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Questions and Answers
What defines a right angle?
What defines a right angle?
Which term describes two angles that share a common side and vertex?
Which term describes two angles that share a common side and vertex?
What is the degree measure of an obtuse angle?
What is the degree measure of an obtuse angle?
Which statement about vertical angles is correct?
Which statement about vertical angles is correct?
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What does an angle bisector do?
What does an angle bisector do?
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What is the sum of the measures of two complementary angles?
What is the sum of the measures of two complementary angles?
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Which pair of angles describes a linear pair?
Which pair of angles describes a linear pair?
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If two angles sum to 180°, what type of angles are they?
If two angles sum to 180°, what type of angles are they?
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Which of the following pairs of angles is supplementary?
Which of the following pairs of angles is supplementary?
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What characteristic do complementary angles share?
What characteristic do complementary angles share?
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What does the Segment Addition Postulate state about points A, B, and C?
What does the Segment Addition Postulate state about points A, B, and C?
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What is the formula used to find the distance between two points in a coordinate plane?
What is the formula used to find the distance between two points in a coordinate plane?
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Which formula is used to find the midpoint between two endpoints?
Which formula is used to find the midpoint between two endpoints?
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What is a segment bisector?
What is a segment bisector?
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Which statement accurately describes parallel lines?
Which statement accurately describes parallel lines?
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What defines a point in geometry?
What defines a point in geometry?
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Which statement accurately describes a line?
Which statement accurately describes a line?
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What characterizes a line segment?
What characterizes a line segment?
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Which of the following statements about coplanar points is true?
Which of the following statements about coplanar points is true?
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What does it mean for segments to be congruent?
What does it mean for segments to be congruent?
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What is Angle 'BAD'?
What is Angle 'BAD'?
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What is the formula for finding the distance between two points on a coordinate plane?
What is the formula for finding the distance between two points on a coordinate plane?
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What is the Midpoint Formula?
What is the Midpoint Formula?
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The angle measured at 90° is called an obtuse angle.
The angle measured at 90° is called an obtuse angle.
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Two angles are complementary if they sum to 180°.
Two angles are complementary if they sum to 180°.
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What defines congruent segments?
What defines congruent segments?
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Define coplanar points.
Define coplanar points.
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How many planes appear in a given figure?
How many planes appear in a given figure?
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A line that extends indefinitely in one direction is called a ______.
A line that extends indefinitely in one direction is called a ______.
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What is a linear pair?
What is a linear pair?
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What is the definition of a right angle?
What is the definition of a right angle?
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What is Angle 'BAD'?
What is Angle 'BAD'?
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Which of the following statements is true regarding parallel lines?
Which of the following statements is true regarding parallel lines?
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What is the Segment Addition Postulate?
What is the Segment Addition Postulate?
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The formula used to find the distance between two points on a coordinate plane is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
The formula used to find the distance between two points on a coordinate plane is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
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What is the Midpoint Formula?
What is the Midpoint Formula?
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What type of angle has a degree measure of 90°?
What type of angle has a degree measure of 90°?
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Complementary angles are angles that sum to 180°.
Complementary angles are angles that sum to 180°.
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What are vertical angles?
What are vertical angles?
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If LM = 22 and MN = 15, what is the value of LN according to the Segment Addition Postulate?
If LM = 22 and MN = 15, what is the value of LN according to the Segment Addition Postulate?
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Match the following geometrical terms with their definitions:
Match the following geometrical terms with their definitions:
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Congruent angles have the same measure.
Congruent angles have the same measure.
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Define an obtuse angle.
Define an obtuse angle.
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Angles that share a common side and vertex are called ______.
Angles that share a common side and vertex are called ______.
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Study Notes
Angle Concepts
- Vertex: The point where two lines of an angle meet.
- Right Angle: An angle measuring exactly 90 degrees.
- Acute Angle: An angle measuring less than 90 degrees.
- Obtuse Angle: An angle measuring greater than 90 degrees and less than 180 degrees.
- Straight Angle: An angle measuring 180 degrees.
- Congruent Angles: Angles with the same measure.
- Adjacent Angles: Two angles sharing a common vertex and side, positioned next to each other.
- Angle Bisector: A line or ray dividing an angle into two equal angles.
- Vertical Angles: Opposite angles formed by two intersecting lines. They are always congruent.
- Angle Addition Postulate: The sum of two adjacent angles equals the measure of the larger angle they make together.
Geometric Concepts
- Point: A specific location with no size or dimension.
- Line: A straight path of points extending infinitely in opposite directions.
- Line Segment: A portion of a line with two defined endpoints.
- Ray: A portion of a line with one endpoint and continuing infinitely in one direction.
- Plane: A flat, two-dimensional surface extending infinitely.
- Collinear: Points lying on the same line.
- Coplanar: Points lying on the same plane.
- Congruent Segments: Line segments with equal length.
Segment Concepts
- Segment Addition Postulate: The sum of the lengths of two smaller segments equals the length of the larger segment they compose.
- Distance Formula: Used to find the distance between two points in a coordinate plane: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2−x1)2+(y2−y1)2, where (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2) are the coordinates of the two points.
- Midpoint Formula: Used to find the midpoint of a line segment: M=(x1+x22,y1+y22)M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})M=(2x1+x2,2y1+y2).
- Segment Bisector: divides a segment into two equal parts.
- Perpendicular Bisector: A line, segment, or ray that intersects a segment at its midpoint and forms a right angle.
- Parallel Lines: Two lines that never intersect.
Angle Relationships
- Complementary Angles: Two angles whose measures add up to 90 degrees. They may or may not be adjacent.
- Supplementary Angles: Two angles whose measures add up to 180 degrees. They may or may not be adjacent.
- Linear Pair: Two adjacent supplementary angles that form a straight line.
Measuring Segments & Angles & Defining Congruence
- Segment Addition Postulate: If points A, B, and C are collinear and B is between A and C, then AB + BC = AC.
- Congruent Segments: Segments with the same length are congruent, denoted by AB ≅ CD.
- Midpoint of a Segment: A point that divides a segment into two equal parts. A line, ray, or segment intersecting a segment at its midpoint bisects it.
Finding Segment Measures using Segment Addition Postulate
- Examples: Use the Segment Addition Postulate to solve problems involving segments and their lengths.
Measuring Distance in the Coordinate Plane (Distance Formula)
- Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2−x1)2+(y2−y1)2 used to find the distance between two points in a coordinate plane.
Measuring Distance in the Coordinate Plane (Distance Formula) Day 2
- Practice: Apply the Distance Formula to calculate distances between points.
Parallel & Perpendicular Lines in the Coordinate Plane
- Parallel Lines: Lines that never intersect.
- Perpendicular Lines: Lines that intersect at a right angle.
Learning Checkpoint 2.1 & Vocabulary Quiz
- Review: Concepts related to measuring segments, angles, and congruence.
Assessment 2.1-2.4
- Evaluation: Assess understanding of the covered geometric concepts.
Review/Practice Activity
- Reinforcement: Practice and reinforce the learned concepts.
Partitioning a Segment & Bisecting a Segment
- Partitioning: Dividing a segment into specific ratios.
- Bisecting: Dividing a segment into two equal parts.
Transversals
- Transversal: A line that intersects two or more other lines.
Triangle Sum Theorem
- Triangle Sum Theorem: The sum of the interior angles of a triangle always equals 180 degrees.
Trig Ratios: SOH CAH TOA Finding Missing sides in Right Triangles
- SOH CAH TOA: Trigonometric ratios (sine, cosine, tangent) used to find missing sides in right triangles.
Trig Ratios: SOH CAH TOA Finding Missing angles in Right Triangles
- Inverse Trigonometric Functions: Used to find missing angles in right triangles when side lengths are known.
Angles of Elevation and Depression
- Angle of Elevation: The angle formed between the horizontal and the line of sight when looking upwards.
- Angle of Depression: The angle formed between the horizontal and the line of sight when looking downwards.
Applications of Trig Ratios/Group Quiz
- Real-World Problems: Apply trigonometry to solve practical problems.
Final Review
- Comprehensive Review: Review all concepts covered in the unit.
All SEMESTER 1 Work due by DECEMBER 15
- Submission Deadline: Submit all assignments and assessments by the deadline.
SEMESTER 1 FINAL EXAMS
- Final Evaluation: Comprehensive evaluation of all concepts covered in Semester 1.
Naming Points, Lines, and Planes
- Practice: Learn to identify and name points, lines, and planes in diagrams.
Points, Lines, & Planes
- Definitions: Define the basic geometric concepts of points, lines, and planes.
Naming Points, Lines, and Planes: Practice!
- Exercises: Practice naming points, lines, and planes using diagrams.
Homework 1: Points, Lines, and Planes
- Practice Problems: Apply the understanding of points, lines, and planes to solve problems.
Homework 2: Segment Addition Postulate
- Practice Problems: Apply the Segment Addition Postulate to solve problems involving segments.
Measuring Segments
- Concept Definition: Understand how to measure segments and express their lengths.
Congruent Segments
- Concept Definition: Understand the concept of congruent segments and their notation.
Examples
- Practice Problems: Utilize the Segment Addition Postulate and midpoint concepts to solve problems.
Segment Addition Postulate Maze
- Interactive Exercise: Use the Segment Addition Postulate to navigate a geometric maze.
Unit 2 Tools & Techniques of Geometric Measurement Note Packet
- This unit focuses on geometric measurement, including segments, angles, and right triangles.
- Understanding geometric measurements is crucial for comparing and analyzing shapes and their properties.
Key Concepts
- Measuring segments and angles using various postulates and formulas.
- Identifying parallel and perpendicular lines and their specific properties.
- Using trigonometric ratios (SOH CAH TOA) to solve right triangle problems.
Enduring Understandings
- Mathematical arguments are built upon logically combining existing facts.
- Geometric figures are characterized by numeric values assigned to their attributes.
- Parallel and perpendicular lines exhibit unique geometric and algebraic characteristics.
- Right triangles play a significant role in relating angle measures to side lengths.
Time Frame
- Unit 2 is divided into seventeen weeks with corresponding topics and deadlines.
Unit 2 Dictionary: Geometry Basics
- Points: Locations without size or shape.
- Lines: Made up of points without thickness or width.
- Line Segments: Measurable parts of a line with two endpoints.
- Rays: Lines extending indefinitely in one direction.
- Planes: Flat surfaces extending infinitely in all directions.
- Collinear Points: Points lying on the same line.
- Coplanar Points: Points residing on the same plane.
- Congruent Segments: Segments with equal lengths.
Segment Addition Postulate
- If three collinear points (A, B, and C) are arranged with B in between A and C, then the sum of the lengths of AB and BC equals the length of AC.
Distance Formula
- The distance between two points on a coordinate plane is calculated using the formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint Formula
- The midpoint between two endpoints on a coordinate plane is found using the formula: $M = ( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$
Segment Bisector
- A line, segment, or plane intersecting a segment at its midpoint.
Perpendicular Bisector
- A line, segment, or ray perpendicular to a segment at its midpoint.
Parallel Lines
- Two lines that never intersect.
Angle
- The intersection of two rays at an endpoint.
Vertex
- The common endpoint of an angle, where the rays intersect.
Right Angle
- An angle measuring 90°.
Acute Angle
- An angle measuring less than 90°.
Obtuse Angle
- An angle measuring greater than 90°.
Straight Angle
- An angle measuring 180°.
Congruent Angles
- Two angles with equal measures.
Adjacent Angles
- Two angles sharing a common side and vertex.
Angle Addition Postulate
- If two adjacent angles (∠ABD and ∠DBC) share a common vertex, then the sum of their measures equals the measure of the larger angle (∠ABC).
Angle Bisector
- A line or ray dividing an angle into two equal parts.
Vertical Angles
- Two angles directly opposite each other on intersecting lines. Vertical angles are always congruent.
Complementary Angles
- Two angles whose measures add up to 90°. They don't need to be adjacent.
Supplementary Angles
- Two angles whose measures add up to 180°. They don't need to be adjacent.
Linear Pair
- Adjacent angles that are supplementary. Together, they form a straight line.
Naming Points, Lines, and Planes: Practice (Practice 1-4)
- These practice exercises involve identifying and naming points, lines, and planes using given diagrams.
Homework 1: Points, Lines, and Planes (Optional but Recommended Practice)
- These practice exercises involve identifying and naming points, lines, and planes, and answering specific questions based on diagrams.
Homework 2: Segment Addition Postulate
- These practice exercises involve applying the Segment Addition Postulate to solve for unknown segment lengths and midpoint values using given diagrams.
Measuring Segments
- The distance between two points A and B can be represented as AB or AB =.
Congruent Segments
- If two segments (AB and CD) have equal lengths, they are considered congruent, represented as AB ≅ CD.
Segment Addition Postulate
- If three collinear points (A, B, and C) are arranged with B in between A and C, then the sum of the lengths of AB and BC equals the length of AC.
Midpoint of a Segment
- The midpoint of a segment divides it into two equal parts.
- A line, ray, or segment that intersects a segment at its midpoint is called a segment bisector and bisects the segment.
Examples (1-11)
- These examples demonstrate using the Segment Addition Postulate, the Midpoint Formula, and other concepts to find unknown lengths and values in geometric figures.
Segment Addition Postulate Maze (Show All Work)
- This maze requires solving segment addition problems to navigate through the maze using given diagrams.
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Description
Test your knowledge on essential angle properties and basic geometric principles. This quiz covers definitions and characteristics of various angles, as well as foundational geometric elements like points and lines. Perfect for students looking to reinforce their understanding of geometry.
