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2.
Perpendicular Bisector A line perpendicular to a segment at the segment’s midpoint. Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of the Perpendicular bisector Theorem – If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
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Converse P.B. Thm H A Ex 1. f = g Ex 2. d = e Ex 3. b = c T T f = g; g = c; i = h ce ab; bg cb Ex 1. a = h Ex 2. i = j Ex 3. cg = bg T
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Angle Bisector Angle Bisector - A ray that divides an angle into two congruent angles. Angle Bisector Theorem – If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Converse of Angle Bisector Theorem – If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle
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Ex 1. c bisects <abc; point L is equidistant from both sides. Ex 2. i bisects <ghi; point J is equidistant from both sides. Ex 3. e bisects <def; point K is equidistant from both sides. H A.B. Thm Converse Ex 1. L is equidistant from both sides; c must be an a.b. Ex 2. J is equidistant from both sides; i must be an a.b. Ex 3. K is equidistant from both sides; f must be an a.b.
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Concurrency To be concurrent means for three or more lines to intersect at one point. The point of concurrency of perpendicular bisectors form a circumcenter. Circumcenter - where three perpendicular bisectors of a triangle are concurrent. It is equidistant from the vertices of the triangle
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Circumcenter D is the circumcenter of triangle abc, it is equidistant from points a, b, and c. Right = circumcenter midpoint of hypotenuse
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Circumcenter L is the circumcenter of triangle jkl, it is equidistant from points j, k, and l. Acute = circumcenter inside
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Circumcenter H is the circumcenter of triangle efg, it is equidistant from points e, f, and g. Obtuse = circumcenter outside
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Concurrency The point of concurrency of angle bisectors forms an incenter. Incenter – The point of concurrency of three angle bisectors of a triangle. It is equidistant from the sides of the triangle. Concurrency Concurrency
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Incenter Incenter always inside triangle D is the incenter of triangle abc, it is equidistant to all sides.
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Incenter Incenter always inside triangle A is the incenter of triangle xyz, it is equidistant to all sides.
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Incenter Incenter always inside triangle D is the incenter of triangle abc, it is equidistant to all sides.
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Concurrency Median – segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side. The point of concurrency of medians is called a centroid. Centroid – Point of concurrency of the medians of a triangle. It is located 2/3 of the distance from each vertex to the midpoint of the opposite side
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Centroid Ex 1. g is the centroid of triangle abc Ex 2. o is the centroid of triangle egc Ex 3. k is the centroid of triangle adg Centroid = Point of balance
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Concurrency Altitude – a perpendicular segment from a vertex to the line containing the opposite side. The point of concurrency of three altitudes of a triangle creates an orthocenter. Orthocenter – point where three altitudes of a triangle are concurrent. Nothing special.
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Orthocenter D is the orthocenter of triangle abc with lines e, d, and f as altitudes.
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Orthocenter D is the orthocenter of triangle abc with lines e, d, and f as altitudes.
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Orthocenter D is the orthocenter of triangle abc with lines e, d, and f as altitudes.
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Midsegment Midsegment of a triangle – a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, forming a midsegment triangle. Triangle Midsegment Theorem – A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.
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Midsegment Ex 1. f BC Ex 2. d AC Ex 3. e BA They are all midsegments and are parallel to the lines written beside them.
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Relationships In any triangle, the longest side is always opposite to the largest angle. Small side = small angle.
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Ex 3.
<a is the smalles angle, so its opposite angle is also the smallest.
Side b is the largest side so its opposite angle is the largest.
Ex 2.
<c is the largest angle so its opposite side is also the largest.
Side a is the smallest side so its opposite angle is also the smallest.
Ex. 1
<a is the largest angle so therefore side a, its opposite side, must be the largest side.
<c has the smallest angle so this means that side c is the smallest side in the triangle.
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Inequality Exterior Angle Inequality – The exterior angle of a triangle is larger than either of the non-adjacent angles.
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Ex 1. m<e is larger than all non adjacent angles such as b and y. Ex 2. m<c is greater than all non adjacent angles such as a and y. Ex 3. m<o is greater than all non adjacent angles such as a and b.
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Inequality Triangle Inequality Theorem – The two smaller sides of a triangle must add up to MORE than the length of the 3 rd side.
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Ex 1. 2.08 + 1.71 is less than 5.86 so it doesn’t form a triangle. Ex 2. 3.02 + 2.98 = 6 But it doesn’t form a triangle, it forms a line/segment. Ex 2. 1.41 + 1.41 is greater than 2 so therefore, it does form a triangle.
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Indirect Proofs
Steps for indirect proofs:
Assume the opposite of the conclusion is true
Get the contradiction
Prove it is true
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Ex 1. Ex 2. Prove: a supplement of an acute angle cannot be another acute angle Contradiction Prove: a scalene triangle cannot have two congruent midsegments. Ex 3. Prove: an isosceles triangle cannot have a base angle that is a right angle Angle addition postulate 89 + 89 does not equal 180 Definition of acute angles Acute angles are less than 90 degrees Definition of supplementary angles Two angles add up to form 180 degrees Given Assuming that the supplement of an acute angle is another acute angle. Definition of a scalene triangle. a scalene triangle doesn’t have congruent sides definition of a midsegment. A midsegment can be congruent if there are congruent sides. Definition of a midsegment A midsegment is half the length of the line it is parallel to Given Assuming that a scalene triangle has two congruent midsegments. Substitution 90 + m<b + 90 have to equal 180. m<b must equal 0 Triangle sum theorem M<a + m<b + m<c have to equal 180 Definition of right angles M<a and m<c are both 90 Isosceles triangle theorem. Angle c is congruent to angle a, so angle c is also a right angle. Given Assuming that triangle abc has a base angle that is a right angle. let angle a be the right angle. Triangle abc is an isosceles triangle.
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Prove: triangle abc cannot have a base angle that is a right triangle Given: triangle abc is an isosceles triangle with base ac. Assuming that triangle abc has a base angle that is a right angle. let angle a be the right angle By the isosceles triangle theorem, angle c is congruent to angle a, so angle c is also a right angle. By the difinitio of right angle m<a = 90 and m<r = 90. By the triangle sum theorem m<a + m<b + m<c =180. By substitution, 90 + m<b + 90 = 180 so m<b = 0.
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HINGE The Hinge Theorem – if two triangles have 2 sides congruent, but the third side is not congruent, then the triangle with the larger included angle has the longer 3 rd side. Converse – if two triangles have 2 congruent sides, but the third side is not congruent, then the triangle with the larger 3 rd side has the longer 3 rd angle.
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Ex 1. As <a is larger than <b, side d is larger than side c Ex 2. As <a is smaller than <b, side e is smaller than side b Ex 3. As <b is greater than <a side f is larger than side b Hinge Converse Ex 1. As side d is larger than side c, <a must be larger than <b Ex 3. As side b is smaller than side f, <a must be smaller than <b. Ex 2. As side b is larger than side e, <b must also be larger than <a.
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Special Relationships 45-45-90 triangle - in all 45-45-90 triangles, both leg are congruent, and the length of the hypotenuse is the length of a leg radical 2. 30-60-90 – In all 30-60-90 triangles, the length of the hypotenuse is twice the length of the short leg; the longer side is the length of the smaller leg radical 3.
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45-45-90 Ex 1. b = 4 √2 Ex 3. b = 6.92 √2 Ex 2. b = 4 √2
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30-60-90 Ex 1. b = 4.62 c = 4 Ex 2. h = 4.62 i = 4 Ex 3. e = 8 f = 6.93