Geometry Concepts

Chapter 6 More About Triangles Identify Medians (centroid)

Identify Altitudes (orthocenter)

Identify Perpendicular Bisectors (circumcenter)

Identify Angle Bisectors (incenter)

Properties of Isosceles Triangle

Use tests for congruence

Use the Pythagorean Theorem and its converse

Find the distance between two points

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Section 6.1/6.2/6.3 Medians, Altitudes, Perpendicular Bisectors, Angle Bisectors

Questions to think about:

•

Definition Characteristics

Example Nonexample

EXAMPLES…

1) If △EFG, FN is a

median. Find EN if EG = 11.

2) If △MNP,

MC and

ND are

medians. What is NC if NP = 18?

3) If △MNP,

MC and ND are

medians. If DP = 7.5, find MP.

4) If △RST, RP and

SQ are medians. If

RQ = 7x – 1, SP =5x- 4 and QT = 6x + 9, find PT.

MEDIAN (centroid)

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Definition Characteristics

Example Nonexample

EXAMPLES…Tell whether each red segment is an altitude.

5) 6)

7) 8)

9) 10)

ALTITUDE (orthocenter)

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Definition Characteristics

Example Nonexample

EXAMPLES…Tell whether each red segment is an perpendicular bisector.

11) 12)

13) 14)

15) 16)

PERPENDICULAR

BISECTOR (circumcenter)

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Definition Characteristics

Example Nonexample

EXAMPLES…

17) If △MNP,

MO bisects

∠NMP. If m∠1 = 33,

find m∠2.

18) If △PQR,

QS bise

cts ∠PQR. If

m∠PQR

= 70, what is

m∠2.

19) If △DEF,

EG bisects

∠DEF. If m∠1

= 43, find

m∠DEF.

20) In △RST,

SU is

an angle bisector. Find m∠UST.

21) In △ABC, AD bisects

∠BAC. Find m∠1 if

m∠BAC = 52.

22) In △ABC, AD bisects ∠BAC. What is m∠CAB

ifm∠1 = 28?

ANGLE BISECTOR (incenter)

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Section 6.1/6.2/6.3 Circumcenter, Centroid, Orthocenter, Incenter

Questions to think about:

•

When three or more lines intersect at one point, they are concurrent. The point at which they intersect if the

point of concurrency.

THEOREM CONCURRENCY OF MEDIANS THEOREM

6.1 The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.

AO = 2(OE) AO + OE + OE = AE

BO = 2(OF) BO + OF + OF = BF

CO = 2(OD) CO + OD + OD = CD

EXAMPLES…

23) If △XYZ, XP , ZN ,

and YM are medians.

Find ZQ if QN = 5.

24) Use diagram in previous problem, if XP = 10.5, what is QP?

25) If △ABC, CD , BF ,

and AE are medians.

If CG = 14, what is DG?

26) Use diagram in previous problem, find the measure

of BF if GF=6.8.

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THEOREM CONCURRENCY OF PERPENDICULAR BISECTORS THEOREM

The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertex.

Solid lines are perpendicular bisectors

SC = TC = RC

THEOREM CONCURRENCY OF ANGLE BISECTORS THEOREM

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.

The angle bisectors are AD, BE and CF.

FG = EG = XG

MEDIAN Start at vertex Ends at midpoint of side opposite of vertex

Centroid

ALTITUDE

(line segment)

Start at vertex Form perpendicular angle on oppsite side of vertex

Orthocenter

PERPENDICULAR BISECTOR

(line, line segment)

form perpendicular bisector Circumcenter

ANGLE BISECTOR

(ray, line segment)

Start at vertex form as angle bisector

Incenter

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Section 6.4 Isosceles Triangle

Questions to think about:

•

Definition Characteristics

Example Nonexample

THEOREM ISOSCELES TRIANGLE THEOREM

6.2 If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

THEOREM

6.3 The median from the vertex angle of an isosceles triangle lies on the perpendicular bisector of the base and the angle bisector of the vertex angle.

EXAMPLES…

ISOSCELES TRIANGLE

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27) Find the value of each variable in isosceles triangle

DEF if EG is an angle bisector.

28) Find the values of the variables.

29) Find the values of the variables.

30) Find the values of the variables.

THEOREM CONVERSE OF ISOSCELES TRIANGLE THEOREM

6.4 If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

EXAMPLES…

31) If △ABC, ∠A ≅ ∠B and m∠A = 48. Find m∠C, AC, and BC.

THEOREM

6.5 A triangle is equilateral if and only if it’s equiangular.

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Section 6.5 Right Triangles

Questions to think about:

•

Definition Characteristics

Example Nonexample

THEOREM Leg Leg THEOREM

6.6 If two legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.

THEOREM Hypotenuse Acute Angle THEOREM

6.7 If the hypotenuse and an acute angle of one triangle are congruent to the hypotenuse and corresponding angle of another right triangle, then the triangles are congruent.

THEOREM Leg Acute Angle THEOREM

6.8 If one leg and an acute angle of a right triangle are congruent to the corresponding leg and angle of another right triangle, then the triangles are congruent.

POSTULATE Hypotenuse-Leg THEOREM

6.1 If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

RIGHT TRIANGLE

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EXAMPLES…Determine whether each pair of right triangles is congruent by LL, HA, LA, or HL. If it is not possible to prove that they are congruent, write not possible.

32) 33)

34) 35)

Section 6.6 The Pythagorean Theorem

Questions to think about:

•

Definition Characteristics

Example Nonexample

THEOREM CONVERSE OF THE PYTHAGOREAN THEOREM

PYTHAGOREAN THEOREM

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6.10 If c is the measure of the longest side of a triangle, a and b are the lengths of the other two sides, and c2 = a2 + b2, then the triangle is a right triangle.

EXAMPLES…

36) The lengths of the three sides of a trianlge are 10, 24, and 26 inches. Determine whether this triangle is a right triangle.

37) The lengths of the three sides of a trianlge are 5, 7, and 9 inches. Determine whether this triangle is a right triangle.

38) The lengths of the three sides of a trianlge are 20, 21, and 28 inches. Determine whether this triangle is a right triangle.

39) The lengths of the three sides of a trianlge are 20, 21, and 28 inches. Determine whether this triangle is a right triangle.

40) The lengths of the three sides of a trianlge are 5, 9, and 11 inches. Determine whether this triangle is a right triangle.

41) The lengths of the three sides of a trianlge are 13, 11, and 25 inches. Determine whether this triangle is a right triangle.

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Section 6.7 Distance on the Coordinate Plane

Questions to think about:

•

THEOREM DISTANCE FORMULA

6.11 If d is the measure of the distance between two points with coordinates (x1, y1) and (x2, y2)

then ( ) ( )212

2

12 yyxxd −+−=

EXAMPLES…

42) Use the distane formular to find the distance between J(-8, 6) and K(1, -3). Round to the nearest tenth, if necessary.

43) Use the distane formular to find the distance between J(0, 3) and K(0, 6). Round to the nearest tenth, if necessary.

44) Use the distane formular to find the distance between J(-3, 4) and K(5, 1). Round to the nearest tenth, if necessary.

45) Use the distane formular to find the distance between J(6, 2) and K(4, -4). Round to the nearest tenth, if necessary.

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46) Determine whether △ABC with vertics A(-3, 2), B(6, 5) and B(3, -1) is isosceles.

47) Lena takes the bus from Mill’s Market to the Candle Shop. Mill’s market is 3 miles west and 2 miles nothh of Belndon Park. The Candle shop is 2 miles east and 4 miles south of Blendon Park. How far is the Candle Shop from Mill’s Market?