Transcript
Five-Minute Check (over Lesson 1–2)Then/NowNew VocabularyKey Concept: Distance Formula (on Number Line)Example 1:Find Distance on a Number LineKey Concept: Distance Formula (in Coordinate Plane)Example 2:Find Distance on Coordinate PlaneKey Concept: Midpoint Formula (on Number Line)Example 3:Real-World Example: Find Midpoint on Number LineKey Concept: Midpoint Formula (in Coordinate Plane)Example 4:Find Midpoint in Coordinate PlaneExample 5:Find the Coordinates of an EndpointExample 6: Use Algebra to Find Measures
Over Lesson 1–2
A. AB. BC. CD. D
A B C D
0% 0%0%0%
What is the value of x and AB if B is between A and C, AB = 3x + 2, BC = 7, and AB = 8x – 1?
A. x = 2, AB = 8
B. x = 1, AB = 5
C.
D. x = –2, AB = –4
Over Lesson 1–2
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. x = 1, MN = 0
B. x = 2, MN = 1
C. x = 3, MN = 2
D. x = 4, MN = 3
If M is between L and N, LN = 3x – 1, LM = 4, and MN = x – 1, what is the value of x and MN?
Over Lesson 1–2
A. AB. BC. CD. D
A B C D
0% 0%0%0%
Find RT.
A.
B.
C.
D.
.
.
in.
in.
Over Lesson 1–2
A. AB. BC. CD. D
A B C D
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What segment is congruent to MN?
A. MQ
B. QN
C. NQ
D. no congruent segments
Over Lesson 1–2
A. AB. BC. CD. D
A B C D
0% 0%0%0%
What segment is congruent to NQ?
A. MN
B. NM
C. QM
D. no congruent segments
Over Lesson 1–2
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. 5
B. 6
C. 14
D. 18
You graphed points on the coordinate plane. (Lesson 0–2)
• Find the distance between two points.
• Find the midpoint of a segment.
Find Distance on a Number Line
Use the number line to find QR.
The coordinates of Q and R are –6 and –3.
QR = | –6 – (–3) | Distance Formula
= | –3 | or 3 Simplify.
Answer: 3
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. 2
B. 8
C. –2
D. –8
Use the number line to find AX.
Find Distance on a Coordinate Plane
Find the distance between E(–4, 1) and F(3, –1).
(x1, y1) = (–4, 1) and (x2, y2) = (3, –1)
Find Distance on a Coordinate Plane
Check Graph the ordered pairs and check by using the Pythagorean Theorem.
A. 4
B.
C.
D.
A. AB. BC. CD. D
A B C D
0% 0%0%0%
Find the distance between A(–3, 4) and M(1, 2).
Find Midpoint on a Number Line
DECORATING Marco places a couch so that its end is perpendicular and 2.5 feet away from the wall. The couch is 90” wide. How far is the midpoint of the couch back from the wall in feet?
First we must convert 90 inches to 7.5 feet. The coordinates of the endpoints of the couch are 2.5 and 10. Let M be the midpoint of the couch.
Midpoint Formula
x1 = 2.5, x2 = 10
Find Midpoint on a Number Line
Simplify.
Answer: The midpoint of the couch back is 6.25 feet from the wall.
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. 330 ft
B. 660 ft
C. 990 ft
D. 1320 ft
DRAG RACING The length of a drag racing strip is
mile long. How many feet from the finish line is
the midpoint of the racing strip?
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. (–10, –6)
B. (–5, –3)
C. (6, 12)
D. (–6, –12)
Find the Coordinates of an Endpoint
Write two equations to find the coordinates of D.
Let D be (x1, y1) and F be (x2, y2) in the Midpoint Formula.
(x2, y2) =
Find the Coordinates of an Endpoint
Answer: The coordinates of D are (–7, 11).
Midpoint Formula
Midpoint Formula
A. AB. BC. CD. D
A B C D
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A. (3.5, 1)
B. (–10, 13)
C. (15, –1)
D. (17, –11)
Find the coordinates of R if N (8, –3) is the midpointof RS and S has coordinates (–1, 5).
Use Algebra to Find Measures
Understand You know that Q is the midpoint of PR, and the figure gives algebraic measures for QR and PR. You are asked to find the measure of PR.
Use Algebra to Find Measures
Use this equation and the algebraic measures to find a value for x.
Plan Because Q is the midpoint, you know
that
Solve
Subtract 1 from each side.
Use Algebra to Find Measures
Check
QR = 6 – 3x Original Measure
Five-Minute Check (over Lesson 1–3)Then/NowNew VocabularyExample 1: Real-World Example: Angles and Their PartsKey Concept: Classify AnglesExample 2: Measure and Classify AnglesExample 3: Measure and Classify Angles
Over Lesson 1–3
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. 2
B. 4
C. 6
D. 8
Use the number line to find the measure of AC.
Over Lesson 1–3
A. AB. BC. CD. D
A B C D
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A. 3
B. 5
C. 7
D. 9
Use the number line to find the measure of DE.
Over Lesson 1–3
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. D
B. E
C. F
D. H
Use the number line to find the midpoint of EG.
Over Lesson 1–3
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. 12
B. 10
C. 5
D. 1
Find the distance between P(–2, 5) and Q(4, –3).
Over Lesson 1–3
A. AB. BC. CD. D
A B C D
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A. (–8, 20)
B. (–4, 15)
C. (–2, –5)
D. (2, 20)
Find the coordinates of R if M(–4, 5) is the midpoint of RS and S has coordinates (0, –10).
Over Lesson 1–3
A. AB. BC. CD. D
A B C D
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A. Location A, 10 units
B. Location A, 12.5 units
C. Location B, 10 units
D. Location B, 12.5 units
A boat located at (4, 1) can dock at two locations. Location A is at (–2, 9) and Location B is at (9, –11). Which location is closest? How many units away is the closest dock?
You measured line segments. (Lesson 1–2)
• Measure and classify angles.
• Identify and use congruent angles and the bisector of an angle.
• ray
• opposite rays
• angle
• side
• vertex
• interior
• exterior
• degree
• right angle
• acute angle
• obtuse angle
• angle bisector
Angles and Their Parts
A. Name all angles that have B as a vertex.
Answer:
Angles and Their Parts
Answer:
B. Name the sides of 5.
A. AB. BC. CD. D
A B C D
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B.
A.
B.
C.
D. none of these
A.
B.
C.
D.
A. AB. BC. CD. D
A B C D
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C. Which of the following is another name for 3?
Measure and Classify Angles
A. Measure TYV and classify it as right, acute, or obtuse.
Answer: mTYV = 90, so TYV is a right angle.
Measure and Classify Angles
Answer: 180 > mWYT > 90, so WYT is an obtuse angle.
A. AB. BC. CD. D
A B C D
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A. 30°, acute
B. 30°, obtuse
C. 150°, acute
D. 150°, obtuse
A. Measure CZD and classify it as right, acute, or obtuse.
A. AB. BC. CD. D
A B C D
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A. 60°, acute
B. 90°, acute
C. 90°, right
D. 90°, obtuse
B. Measure CZE and classify it as right, acute, or obtuse.
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. 30°, acute
B. 30°, obtuse
C. 150°, acute
D. 150°, obtuse
C. Measure DZX and classify it as right, acute, or obtuse.
Measure and Classify Angles
INTERIOR DESIGN Wall stickers of standard shapes are often used to provide a stimulating environment for a young child’s room. A five-pointed star sticker is shown with vertices labeled. Find mGBH and mHCI if GBH HCI, mGBH = 2x + 5, and mHCI = 3x – 10.
Measure and Classify Angles
GBH HCI Given
mGBH mHCI Definition of congruent angles
2x + 5 = 3x – 10 Substitution
2x + 15 = 3x Add 10 to each side.
15 = x Subtract 2x from each side.
Step 1 Solve for x.
Measure and Classify Angles
Step 2 Use the value of x to find the measure of either angle.
.
Answer: mGBH = 35, mHCI = 35
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. mBHC = 105, mDJE = 105
B. mBHC = 35, mDJE = 35
C. mBHC = 35, mDJE = 105
D. mBHC = 105, mDJE = 35
Find mBHC and mDJE if BHC DJE, mBHC = 4x + 5, and mDJE = 3x + 30.
Five-Minute Check (over Lesson 1–4)Then/NowNew VocabularyKey Concept: Special Angle PairsExample 1: Real-World Example: Identify Angle PairsKey Concept: Angle Pair RelationshipsExample 2: Angle MeasureKey Concept: Perpendicular LinesExample 3: Perpendicular LinesKey Concept: Interpreting DiagramsExample 4: Interpret Figures
Over Lesson 1–4
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. A
B. B
C. C
D. D
Refer to the figure. Name the vertex of 3.
Over Lesson 1–4
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. G
B. D
C. B
D. A
Refer to the figure. Name a point in the interior of ACB.
Over Lesson 1–4
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. DB
B. AC
C. BD
D. BC
Refer to the figure. Which ray is a side of BAC?
Over Lesson 1–4
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. ABG
B. ABC
C. ADB
D. BDC
Refer to the figure. Name an angle with vertex B that appears to be acute.
Over Lesson 1–4
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. 41
B. 35
C. 29
D. 23
Refer to the figure. If bisects ABC, mABD = 2x + 3, and mDBC = 3x – 13, find mABD.
Over Lesson 1–4
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. 20°
B. 40°
C. 60°
D. 80°
OP bisects MON and mMOP = 40°. Find the measure of MON.
You measured and classified angles. (Lesson 1–4)
• Identify and use special pairs of angles.
• Identify perpendicular lines.
• adjacent angles
• linear pair
• vertical angles
• complementary angles
• supplementary angles
• perpendicular
Identify Angle Pairs
A. ROADWAYS Name an angle pair that satisfies the condition two angles that form a linear pair.
A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays.
Sample Answers: PIQ and QIS, PIT and TIS, QIU and UIT
Identify Angle Pairs
B. ROADWAYS Name an angle pair that satisfies the condition two acute vertical angles.
Sample Answers: PIU and RIS, PIQ and TIS, QIR and TIU
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. CAD and DAE
B. FAE and FAN
C. CAB and NAB
D. BAD and DAC
A. Name two adjacent angles whose sum is less than 90.
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. BAN and EAD
B. BAD and BAN
C. BAC and CAE
D. FAN and DAC
B. Name two acute vertical angles.
Angle Measure
ALGEBRA Find the measures of two supplementary angles if the measure of one angle is 6 less than five times the measure of the other angle.Understand The problem relates the measures of two
supplementary angles. You know that the sum of the measures of supplementary angles is 180.
Plan Draw two figures to represent the angles.
Angle Measure
6x – 6 = 180 Simplify.
6x = 186 Add 6 to each side.
x = 31 Divide each side by 6.
Solve
Angle Measure
Use the value of x to find each angle measure.
mA = x mB = 5x – 6 = 31 = 5(31) – 6 or 149
Answer: mA = 31, mB = 149
Check Add the angle measures to verify that the angles are supplementary. mA + mB = 180
31 + 149 = 180180 = 180
A. AB. BC. CD. D
A B C D
0% 0%0%0%
A. 1°, 1°
B. 21°, 111°
C. 16°, 74°
D. 14°, 76°
ALGEBRA Find the measures of two complementary angles if one angle measures six degrees less than five times the measure of the other.
Perpendicular Lines
ALGEBRA Find x and y so thatKO and HM are perpendicular.
Perpendicular Lines
90 = (3x + 6) + 9x Substitution
90 = 12x + 6 Combine like terms.
84 = 12x Subtract 6 from each side.
7 = x Divide each side by 12.
Perpendicular Lines
To find y, use mMJO.
mMJO = 3y + 6 Given
90 = 3y + 6 Substitution
84 = 3ySubtract 6 from each side.
28 = y Divide each side by 3.
Answer: x = 7 and y = 28
A. AB. BC. CD. D
A B C D
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A. x = 5
B. x = 10
C. x = 15
D. x = 20
Interpret Figures
A. Determine whether the following statement can be justified from the figure below. Explain.mVYT = 90
Interpret Figures
B. Determine whether the following statement can be justified from the figure below. Explain.TYW and TYU are supplementary.
Answer: Yes, they form a linear pair of angles.
Interpret Figures
C. Determine whether the following statement can be justified from the figure below. Explain.VYW and TYS are adjacent angles.
Answer: No, they do not share a common side.
A. AB. B
A. yes
B. no
A. Determine whether the statement mXAY = 90 can be assumed from the figure.
A B
0%0%
A. AB. B
A. yes
B. no
B. Determine whether the statement TAU is complementary to UAY can be assumed from the figure.
A B
0%0%
A. AB. B
A. yes
B. no
C. Determine whether the statement UAX is adjacent to UXA can be assumed from the figure.
A B
0%0%
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