Properties and Attributes of Triangles

Chapter 5 JournalChristian Aycinena 9-5

PERPENDICULAR BISECTOR:First of all, we need to know what is equidistant, or a point that is the same distance from two or more objects.Perpendicular Bisector:A line that bisect a segment and is perpendicular to that segment. Theorem:Any point that lies on the perpendicular bisector is equidistant to both of the endpoints of the segments.Converse:If it is equidistant from both of the endpoints of the segment, then it is on the perpendicular bisector.

We know that CD is perpendicular to AB and that AO=OB, so using the perpendicular bisector theorem we know that CB=AC(So if CB=3,then AC=3), also that EB=AE(If EB=1, then AE=1) finally AD =DB

We know that AC=CB, EA=EB and AD=DB, so using the converse of the perpendicular bisector theorem we could prove by each segment that CD is perpendicular to AB and also that AO=OB

3cm

1 cmE

E

Angle Bisector:A ray or line that cuts and angle into two congruent angles. It always lies on the interior of the angle.Theorem: Any point that lies on the angle bisector is equidistant to both of the sides of the angle. Converse:States that if a point is in the interior of an angle is equidistant from the sides of the angle then it is on the angle bisector.

PA

S

TBy this we know that PS=PT=1.5(both), also XY=YZ and CE=ED=4

Ex. 2 Ex. 3 From daily Life.

ANGLE BISECTOR

X

Y

Z

1.5cm

4cm

C

D

E

Concurrent means a point where three or more lines intersect. The point of concurrency is the point where they intersect.

Point of Concurrency

The point of concurrency where the three perpendicular bisectors of a triangle meet is called the circumcenter of the triangleThe circumcenter is equidistant to the three vertices of the triangle.

MORE: Circumscribed: every vertex of the polygon(triangle in this case) lies on the circle

CIRCUMCENTER

Ex. 2.

Ex. 3- Real Life

The circumcenter of triangle ABC is the center of the circumscribed circle.

Ex. 1.

This is an example of Circumcenter.

Ex. 4

More Examples:

Acute Triangle Circumcenter

Right Triangle Circumcenter Obtuse Triangle Circumcenter

In an acute triangle circumcenter is on the inside of the triangleOn a right triangle it is on the midpoint of the hypotenuse.In an obtuse it is outside the triangle.

A triangle has three angle bisector, the angle bisector are also concurrent and the point where the angle bisectors of a triangle intersect is the incenter.

=

Always occurs on the inside of the triangle.Incenter is equidistant to any of the sides of the triangle.Inscribed: A circle in which each side of the polygon is tangent to the circle.

INCENTER

Ex. 1. The incenter inscribes a circle in a triangle and touches the three sides.

Ex. 2

Ex. 3

Ex 4

CENTROIDA median of a triangle is a segment that goes from the vertex of a triangle to the opposite midpoint. Every triangle has three medians and they are concurrent.

A

MedianCentroid

B D CThe centroid is a point of concurrency of the medians of a triangle. The centroid is always inside the triangle and is also called the center of gravity.The concurrency of medians of a triangle is located at 2/3 of the distance from each vertex to the midpoint of the opposite side.

BG=2/3BFAG=2/3ADCG=2/3CE

Ex. 1.

Ex. 2Real life.

Ex. 3.

ORTHOCENTERAltitude is a perpendicular segment from a vertex to the line containing the opposite side of the triangle.

The point in which altitudes meet is called the orthocenter.

•If the triangle is acute the orthocenter is on the inside of the triangle.

•If it is right triangle the orthocenter is on the vertex of the right angle.

•If it is obtuse triangle it is on the outside of the triangle.

Is a segment that joins the midpoints of two sides of the triangle.Every Triangle has three mid segments which form the mid segment triangle.

Midsegments: AB, BC, CAMidsegment triangle: ABC

The mid segment theorem says that a mid segment of a triangle is parallel to a side of the triangle, and its length is half the length of its parallel side.

Ex. 1, Shows DE as a midesegment, so DE is parallel to BC and DE=1/2BC

Ex. 2, Shows the midesegment.

Ex. 3 A real life.

MIDSEGMENT

In any triangle the longest side is always opposite the largest angle. The second largest side is always opposite the second largest angle. The shortest side is opposite the smallest angle. In a triangle the largest angle is always opposite the longest side. The second largest side is always opposite the second longest side. The smallest angles is opposite the shortest side.

ANGLE-SIDE RELATIONSHIP

Examples:

Ex. 1. Angle R is the smallest since it is in front of the smallest side and angle Q is the largest angle since it is in front of largest side.

Ex.2. Angle C is the largest angle since it is in front of largest side Angle B is the smallest since it is in front of the shortest side

A

B C

Z

Y X

4.5cm4.2cm

7 cm

Ex.3-Real life. Angle Z is the largest angle since it is in front of largest side Angle X is the smallest since it is in front of the shortest side

Ex. 1: Side CB is the largest since it is in front of the larger angle and side AB is the smallest since it is in front of the smallest angle

Ex. 2. Side AB is the smallest since it is in front of the smallest angle. Side AC is the largest since it is in front of the larger angle

8030

70

Z X

Y

Side ZY is the largest since it is in front of the larger angle and side YX is the smallest since it is in front of the smallest angle

The exterior angle in a triangle is bigger than either of the non adjacent interior angles.

Ex 1: W is an exterior angle in this triangle. Ands is greater than <X and greater than <Y.

EXTERIOR ANGLE INEQUALITY

In the figure, is an exterior angle.So, and

Angle OXZ is an exterior angle, so it is greater than XYZ and XZY.

TRIANGLE INEQUALITY

The sum of any two side lengths of a triangle is greater that the third side length.

AB+BC>ACBC+AB>ABAC+AB>BC

A

BC

Tell weather a triangle can have sides with this lengths: 3,5,7-YES8, 8,10-YES9,1,5-NO

INDIRECT PROOFIt is used when it is not possible to prove something directly.Writing one:

1. Assume that what you are proving is FALSE.2. Use that as your given, and start proving it.3. When you come to a contradiction you have prove that it is

true

EX:Triangle JKL is a right triangleProve: Triangle JKl does not have an obtuse angle.1. Assume Triangle JKL has an obtuse angle. Let <k be obtuse.-Negation}2. m<K +m<L =90—Acute <´s of a right triangle are comp.3. m<k= 90- m<L –Subtraction Prop. Of =4. m<k > 90– Def of obtuse <5. _90-m<L >90 Substitute 90 – m<L for m<K6. m<l < 0—Substract 90 from both sides and solve for m<LBy the Protractor Postulate, a triangle can´t have an angle with measure less than 0. Therefore triangle JKL does not have an obtuse angle.

K

J L

Examples: Examples #1Triangle LMN has at most one right angle. Step 1: Assume Triangle LMN has more than one right angle. That is, assume that angle L and angle M are both right angles. Step 2: If M and N are both right angles, then m∠L = m∠M = 90 Step 3: m∠L + m∠M + m∠N = 180 [The sum of the measures of the angles of a triangle is 180.]

Step 4: Substitution gives 90 + 90 + m∠N = 180.Step 5: Solving gives m∠N = 0.Step 6: This means that there is no Triangle LMN, which contradicts the given statement.Step 7: So, the assumption that ∠L and ∠M are both right angles must be false. Step 8: Therefore, Triangle LMN has at most one right angle.

Example #2The supplement of an acute angle cannot be an acute angle.Step 1: The supplement of an acute angle can be acute-NEGATIONStep 2: Supplementary angles add up to 180- Definition of supp.Step 3: Acute angles measure less than 90-Definition of Acute <´sStep 4: The sume of “ acute angles is less than 180-Sum Add Post.5. Since this is our contradiction the supplement of an acute can´t be acute.

HINGE THEOREMTheorem:If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side across form the larger included angle.Converse: If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the lager included angle is across form the longer third side.

Examples:

89 90

ACEF

AC < EF

6345

ZX < LM

2028

AY

m<Y < m<A

5.55.7

F M

m<F > m<M

45-45-90 TRIANGLE THEOREMIn a 45-45-90 triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times square root of 2

8 x

X= 8 times the square root of 2

16X So you divide by

the square root of two and X=32

30-60-90 TRIANGLE THEOREMIn a 30-6-90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times square root of 3

8

X

Y

X= 4Y= 4 times square root of 3

Geometry Journal Chapter 5 Name Christian AycinennaIn your own wordsrespond to the following:_____(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)