Chapter 1-4 Notes - mrburdickmath.weebly.com · Chapter 1-4 Notes Author: Zack Burdick Created...
Transcript of Chapter 1-4 Notes - mrburdickmath.weebly.com · Chapter 1-4 Notes Author: Zack Burdick Created...
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Chapter 1-4 Notes
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Definitions
• Congruent
• When two geometric figures have the same
• We label congruence with:
size and shape
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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
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Definitions
• Midpoint
• A point on a line segment that divides it into two segmentscongruent
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Postulates
• Midpoint Postulate
• Any line segment will have midpointexactly one
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Examples
• Example 1
Is M a midpoint of 𝑨𝑩?
AM + MB = AB
AM + 16 = 34
– 16 – 16
AM = 18No, M is not the midpoint
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Formulas
• Midpoint Formula
• For two points (𝒙𝟏, 𝒚𝟏) and 𝒙𝟐, 𝒚𝟐 , the midpoint is (𝒙𝟏 + 𝒙𝟐
𝟐,𝒚𝟏 + 𝒚𝟐
𝟐)
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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)
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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)
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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)
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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
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Examples
• Example 5
Use a ruler to draw a bisector of the segment below.
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Definitions
• Perpendicular Bisector
• A line, ray, or segment that passes through the of another
segment and intersects the segment at a
midpoint
right angle
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Postulates
• Perpendicular Bisector Postulate
• For every line segment, there is perpendicular bisector that passes
through the midpoint.
one
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Examples
• Example 6
Which line is the perpendicular bisector of 𝑴𝑵?
𝑶𝑸
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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
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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
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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
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Definitions
• Angle Bisector
• A ray that divides an angle into two congruent angles
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Postulates
• Angle Bisector Postulate
• Every angle has exactly one angle bisector
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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
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•HW: #98-102, 107-110, 113-120, 121-127 odd
T
U
S
R
Q
V
P