Chapter 1-4 Notes

Definitions

• Congruent

• When two geometric figures have the same

• We label congruence with:

size and shape

Notes

• What is the difference between equal and congruent?

• Equality is used with

• EX:

• Congruence is used with

• EX:

measurement

AB = 5cm

figures

ABC DEF

Definitions

• Midpoint

• A point on a line segment that divides it into two segmentscongruent

Postulates

• Midpoint Postulate

• Any line segment will have midpointexactly one

Examples

• Example 1

Is M a midpoint of 𝑨𝑩?

AM + MB = AB

AM + 16 = 34

– 16 – 16

AM = 18No, M is not the midpoint

Formulas

• Midpoint Formula

• For two points (𝒙𝟏, 𝒚𝟏) and 𝒙𝟐, 𝒚𝟐 , the midpoint is (𝒙𝟏 + 𝒙𝟐

𝟐,𝒚𝟏 + 𝒚𝟐

𝟐)

Examples

• Example 2

Find the midpoint between (9, -2) and (-5, 14).

M = (𝒙𝟏+𝒙𝟐

𝟐,𝒚𝟏+𝒚𝟐

𝟐)

M = (𝟗+(−𝟓)

𝟐,−𝟐+𝟏𝟒

𝟐)

M = (𝟒

𝟐,𝟏𝟐

𝟐)

M = (𝟐, 𝟔)

Examples

• Example 3

Find the midpoint between (4, -7) and (2, 6).

M = (𝒙𝟏+𝒙𝟐

𝟐,𝒚𝟏+𝒚𝟐

𝟐)

M = (𝟒+𝟐

𝟐,−𝟕+𝟔

𝟐)

M = (𝟔

𝟐,−𝟏

𝟐)

M = (𝟑,−𝟎. 𝟓)

Examples

• Example 4

If M(3, -1) is the midpoint of 𝑨𝑩 and B(7, -6), find A.

M = (𝒙𝟏+𝒙𝟐

𝟐,𝒚𝟏+𝒚𝟐

𝟐)

(3, -1) = (𝒙𝟏+𝟕

𝟐,𝒚𝟏+(−𝟔)

𝟐)

3 = 𝒙𝟏+𝟕

𝟐-1 =

𝒚𝟏−𝟔

𝟐*22* *22*

6 = 𝒙𝟏 + 𝟕 -2 = 𝒚𝟏 − 𝟔−𝟕 − 𝟕 +𝟔 + 𝟔– 1 = 𝒙𝟏 4 = 𝒚𝟏

A(-1, 4)

Definitions

• Segment Bisector

• A line, segment, or ray that passes through a of another segment.

• A bisector cuts a line segment into two parts

midpoint

congruent

Examples

• Example 5

Use a ruler to draw a bisector of the segment below.

Definitions

• Perpendicular Bisector

• A line, ray, or segment that passes through the of another

segment and intersects the segment at a

midpoint

right angle

Postulates

• Perpendicular Bisector Postulate

• For every line segment, there is perpendicular bisector that passes

through the midpoint.

one

Examples

• Example 6

Which line is the perpendicular bisector of 𝑴𝑵?

𝑶𝑸

Examples

• Example 7

Find x and y.

3x – 6 = 21

+ 6 + 6

3x = 27 𝟑 𝟑x = 9

4y – 2 = 90+ 2 + 2

4y = 92 𝟒 𝟒y = 23

Examples

• Example 8

What is the measure of each angle?

mABC = mXYZ

5x + 7 = 3x + 23– 3x – 3x2x + 7 = 23

– 7 – 72x = 16 𝟐 𝟐

x = 8

Example 8 continued

mABC = 5x + 7 mXYZ = 3x + 23

mABC = 5(8) + 7

mABC = 40 + 7

mABC = 47

mXYZ = 3(8) + 23

mXYZ = 24 + 23

mXYZ = 47

Definitions

• Angle Bisector

• A ray that divides an angle into two congruent angles

Postulates

• Angle Bisector Postulate

• Every angle has exactly one angle bisector

Examples

• Example 9

If mROT = 165, what is mSOP and mPOT? Is 𝑶𝑷 the angle bisector of SOT?

mROT = mROS + mSOP + mPOT

165 = 57 + mSOP + mSOP

165 = 57 + 2*mSOP– 57 – 57108 = 2*mSOP 𝟐 𝟐 54 = mSOP

Yes, 𝑶𝑷 is the angle bisector

•HW: #98-102, 107-110, 113-120, 121-127 odd

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