inequality In a triangle inequality the sum of two sides is always greater than the third side length. Triangle inequality examples 1. Can 8,6,10 be the measures of a triangle? If so tell if it is acute obtuse or right. 8+6>10 14>10 Yes 8²+6²=10² 64+36=100 100=100 It is a right triangle 2. Can 5,6,11 be a triangle? 2. 3. 5+6 11….11= 11. No, because the two short sides have to add up a greater number than the longer side. The measures of the sides of a triangle are 5 and 9. Find all the possible measures of the third side. 2. 3. 4. 5+9= 14 9-5=3 x>3, x<14… possible measures are 4,5,6,7,8,9,10,11,12,13 Indirect Proofs To write an indirect proof we first assume that the opposite of the statement that we want to prove is true. Then we work until we get a false conclusion. Later we conclude that if the opposite is false, then our statement must be true. Examples of indirect proofs 1) Prove that the three angles of a triangle add up to 180 degrees by indirect proof. • Given triangle ABC assume that m<A + m<B+ m<C doesn’t equal 180 degrees. • 1) trace a parallel line to BC through A • Euclides postulate • 2) m< DAB + m<A+ m< CAF = 180 • Angle addition postulate 3) < DAB congruent to <B <CAF congruent to <C - alt. interior angles 4) m< DAB= m< B, m<CAF = m<F -deff. of congruent angles 5) m<B +m<A+m<C= 180 - substitution prop. -This is a statement that contradicts our original assumption. Then our original assumption must be false. 2) Write an indirect proof that a right triangle cannot have an obtuse angle. Given: Triangle JKL is a right triangle Prove: Triangle JKL does not have an obtuse angle Assume triangle JKL does have an obtuse angle. m<K+m<L = 90 the acute angles of a rt. Triangle are complementary m<K= 90 – m<L subtr. Prop. of = m<K > 90 def of obtuse. 90-m<L > 90 subs. 90 –m<L for m<K m<L < 0 degrees subtract 90 from both sides and solve for m<L By the protractor postulate a triangle cannot have an angle less than 0. So the original conjecture is true and an right triangle cannot have an obtuse angle. 3) Write an indirect proof that the supplement of an acute angle cannot have and acute angle. Statement Reason Assuming that the supplement of an acute angle is also an acute angle. given Two angles add up to form 180 degrees. Deff of supplementary angles. If the angle is less than 90 Deff. Of acute angles 89 + < 89 doesn’t equal 180 addition postulate The last< step is the contradiction Angle of our assumption Hinge Theorem… If two triangles have two congruent sides, then the one with the biggest angle between those sides will have the biggest third side. …and its converse. If two triangles have two congruent sides and one different side the biggest third side will have the biggest angle between the 2 congruent sides. Right Triangles 45-45-90: Two legs are congruent and the hypotenuse equals the measurement of one leg times the squared root of 2. 30-60-90: In this type of right triangle there is a minor leg in front of the 30 degrees angle. A major leg in front of the 60 degrees angle and the hypotenuse. The hypotenuse is twice the minor leg, and the major leg equals the minor leg times the squared root of 3. 45-45-90 30-60-90 Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either opposite interior angle. The end!!!