Marcela Janssen End of presentation! JOURNAL CHAPTER 5 INDEX Perpendicular Bisectors Angle Bisectors Concurrency, concurrency of perpendicular bisectors and circumcenter concurrency of angle bisectors of a triangle theorem and incenter Median, centroid and the concurrency of medians of a triangle theorem. To return here search for this: Altitude, orthocenter and concurrency of altitudes of a triangle theorem Midsegment and midsegment theorem Angle-side relationship theorem exterior angle inequality triangle inequality Indirect proof Hinge theorem and its converse What is a perpendicular bisector? A perpendicular is a line perpendicular to a segment at the segment’s midpoint. Perpendicular Bisector Theorem and its Converse. Perpendicular Bisector Theorem: If a segment is bisected by a perpendicular line, then any point on the perpendicular bisector is equidistant to the endpoint of the segment. Converse: If a point is equidistant to the segment, then lies on a perpendicular bisector. Example 1 Given that the line t is the perpendicular bisector of line JK, JG = x + 12, and KG = 3x – 17, find KG. Answer: KG = 26.5 Example 2 Given: seg AF is congruent to seg FC, <ABE is congruent to <EBC Which line is a perpendicular bisector in ABC? Answer – seg. GF Example 3 Given: L is the perpendicular bisector of seg AB Prove: XA=XB statement Reason L is the perpendicular bisector of seg AB Given Y is the midpoint of seg AB Def. of perpendicular bisector <AYX and < BYX are right angles <AYX is congruent to <BYX Def. of perpendicular Segment AY is congruent to seg BY Def. of a midpoint Seg XY is congruent to seg XY Reflexive P. Triangle AYX is congruent to triangle BYX SAS Seg XA is congruent to seg XB CPCT XA=XB Def. of congruency What is an angle bisector? An angle bisector is a ray that divides an angle into two congruent angles. Angle bisector theorem and its converse. Angle Bisector Theorem: If an angle is bisected by a ray/line, then any point on the line is equidistant from both sides of the angle. Converse: If a point in the interior of an angle is equidistant form both sides of the angle, then it lies on the angle bisector. Example 1 Given that m<RSQ = m<TSQ and TQ = 1.3, find RQ. Answer: RQ = 1.3 Example 2 Given that RQ = TQ, m<QSR = (9a + 48) o , and m<QST = (6a + 50) o, find m<QST. (9a + 48) o = (6a + 50) o 9a - 6a = 50 – 48 3a = 2 a = 2/3 6/1 (2/3) + 50 12/3 + 50 4 + 50 < QST = 54o Example 3 Ray MO bisects <LMN, m< LMO = 15x – 28, and m<NMO = x + 70. Solve for x and find m<LMN. 15x 15x 14x X= + 8 = x + 70 – 70 = 70 – 8 = 62 62/14 What does concurrent means? Concurrent is when three or more lines intersect at one point. Concurrency of Perpendicular bisectors of a triangle theorem Concurrency of Perpendicular Bisectors of a Triangle Theorem: The circumcenter of a triangle is equidistant from the vertices of the triangle. What is a circumcenter? Circumcenter is the point of concurrency of the three perpendicular bisectors of a triangle. Concurrency of angle bisectors of a triangle theorem. The Concurrency of angle bisectors of a triangle theorem is when a bisector cuts a triangle in half, therefore making a perpendicular line, making a 90 degrees angle. What is an incenter? The incenter of a triangle is the point of concurrency of the three angle bisectors of a triangle. What is a median? A median is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. What is a centroid? A centroid is the point of concurrency of the three medians of a triangle. Concurrency of medians of a triangle theorem. The centroid of the triangle located two thirds of the distance from each of the vertex to the midpoint of the opposite side. What is altitude? Altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. What is orthocenter? The orthocenter is the point of concurrency of the three altitudes of a triangle. Concurrency of altitudes of a triangle theorem All the lines that hold the altitudes of a triangle are concurrent. What is a midsegment? A midsegment of a triangle is a segment that joins the midpoints of two sides of a triangle. Midsegment Theorem A midsegment of a triangle is a parallel to the side of a triangle, and its length of that side. Angle-Side Relationship Theorem In any triangle, the side that is opposite to the biggest angle will have the biggest length; the side opposite to the smallest angle will be the smallest length. Exterior Angle Inequality The exterior angle inequality states that the exterior angle of a is greater than the other non-adjacent interior angles. Triangle Inequality Theorem The sum of any two sides length of a triangle is greater than the third side length. AB + BC > AC BC + AC > AB AC + AB > BC Steps of an indirect proof: 1. Assume that what you are trying to prove is false. 2. Try to prove it by using previews knowledge. 3. When you come to a contradiction, you have proven the theory true! Example Given: RS > RQ Prove: m RQS > m S Proof: Locate P on RS so that RP=RQ. So RP = RQ by def. of congruent segments. Then <1 = <2 by isosceles triangle, and m<1 = m<2 by def. of congruent angles. By the Angle Addition postulate, m<RQS = .m< 1 + m< 3. So m<RQS > m<2 by the Comparison Property. Then m<RQS > m<2 by substitution. By the Exterior Angle Theorem, m<2 = m<3 + m<S. So m<2 > m<S by Comparison Property. Therefore m<RQS > m<S by Trans. property of inequality Hinge Theorem and its converse. Hinge Theorem: If two sides of two triangles are congruent and the angle between them is not congruent then the triangle with the larger angle will have the longer 3rd side. Converse: If the triangle with the larger angle is the one that haves the longer side, then the two sides of the triangle are congruent and the angle between them is not congruent. Ready for the Chapter 5 Test?