Chapter 5 - Journal

Marcela Janssen

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!JOURNAL CHAPTER 5 INDEX

Perpendicular Bisectors

Angle Bisectors Concurrency, concurr

ency 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.

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 it

s converse

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Perpendicular bisectors

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 1Given 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 2Given: seg AF is congruent to seg FC, <ABE is congruent to <EBCWhich line is a perpendicular bisector in ABC?Answer – seg. GF

Example 3Given: L is the perpendicular bisector of seg ABProve: 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

Angle Bisector

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 1Given that m<RSQ = m<TSQ and TQ =

1.3, find RQ.Answer: RQ = 1.3

Example 2Given 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 – 483a = 2a = 2/3

6/1 (2/3) + 5012/3 + 50

4 + 50< QST = 54o

Example 3Ray MO bisects <LMN, m< LMO = 15x – 28, and m<NMO = x + 70. Solve for x and find m<LMN.

15x + 8 = x + 7015x – 70 = 70 – 814x = 62X = 62/14

Concurrency, concurrency of perpendicular bisectors and

circumcenter

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 and incenter.

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.

Median, centroid and the concurrency of medians of a

triangle theorem.

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.

Altitude, orthocenter and concurrency of altitudes of

a triangle theorem

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.

Midsegment and midsegment theorem

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

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

Exterior Angle InequalityThe exterior angle inequality states that

the exterior angle of a is greater than the other non-adjacent interior angles.

triangle inequality

Triangle Inequality TheoremThe 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

Indirect Proof

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!

ExampleGiven: RS > RQProve: 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 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.

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