Salvador Amaya 9 5 JOURNAL Perpendicular Bisector A

Salvador Amaya 9 -5 JOURNAL

Perpendicular Bisector �A line or ray that is at the midpoint of a segment and is perpendicular to that same segment.

Perpendicular Bisector Theorem � If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of a segment.

Examples Ex 1: DC=EC Ex 2: DB=EB Ex 3: DA=EA

Converse of Perpendicular Bisector Theorem � Converse: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a line.

Examples Ex 1: Since QE=WE, TE is a perpendicular bisector of QW Ex 2: Since QR=WR, TE is a perpendicular bisector of QW Ex 3: Since QT=WT, TE is a perpendicular bisector of QW

Angle Bisector �A line or ray that divides an angle in two congruent angles.

Angle Bisector Theorem � Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

Examples Ex 1: ZC=XC Ex 2: ZB=XB Ex 2: ZV=XV

Converse of Angle Bisector Theorem � If a point inside an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.

Examples H Ex 1: Since DS=FS, AS is an angle bisector of <DHF Ex 2: Since DG=FG, AS is an angle bisector of <DHF Ex 3: Since DA=FA, AS is an angle bisector of <DHF

Concurrent � Three point. or more lines that intersect at one

Examples

Concurrency of Perpendicular Bisectors of a Triangle Theorem � The perpendicular bisectors of a triangle meet at a point which is equidistant to all the vertices.

Examples 2 12 x 2 2 12 x 5 5 5 12 x

Circumcenter � Where meet. all three perpendicular bisectors

Examples

Concurrency of Angle Bisectors Theorem � The angle bisectors of a triangle meet at a point which is equidistant to all the sides.

Examples 3 3 3 8 x 8 x 8 x 6 6 6

Incenter � Where all three angle bisectors meet.

Examples

Median � Segment that has an endpoint in the vertex of an angle and the midpoint of its opposite side.

Examples

Centroid � Where all three medians meet.

Examples

Concurrency of Medians of a Triangle Theorem � The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

Examples

Altitude � Segment from the vertex of a triangle perpendicular to the opposite segment.

Examples

Orthocenter � Where all three altitudes meet.

Examples

Midsegment � Segment from the midpoint of a segment to the midpoint of the opposite segment.

Midsegment Theorem � The midsegment that connects two sides of a triangle is parallel to the third side and is half as long.

Examples Ex 1: Segment f is parallel to segment a and half as long. Ex 2: Segment d is parallel to segment b and half as long. Ex 3: Segment e is parallel to segment c and half as long.

Side-Angle Relationships � The angle opposite to the smaller side is the smaller angle. The angle opposite to the next smaller side is the next smaller angle. The angle opposite to the longer side is the largest angle. � The side opposite to the smaller angle is the smaller side. The side opposite to the next smaller angle is the next smaller side. The side opposite to the largest angle is the longest side.

What are the sides in order from largest to smallest? Examples b, a, c 58° 71° 51°

Examples What are the angles in order from largest to smallest? 10 11 12 A, B, C

Which angle is bigger: B or C? Examples 35 B because its opposite side is bigger. 36

Exterior Angle Inequality � The exterior angle in a triangle is bigger than any of the non adjacent interior angles.

Examples Ex 1: <FAC is greater than <ABC and <ACB Ex 2: <ACH is greater than <ABC and <BAC Ex 3: <ABG is greater than <ACB and <BAC

Triangle Inequality � The longest side of a triangle has to be greater than the sum of the two shorter sides for the sides to form a triangle.

Examples � Can this sides make a triangle? � 3. 01, 3. 99, 8 � No because 3. 01+3. 99=7, which is not greater than 8.

Examples � Can this sides make a triangle? � 4, 4, 8 � No because 4+4=8 is equal to 8, so these won’t form a triangle, they will form a line.

Examples � Can this sides make a triangle? � 2. 01, 2. 57, 4 � Yes because 2. 01+2. 57=4. 58, which is greater than 4.

Indirect Proof � 1. Write the false of the statement. � 2. That false is the given. � 3. Solve as a normal proof until you come to a contradiction.

Examples Prove that: a triangle can’t have two right triangles. Step Statement Reason Write the false A triangle can have two right angles Given Solve as normal m<1=90, m<2=90 Definition of right angles m<1+m<2+m<3=180 Definition of a triangle 90+90+m<3=180 Substitution m<3=0 Subtraction Contradiction An angle can’t measure 0

Examples Prove that: The client is not guilty: A lawyer is defending his client, who was accused of a Murder in Guatemala at 2: 00 PM, but another person saw him in Suchitepequez at the same time. Step Statement Reason Write the false The client is guilty Given Solve as normal The client was at Guate and Suchi at the same time Given Contradiction Someone can’t be in two parts at the same time

Examples Prove that: A triangle can’t have 2 obtuse angles. Step Statement Reason Write the false A triangle can have two obtuse angles Given Solve as normal m<1=more than 90, m<2=more than 90 Definition of obtuse angles m<1+m<2+m<3=180 Definition of a triangle The least an obtuse angle can be is 91 Definition of obtuse angles m<1+m<2+m<3=180 Definition of a triangle 91+91+m<3=180 Substitution m<3=-2 Subtraction Contradiction An angle can’t measure negative

Hinge theorem � If two sides of a triangle are congruent to 2 sides of another triangle, and the included angle of those 2 sides in the first triangles is greater than the included angle of the 2 sides in the second triangle, then the 3 rd side of the 1 st triangle is longer than the 3 rd side of the 2 nd angle.

Examples � Which � CD side is longer: AB or CD? because the sides in its triangle have the largest included angle.

Examples � Which � ZX side is longer: QW or ZX because the sides in its triangle have the largest included angle.

Examples � Which � ER side is longer: ER or DF? because the sides in its triangle have the largest included angle.

Converse of the Hinge Theorem � If two sides of a triangle are congruent to 2 sides of another triangle, and the 3 rd sides are not congruent, then the longer side is across the largest included angle.

Examples � Which � <a angle is larger: a or b? because its opposite side is longer.

Examples � Which � <d angle is larger: c or d? because its opposite side is longer.

Examples � Which angle is larger: q or w? � <q because Its opposite side is longer.

30 -60 -90 Triangle Relationships Hypotenuse is twice as long as the short leg. � Long leg is measure of short leg times √ 3. � Or… � Short leg is half as long as the hypotenuse. � Or… � Short leg is lenght of long leg divided by √ 3. �

Examples � What is x and y if the length of the short leg is 5? � x= 5(√ 3)= 5 √ 3 � y= 5 x 2= 10

Examples � What is x and y if the length of the hypotenuse is 14? � x= 14/2=7 � y= 7(√ 3)= 7√ 3

Examples � What is x and y if the length of the long leg is 12? � x= 12/√ 3= 12√ 3/3= 4√ 3 � y= 4√ 3(2)= 8√ 3

45 -45 -90 Triangle Relationships � Both legs are congruent. � Hypotenuse is length of leg times √ 2. � Or… � Leg is lenght of hypotenuse divided by √ 2.

Examples � What is x if length of a leg is 13? � x= 13(√ 2)= 13√ 2

Examples � What is x if the length of the hypotenuse is 8? � x= 8/√ 2= 8√ 2/2= 4√ 2

Examples � What is x if the length of the hypotenuse is 9? � x= 9/√ 2= 9√ 2/2= 4. 5√ 2

_____(0 -10 pts. ) Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. Give 3 examples of each. _____(0 -10 pts. ) Describe what an angle bisector is. Explain the angle bisector theorem and its converse. Give at least 3 examples of each. _____(0 -10 pts. ) Describe what concurrent means. Explain the concurrency of Perpendicular bisectors of a triangle theorem. Explain what a circumcenter is. Give at least 3 examples of each. _____(0 -10 pts. ) Describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter is. Give at least 3 examples of each. _____(0 -10 pts. ) Describe what a median is. Explain what a centroid is. Explain the concurrency of medians of a triangle theorem. Give at least 3 examples of each. _____(0 -10 pts. ) Describe what an altitude of a triangle is. Explain what an orthocenter is. Explain the concurrency of altitudes of a triangle theorem. Give at least 3 examples. _____(0 -10 pts. ) Describe what a midsegment is. Explain the midsegment theorem. Give at least 3 examples. _____(0 -10 pts. ) Describe the relationship between the longer and shorter sides of a triangle and their opposite angles. Give at least 3 examples. _____(0 -10 pts. ) Describe the exterior angle inequality. Give at least 3 examples. _____(0 -10 pts. ) Describe the triangle inequality. Give at least 3 examples. _____(0 -10 pts. ) Describe how to write an indirect proof. Give at least 3 examples. _____(0 -10 pts. ) Describe the hinge theorem and its converse. Give at least 3 examples. _____(0 -10 pts. ) Describe the special relationships in the special right triangles (30 -60 -90 and 45 -45 -90). Give at least 3 examples of each. _____(0 -5 pts. ) Neatness and originality bonus. ______Total points earned (120 possible)