Geometry Chapter 5 - Properties and Attributes of Triangles

Segments in Triangles

Lesson 1: Perpendicular and Angle Bisectors

● equidistant

Triangle congruence theorems can be used to prove theorems about equidistant points.

Distance and Perpendicular Bisectors

Theorem Hypothesis Conclusion

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 a segment.

● Locus

The perpendicular bisector of a segment can be defined as the locus of points in a plane that are

equidistant from the endpoints of the segment.

Applying the Perpendicular Bisector Theorem and Its Converse

Ex1: Find each measure

A. NM = B. BC = C. TU =

Remember that the distance between a point and a line is the length of the perpendicular segment from

the point to the line.

A BY

X

l

A B

X

l

Y

M

N

2.6 A

C

B

D

38

3812

3x + 9 7x - 17

U

T

Distance and Angle Bisectors

Theorem Hypothesis Conclusion

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 the 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.

Applying the Angle Bisector Theorems

Ex2: Find each measure

A. BC = B. m∠EFH, given that

m∠EFG = 50°

C. m∠MKL

M

B

B

C

C

A

A

P

P

B

B

C

C

A

A

P

Ex4:

Write an equation in point-slope form for

the perpendicular bisector of the segment with

endpoints C(6, -5), and D(10, 1).

Geometry

Lesson 2: Bisectors of Triangles

Since a triangle has three sides, it has three perpendicular bisectors. When you construct the

perpendicular bisectors, you find that they have an interesting property.

● concurrent

● point of concurrency

● circumcenter of the triangle

Circumcenter Theorem

The circumcenter of a triangle is equidistant

from the vertices of the triangle.

The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

Acute triangle Obtuse triangle Right triangle

P

P

C

C

B

B

A

The circumcenter of ΔABC is the center of its circumscribed circle.

● Circumscribed circle

•

Using Properties of Perpendicular Bisectors

Ex1: DG , EG , and FG are the

perpendicular bisectors of ΔABC. Find GC.

Ex2: Find the circumcenter of ΔHJK with

vertices H(0, 0), J(10, 0), and K(0, 6).

A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also

concurrent.

● incenter of a triangle

Incenter Theorem

The incenter of a triangle is equidistant from the sides

of the triangle.

A

B

C

P•

P

P

C

C

B

B

A

Unlike the circumcenter, the incenter is always inside the triangle.

Acute triangle Obtuse triangle Right triangle

The incenter is the center of the triangle's inscribed circle.

● Inscribed circle

Using Properties of Angle Bisectors

Ex3: MP and LP are angle bisectors of

ΔLMN. Find each measure.

A. the distance from P to MN

.

B. m∠PMN

Ex4: A city planner wants to build a new library

between a school, a post office, and a hospital.

Draw a sketch to show where the library should be

placed so it is the same distance from all three

buildings.

Geometry

Lesson 3: Medians and Altitudes of Triangles

● median of a triangle

Every triangle has three medians, and the medians are concurrent.

● centroid of the triangle

The centroid is always inside the triangle. The centroid is also called the center of gravity because it is

the point where a triangluar region will balance.

A

B

C

P•

S L

P

A B

C

D

Centroid Theorem

The centroid of a triangle is located 2

3 of the distance from each vertex to the midpoint of the

opposite side.

*Remember, the centroid is closer to each side than to the verte

Using the Centroid to Find Segment Lengths

Ex1: In ΔLMN, RL = 21, and SQ = 4. Find

A. LS =

B. NQ =

Ex2: A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what

coordinates will the triangular region balance?

c

P

P

C

C

B

B

A

● altitude of a triangle

Every triangle has three altitudes. An altitude

can be inside, outside, or on the triangle.

● orthocenter of a triangle

Geometry

Lesson 4: The Triangle Midsegment Theorem

● midsegment of a triangle

midsegments:

midsegment triangle:

Every triangle has three midsegments, which form the midsegment triangle.

Q

RP

Examining Midsegments in the Coordinate Plane

Ex1: The vertices of ΔXYZ are X(-1, 8), Y(9, 2), and Z(3, -4). M and N are the midpoints of XZ YZ . Show that

A. MN // XY

B. MN = 1

2 XY.

The relationship shown in Example 1 is true for the midsegment of every 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.

A

C

B

Using the Triangle Midsegment Theorem

Ex2: Find each measure.

A. BD

B. m∠CBD

Ex3: In an A-frame support, the distance PQ is 46

inches. What is the length of the support ST if

S and T are at the midpoints of the sides?

Geometry

Relationships in TrianglesLesson 5: Indirect Proof and Inequalities in One Triangle

You have written proofs using direct reasoning. That is, you began with a true hypothesis and built a

logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that

the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof

is also called a proof by contradiction.

Writing an Indirect Proof

1. Identify the conjecture to be proven.

2. Assume the opposite (the negation) of the conclusion is true.

3. Use direct reasoning to show that the assumption leads to a contradiction.

4. Conclude that since the assumption is false, the original conjecture must be true.

Writing an Indirect Proof

Ex1A: Write an indirect proof that a right triangle cannot have an obtuse angle.

Ex1B:Write an indirect proof that if a > 0, then 1

a > 0.

Angle-Side Relationships in Triangles

Theorem Hypothesis Conclusion

If two sides of a triangle are not

congruent, then the larger angle

is opposite the longer side

If two angles of a triangle are not

congruent, then the longer side is

opposite the larger angle.

Ordering Triangle Side Lengths and Angle Measures

Ex2A: Write the angles in order from smallest to

largest.

B. Write the sides in order from shortest to

longest.

A triangle is formed by three segments, but not every set of three segments can form a triangle.

Triangle Inequality

The sum of any two side lengths of a triangle

is greater than the third length.

A B

C

X

Y

Z

A

B C

Applying the Triangle Inequality Theorem

Tell whether a triangle can have sides with the given lengths. Explain.

Ex1A: 3, 5, 7 B. 4, 6.5, 11 C. n + 5, n2 , 2n, when n = 3

Finding Side Lengths

Ex4: The lengths of two sides of a triangle are 8 in. and 13 in. Find the range of possible lengths for

the third side.

Ex5: The figure shows the approximate distances between cities in California. What is the range of

distances from San Francisco to Oakland?

Geometry

Lesson 6: Inequalities in Two Triangles

Inequalities in Two Triangles

Theorem Hypothesis Conclusion

Hinge Theorem 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 is across from the larger

included angle.

A

D

B

C

E

F

Using the Hinge Theorem

Ex1A:Compare m∠BAC and

m∠DAC.

B: Compare EF and FG. C: Find the range of values for

k.

Ex2: John and Luke leave school at the same time. John rides his bike 3 blocks west and 4 blocks

north. Luke rides 4 blocks east and then 3 blocks at a bearing of N 10° E. Who is farther from

school?

Proving Triangle Relationships

Ex3: Write a two-column proof.

Given: AB ≅ CD ,

m∠ABD > m∠CDB

Prove: AD > CB

Statement Reason