Geometry Honors T.E.A.M.S. Geometry Honors Summer Assignment
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Dear Parents and Students:
All students entering Geometry or Geometry Honors are required to complete this assignment. This assignment is a review of essential topics to strengthen math skills for the upcoming school year.
If you need assistance with any of the topics included in this assignment, we strongly recommend that you to use the following resource: http://www.khanacademy.org/.
If you would like additional practice with any topic in this assignment visit: http://www.math- drills.com.
Below are the POLICIES of the summer assignment:
The summer assignment is due the first day of class. On the first day of class, teachers will collect the summer assignment. Any student who does not have the assignment will be given one by the teacher. Late projects will lose 10 points each day. Summer assignments will be graded as a quiz. This quiz grade will consist of 20% completion and 80% accuracy. Completion is defined as having all work shown in the space provided to receive full credit, and a parent/guardian signature. Any student who registers as a new attendee of Teaneck High School after August 15th will have one extra week to complete the summer assignment. Summer assignments are available on the district website and available in the THS guidance office.
HAVE A GREAT SUMMER!
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An Introduction to the Basics of Geometry
Directions: Read through the definitions and examples given in each section, then complete the practice questions, found on pages 20 to 26. Those pages will be collected by your Geometry Teacher on the first day of school.
Section 1: Points, Lines and Planes
Undefined term: words that do not have formal definitions, but there is an agreement about what they mean. In Geometry, the words point, line and plane are undefined terms.
Undefined Term Meaning Example/Picture and symbols
A point has no dimension but has location. A dot is Point used to represent a point.
A line has one dimension. It is represented by a line with two arrowheads, showing that it extends in two directions without end. Line Through any two points there is exactly one line. You can use any two points on a line to name it, or it can be named by a lowercase letter written by the line.
A plane has two dimensions. It is represented by a shape that looks like a floor or a wall, but it extends without end.
Plane Through any three points not on the same line, there is exactly one plane. You can use three points that are not on the same line to name a plane, or you can use a capital letter (without a point next to it) to name a plane.
Collinear points: points that lie on the same line. Coplanar points: points that lie in the same plane.
Example 1: Naming Points, Lines and Planes
a. Give two other names for 푃푄⃡ and plane R.
Answer: Other names for 푃푄⃡ are 푄푃⃡ and line n. Other names for plane R are plane SVT and plane PTV.
b. Name three points that are collinear. Name four points that are coplanar. Answer: Points S, P, and T lie on the same line, so they are collinear. Points S, P, T, and V lie in the same plane, so they are coplanar.
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Defined Terms: Segment and Ray
The definitions below use line AB (written as 퐴퐵⃡ ) and points A and B.
Defined Term Definition Example/Picture and Symbols
The line segment AB, or segment AB (written as 퐴퐵̅̅̅̅) consists of the endpoints A and B and all points on 퐴퐵⃡ that are between A and B. The endpoints are like stop and start points. Unlike lines, segments do not continue on forever in both Segment directions and they can be measured.
퐴퐵̅̅̅̅ (read “segment AB”) Note that 퐴퐵̅̅̅̅ can also be called 퐵퐴̅̅̅̅.
The ray AB (written as 퐴퐵 ) consists of the endpoint A and all points on 퐴퐵⃡ that lie on the same side of A as B. In other words, rays have a starting point (called an endpoint) and Ray continue in the direction of the other point.
Note that 퐴퐵 and 퐵퐴 are two different rays because they are Top: 퐴퐵 (read “ray AB”) going in different directions.
Bottom: 퐵퐴 (read “ray BA”)
If point C lies on 퐴퐵⃡ between A and B, then 퐶퐴 and 퐶퐵 are opposite rays. Opposite Rays They have the same point but go in opposite directions to form 퐶퐴 and 퐶퐵 are opposite rays. a line.
Two or more geometric figures intersect when they have one or more points in common.
The intersection of the figures is the set of all points they have in common. Intersection
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Example 2: Naming Segments, Rays and Opposite Rays
a. Give another name for 퐺퐻̅̅̅̅ .
Answer: Another name for 퐺퐻̅̅̅̅ is 퐻퐺̅̅̅̅.
b. Name all rays with endpoint 퐽. Which of these rays are opposite rays?
Answer: The rays with endpoint 퐽 are 퐽퐸 , 퐽퐺 , 퐽퐹 , and 퐽퐻 . The pairs of opposite rays with endpoint 퐽 are 퐽퐸 and 퐽퐹 , and 퐽퐺 and 퐽퐻 .
Practice Questions for Section 1 can be found on Page 20.
Section 2: Measuring Segments
In Geometry, a rule that is accepted without proof is called a postulate or an axiom. A rule that can be proved is called a theorem.
The Ruler Postulate: The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB (notice there is no symbol above the 2 letters), is the absolute value of the difference of the coordinates of A and B.
Congruent Segments: Line segments that have the same length are called congruent segments. You can say “the length of 퐴퐵̅̅̅̅ is equal to the length of 퐶퐷̅̅̅̅,” or you can say “퐴퐵̅̅̅̅ is congruent to 퐶퐷̅̅̅̅.” The symbol ≅ means “is congruent to.”
In the diagram above, of 퐴퐵̅̅̅̅ and 퐶퐷̅̅̅̅ have tick marks on them, indicating 퐴퐵̅̅̅̅ ≅ 퐶퐷̅̅̅̅. When there is more than one pair of congruent segments, use multiple tick marks.
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When three points are collinear, you can say that one point is between the other two.
Segment Addition Postulate
If B is between A and C, then AB + BC=AC.
If AB+BC=AC, then B is between A and C.
Example 1: Comparing Segments for Congruence
a. Plot J(−3, 4), K(2, 4), L(1, 3), and M(1, −2) in a coordinate plane. Then determine whether ̅퐽퐾̅̅ and 퐿푀̅̅̅̅ are congruent.
Answer:
Plot the points, as shown. To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints. 퐽퐾 = |−3 − 2| = 5, Ruler Postulate To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints. 퐿푀 = |3 − (−2)| = 5, Ruler Postulate 퐽퐾 = 퐿푀. So, ̅퐽퐾̅̅ ≅ 퐿푀̅̅̅̅.
Example 2: Using the Segment Addition Postulate
a. Find 퐷퐹.
Answer: Use the Segment Addition Postulate to write an equation. Then solve the equation to find 퐷퐹.
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b. Find 퐺퐻.
Answer: Use the Segment Addition Postulate to write an equation. Then solve the equation to find 퐺퐻.
Example 3: Using the Segment Addition Postulate
The cities shown on the map lie approximately in a straight line. Find the distance from Tulsa, Oklahoma, to St. Louis, Missouri.
Answer:
1. Understand the Problem. You are given the distance from Lubbock to St. Louis and the distance from Lubbock to Tulsa. You need to find the distance from Tulsa to St. Louis. 2. Make a Plan. Use the Segment Addition Postulate to find the distance from Tulsa to St. Louis. 3. Solve the Problem. Use the Segment Addition Postulate to write an equation. Then solve the equation to find 푇푆.
So, the distance from Tulsa to St. Louis is 361 miles.
Practice Questions for Section 2 can be found on Page 21.
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Section 3: Using Midpoint and Distance Formulas
Midpoints and Segment Bisectors
The midpoint of a segment is the point that divides the segment into two congruent segments.
A segment bisector is a point, ray, line, line segment, or plane that intersects the segment at its midpoint. A midpoint or a segment bisector bisects a segment. (“Bi” means two, “sect” means sections).
Example 1: Finding Segment Lengths
In the skateboard design, 푉푊̅̅̅̅̅ bisects 푋푌̅̅̅̅ at point T, and 푋푇 = 39.9 cm.
Find 푋푌.
Answer:
Point 푇 is the midpoint of 푋푌̅̅̅̅, so 푋푇 = 푇푌 = 39.9 cm.
푋푌 = 푋푇 + 푇푌 Segment Addition Postulate 푋푌 = 39.9 + 39.9 Substitute 푋푌 = 79.8 Add
So, 푋푌 = 79.8 cm
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Example 2: Using Algebra with Segment Lengths
Point M is the midpoint of 푉푊̅̅̅̅̅. Find the length of 푉푀̅̅̅̅̅.
Answer:
1. Write and solve an equation. Use the fact that 푉푀̅̅̅̅̅ = ̅푀푊̅̅̅̅̅.
푉푀 = 푀푊 Write the equation 4푥 − 1 = 3푥 + 3 Substitute 푥 − 1 = 3 Subtract 3푥 from both sides 푥 = 4 Add 1 to each side
2. Evaluate 푉푀 = 4푥 − 1 when 푥 = 4 푉푀 = 4(4) − 1 = 15 So the length of 푉푀̅̅̅̅̅ is 15 units.
Using the Midpoint Formula
The coordinates of the midpoint of a segment are the averages of the x-coordinates and of the y-coordinates of the endpoints.
If A(푥1, 푦1), and B(푥2, 푦2) are points in a coordinate plane, then the midpoint M of AB has coordinates
Example 3: Using the Midpoint Formula
a. The endpoints of 푅푆̅̅̅̅ are R(1, −3) and S(4, 2). Find the coordinates of the midpoint M. Answer:
1+4 −3+2 5 −1 Use the midpoint formula: 푀 ( , ) = 푀 ( , ) 2 2 2 2
5 1 So the coordinates of 푀 are ( , − ). 2 2
b. The midpoint of ̅퐽퐾̅̅ is 푀(2, 1). One endpoint is 퐽(1, 4). Find the coordinates of endpoint 퐾. Answer:
Let (x, y) be the coordinates of endpoint 퐾. Use the Midpoint Formula.
Step 1: Find x Step 2: Find y 1 + 푥 4 + 푦 = 2 = 1 2 2 1 + 푥 = 4 4 + 푦 = 2 푥 = 3 푦 = −2
The coordinates of endpoint 퐾 are (3, -2).
Using the Distance Formula
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If A(푥1, 푦1), and B(푥2, 푦2) are points in a coordinate plane, then the distance between A and B is
2 2 퐴퐵 = √(푥2 − 푥1) + (푦2 − 푦1)
Example 4: Using the Distance Formula
Your school is 4 miles east and 1 mile south of your apartment. A recycling center, where your class is going on a field trip, is 2 miles east and 3 miles north of your apartment. Estimate the distance between the recycling center and your school.
Answer:
You can model the situation using a coordinate plane with your apartment at the origin (0, 0). The coordinates of the recycling center and the school are 푅(2, 3) and 푆(4, −1), respectively. Use the Distance Formula. Let
(푥1, 푦1) = (2, 3) and (푥2, 푦2) = (4, −1).
2 2 Distance Formula 푅푆 = √(푥2 − 푥1) + (푦2 − 푦1) 푅푆 = √(4 − 2)2 + (−1 − 3)2 Substitute 푅푆 = √(2)2 + (−4)2 Subtract 푅푆 = √4 + 16 Evaluate Powers 푅푆 = √20 Add 푅푆 ≈ 4.5 Use Calculator
So, the distance between your school and the recycling center is about 4.5 miles.
Practice Questions for Section 3 can be found on Page 22.
Section 4: Perimeter and Area in the Coordinate Plane
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Polygons
In geometry, a figure that lies in a plane is called a plane figure. Recall that a polygon is a closed plane figure formed by three or more line segments called sides. Each side intersects exactly two sides, one at each vertex, so that no two sides with a common vertex are collinear. You can name a polygon by listing the vertices in consecutive order.
A polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that is not convex is concave. Polygon Types Number Type of Polygon of sides 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon *One way to determine that a polygon is convex is to 9 Nonagon imagine turning it on all sides and pouring waters over it 10 Decagon each time. If the water will roll off the polygon no 11 Undecagon matter which side it is sitting on, then it is convex. 12 Dodecagon N n-gon (example – 23 sides 23-gon)
Example 1: Classifying Polygons
Classify the polygon by the number of sides. Tell whether it is concave or convex.
a. b.
Answer: The polygon has four sides. So, it is a Answer: The polygon has six sides. So, it is a quadrilateral. The polygon is concave. hexagon. The polygon is convex.
Finding Perimeter and Area in the Coordinate Plane
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You can use the formulas given below and the Distance Formula to find the perimeters and areas of polygons in the coordinate plane.
Example 2: Finding Perimeter in the Coordinate Plane
Find the perimeter of △ 퐴퐵퐶 with vertices 퐴(−2, 3), 퐵(3, −3), and 퐶(−2, −3).
Answer: Step 1 Draw the triangle in a coordinate plane. Then find the length of each side.
Side 푨푩̅̅̅̅ 2 2 Distance Formula 퐴퐵 = √(푥2 − 푥1) + (푦2 − 푦1) 퐴퐵 = √(3 − (−2))2 + (−3 − 3)2 Substitute 퐴퐵 = √(5)2 + (−6)2 Subtract 퐴퐵 = √25 + 36 Evaluate powers 퐴퐵 = √61 Add 퐴퐵 ≈ 7.81 Use Calculator Side 푩푪̅̅̅̅ 퐵퐶 = ∣ −2 − 3 ∣ = 5 Ruler Postulate Side 푨푪̅̅̅̅ 퐴퐶 = ∣ 3 − (− 3) ∣ = 6 Ruler Postulate
Step 2 Find the sum of the side lengths.
퐴퐵 + 퐵퐶 + 퐶퐴 ≈ 7.81 + 5 + 6 = 18.81
So, the perimeter of △ 퐴퐵퐶 is about 18.81 units.
Example 3: Finding Area in the Coordinate Plane
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Find the area of △ 퐷퐸퐹 with vertices 퐷(1, 3), 퐸(4, −3), and 퐹(−4, −3).
Answer:
Step 1 Draw the triangle in a coordinate plane by plotting the vertices and connecting them.
Step 2 Find the lengths of the base and height.
Base
The base is ̅퐹퐸̅̅̅, which is a horizontal segment so we can use the Ruler Postulate:
퐹퐸 = |−4 − 4| = 8
Height
The height is the distance from point 퐷 to ̅퐹퐸̅̅̅. By counting grid lines, you can determine the height is 6 units.
Step 3 Substitute the values for the base and height into the formula for the area of a triangle.
1 퐴 = 푏ℎ 2 1 퐴 = 8 ∙ 6 2
퐴 = 24
So, the area of △ 퐷퐸퐹 is 24 square units.
Practice Questions for Section 4 can be found on Page 23.
Section 5 – Angles and their Measures
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Naming Angles
An angle is a set of points consisting of two different rays that have the same endpoint, called the vertex. The rays are the sides of the angle.
You can name an angle in several different ways.
Use its vertex, such as ∠A. Use a point on each ray and the vertex, such as ∠BAC or ∠CAB. (notice the vertex is always in the middle when naming this way). Use a number, such as ∠1.
The region that contains all the points between the sides of the angle is the interior of the angle. The region that contains all the points outside the angle is the exterior of the angle.
Example 1: Naming Angles
A lighthouse keeper measures the angles formed by the lighthouse at point M and three boats. Name three angles shown in the diagram.
Answer:
∠JMK or ∠KMJ
∠KML or ∠LMK
∠JML or ∠LMJ
*Common Error: When a point is the vertex of more than one angle, you cannot use the vertex alone to name the angle.
Measuring and Classifying Angles
A protractor helps you approximate the measure of an angle. The measure is usually given in degrees. 14 | P a g e
Protractor Postulate
Consider 푂퐵⃡ and a point 퐴 on one side of 푂퐵⃡ . The rays of the form 푂퐴 can be matched one to one with the real numbers from 0 to 180.
The measure of ∠AOB, which can be written as 푚∠퐴푂퐵, is equal to the absolute value of the difference between the real numbers matched with 푂퐴 and 푂퐵 on a protractor.
In the diagram, 푚∠퐴푂퐵 = 140°, because 푂퐴 passes through the 40°/140° line and 푂퐵 passes through the 180°/0° line.
Using the outer numbers |180 – 40| = 140° Using the inner numbers |0 – 140|=140°.
Types of Angles
Acute angle Right angle Obtuse angle Straight angle
(a small square drawn
at the vertex of an angle symbolizes a right angle) Measures greater than Measures greater than Measures 90° Measures 180° 0° and less than 90° 90° and less than 180°
Example 2: Measuring and Classifying Angles
Find the degree measure of each of the following angles. Classify each angle as acute, right, or obtuse.
a. ∠AOB 푂퐴 lines up with 0° on the inner scale, and 푂퐵 passes through 35° on the inner scale, so 푚∠퐴푂퐵 = 35°. It is an acute angle. b. ∠BOE 푂퐵 lines up with 35° on the inner scale, and 푂퐸 passes through 145° on the inner scale, so 푚∠퐵푂퐸 = |35 − 145|° = 110°. It is an obtuse angle.
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Congruent angles are two angles with equal measures. If 푚∠퐴퐵퐶 = 푚∠퐷퐸퐹, then ∠퐴퐵퐶 ≅ ∠퐷퐸퐹. (Angles are congruent when their measures are equal.
An angle bisector is a ray between two sides of an angle the creates two congruent angles. If 푆푉 bisects ∠푅푆푇 , then
∠푅푆푉 ≅ ∠푉푆푇
Example 3: Identifying Congruent Angles
Use the diagram to answer the questions.
a. Identify the angles congruent to ∠ADG. Because ∠퐵퐸퐻 and ∠퐶퐹퐼 have matching arcs, ∠퐴퐷퐺 ≅ ∠퐵퐸퐻 ≅ ∠퐶퐹퐼.
Reading: In diagrams, matching b. Identify the angles congruent to ∠DAG. arcs indicate congruent angles. Because ∠퐷퐴퐺, ∠퐴퐺퐷, ∠퐸퐵퐻, ∠퐸퐻퐵, ∠퐹퐶퐼 and ∠퐹퐼퐶 have matching arcs, so When there is more than one pair ∠퐷퐴퐺 ≅ ∠퐴퐺퐷 ≅ ∠퐸퐵퐻 ≅ ∠퐸퐻퐵 ≅ ∠퐹퐶퐼 ≅ ∠퐹퐼퐶. of congruent angles, use multiple arcs.
Angle Addition Postulate
If P is in the interior of ∠RST, then the measure of ∠RST is equal to the sum Words of the measures of ∠RSP and ∠PST.
If P is in the interior of ∠RST, then Symbols m∠RST = m∠RSP + m∠PST.
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Example 4: Using the Angle Addition Postulate to Find Angle Measures
Given that 푚∠퐿퐾푁 = 145°, find 푚∠퐿퐾푀 and 푚∠푀퐾푁.
Step 1 Write and solve an equation to find the value of x.
푚∠퐿퐾푁 = 푚∠퐿퐾푀 + 푚∠푀퐾푁 Angle Addition Postulate 145° = (2푥 + 10)° + (4푥 − 3)° Substitute 145 = 6푥 + 7 Simplify (Combine Like Terms) 138 = 6푥 Subtract 23 = 푥 Divide
Step 2 Evaluate the given expressions when x = 23.
푚∠퐿퐾푀 = (2푥 + 10)° = (2 ⋅ 23 + 10)° = 56° 푚∠푀퐾푁 = (4푥 − 3)° = (4 ⋅ 23 − 3)° = 89°
So, 푚∠퐿퐾푀 = 56° and 푚∠푀퐾푁 = 89°.
Example 5: Using a Bisector to Find Angle Measures
푄푆 bisects ∠푃푄푅, and 푚∠푃푄푆 = 24°. Find 푚∠푃푄푅.
Step 1 Draw a diagram.
Step 2 Because 푄푆 bisects ∠푃푄푅, 푚∠푃푄푆 = 푚∠푅푄푆. So, 푚∠푅푄푆 = 24°.
Use the Angle Addition Postulate to find 푚∠푃푄푅.
푚∠푃푄푅 = 푚∠푃푄푆 + 푚∠푅푄푆 Angle Addition Postulate 푚∠푃푄푅 = 24° + 24° Substitute Angle Measures 푚∠푃푄푅 = 48° Add
So, 푚∠푃푄푅 = 48°.
Practice Questions for Section 5 can be found on Page 25.
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Section 6 – Describing Pairs of Angles
Complementary and Supplementary Angles
Complementary angles Supplementary angles
Two positive angles whose measures have a sum Two positive angles whose measures have a sum of 90°. Each angle is the complement of the other. of 180°. Each angle is the supplement of the other.
Adjacent Angles
Angles can be adjacent angles or nonadjacent angles. Adjacent angles are two angles that share a common vertex and side, but have no common interior points.
Example 1: Identifying Pairs of Angles
In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.
Answer:
Because 37° + 53° = 90°, ∠BAC and ∠RST are complementary angles. Because 127° + 53° = 180°, ∠CAD and ∠RST are supplementary angles. Because ∠BAC and ∠CAD share a common vertex and side, they are adjacent angles.
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Example 2: Finding Angle Measures
a. ∠1 is a complement of ∠2, and m∠1 = 62°. Find m∠2. Answer: If ∠1 is a complement of ∠2, then 푚∠1 + m∠2 = 90 Definition of Complementary Angles 62 + m∠2 = 90 Substitute m∠2 = 28° Subtract
b. ∠3 is a supplement of ∠4, and m∠4 = 47°. Find m∠3. Answer: If ∠3 is a complement of ∠4, then 푚∠3 + m∠4 = 180 Definition of Supplementary Angles 푚∠3 + 47 = 180 Substitute m∠3 = 133° Subtract
Linear Pairs and Vertical Angles
Two adjacent angles are a linear pair when their Two angles are vertical angles when their sides noncommon sides are opposite rays. The angles in form two pairs of opposite rays. a linear pair are supplementary angles.
∠1 and ∠2 are a linear pair. ∠3 and ∠6 are vertical angles. ∠4 and ∠5 are vertical angles.
Example 4: Identify Angle Pairs
Identify all the linear pairs and all the vertical angles in the figure.
Answer:
To find vertical angles, look for angles formed by intersecting lines. ∠1 and ∠5 are vertical angles.
To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. ∠1 and ∠4 are a linear pair. ∠4 and ∠5 are also a linear pair.
Practice Questions for Section 6 can be found on Page 26.
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Name: ______Teacher:______
Practice Questions
Section 1 Questions – Points. Line and Planes:
In exercises 1 – 4, use the diagram at right.
1. Give two other names for 퐶퐷⃡
2. Give another name for plane M.
3. Name three points that are collinear. Then name a fourth point that is not collinear with these three points.
4. Name a point that is not coplanar with points A, C, E.
In exercises 5 – 7, use the diagram at right.
5. What are two other names for 푃푄⃡ ?
6. What is another name for 푅푆̅̅̅̅?
7. Name all rays with endpoint T. Which of these rays are opposite rays?
In Exercises 9 and 10, sketch the figure described.
8. 퐴퐵̅̅̅̅ and 퐵퐶̅̅̅̅ 9. line k in plane M.
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Section 2 Questions – Measuring Segments:
In Exercises 1–3, plot the points in the coordinate plane. Then determine whether 퐴퐵̅̅̅̅ and 퐶퐷̅̅̅̅ are congruent.
1. A(-5, 5), B(-2, 5) 2. A(4, 0), B(4, 3) 3. A(-1, 5), B(5, 5) C(2, -4), D(-1, -4) C(-4, -4), D((-4, 1) C(1, 3), D(1, -3)
In exercises 4 – 6, find VW.
4. 5. 6.
7. A bookstore and a movie theater are 6 kilometers apart along the same street. A florist is located between the bookstore and the theater on the same street. The florist is 2.5 kilometers from the theater. How far is the florist from the bookstore? (Hint: draw a line segment and label 3 points with each location)
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Section 3 Questions – Using Midpoint and Distance Formulas:
In Exercises 1–3, identify the segment bisector of 퐴퐵̅̅̅̅. Then find AB.
1. 2. 3.
In Exercises 4-6, identify the segment bisector of ̅퐸퐹̅̅̅. Then find EF.
4. 5. 6.
In Exercises 7–9, the endpoints of 푃푄̅̅̅̅ are given. Find the coordinates of the midpoint M and the length of 푃푄̅̅̅̅.
7. P(-4, 3) and Q(0, 5) 8. P(-2, 7) and Q(10, -3) 9. P(3, -15) and Q(9, -3)
In Exercises 10–12, the midpoint M and one endpoint of JK are given. Find the coordinates of the other endpoint.
9 10. J(7, 2) and M(1, -2) 11. J(5, -2) and M(0, -1) 12. J(2, 16) and M(− , 7) 2
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Section 4 Questions – Perimeter and Area in the Coordinate Plane:
In Exercises 1–4, classify the polygon by the number of sides. Tell whether it is convex or concave.
1. 3.
4.
2.
In Exercises 5–8, find the perimeter and area of the polygon with the given vertices.
5. X(2, 4), Y(0, -2), Z(2, -2)
6. P(1, 3), Q(1, 1), R(-4, 2)
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7. J(-4, 1), K(-4, -2), L(6, -2), M(6, 1)
8. D(5, -3), E(5, -6), F(2, -6), G(2, -3)
In Exercises 9–14, use the diagram.
9. Find the perimeter of ΔABD.
10. Find the perimeter of ΔBCD.
11. Find the perimeter of quadrilateral ABCD.
12. Find the area of ΔABD.
13. Find the area of ΔBCD.
14. Find the area of quadrilateral ABCD.
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Section 5 Questions – Measuring Angles:
In Exercises 1–3, name three different angles in the diagram.
1. 2. 3.
In Exercises 4–9, find the indicated angle measure(s).
4. Find 푚∠퐽퐾퐿. 7. Find 푚∠퐶퐴퐷 and 푚∠퐵퐴퐷.
5. 푚∠푅푆푈 = 91°. Find 푚∠푅푆푇. 8. 퐸퐺 bisects ∠퐷퐸퐹. Find 푚∠퐷퐸퐺 and 푚∠퐺퐸퐹.
6. ∠푈푊푋 is a straight angle. Find 9. 푄푅 bisects ∠푃푄푆. Find 푚∠푃푄푅 and 푚∠푃푄푆. 푚∠푈푊푉 푎푛푑 푚∠푋푊푉.
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Section 6 Questions – Describing Pairs of Angles:
In Exercises 1 and 2, use the figure.
1. Name the pair(s) of adjacent complementary angles.
2. Name the pair(s) of nonadjacent supplementary angles.
In Exercises 3 and 4, find the angle measure.
3. ∠A is a complement of ∠B and m∠=36 .° Find m∠B.
4. ∠C is a supplement of ∠D and m∠D=117 .° Find m∠C.
In Exercises 5 and 6, find the measure of each angle.
5. 6.
In Exercises 7–9, use the figure.
7. Identify the linear pair(s) that include ∠1.
8. Identify each vertical angle pair.
9. Are ∠6 and ∠7 a linear pair? Explain.
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