Download Geometry

Geometry
Unit 2
Title
Lines, Angles, and Triangles
Big Ideas/Enduring Understandings
Module 4
Parallel lines and Perpendicular Lines can be used to solve real-world
problems.
Module 5
Triangle congruence can be used to solve real-world problems.
Suggested Time Frame
1st and 2nd Six Weeks
30 Days
Guiding Questions
Module 4
How can you find the measure of angles formed by intersecting lines?
How can you prove and use theorems about angles formed by transversals
that intersect parallel lines?
How can you prove that two lines are parallel?
What are the key ideas about perpendicular bisectors of a segment?
How can you find the equation of a line that is parallel or perpendicular to a
given line?
How can you use parallel and perpendicular lines to solve real-world
problems?
Module 5
How can you show that two triangles are congruent?
What does the ASA Triangle Congruence Theorem tell you about triangles?
What does the SAS Triangle Congruence Theorem tell you about triangles?
What does the SSS Triangle Congruence Theorem tell you about triangles?
How can you use triangle congruence criteria to solve real-world problems?
Module 6
Applications of Triangle Congruence can be used to solve realworld problems.
Module 7
Geometry Unit 2
Updated April 2015
Module 6
How can you be sure that the result of a construction is valid?
What does the AAS Triangle Congruence Theorem tell you about two
triangles?
What does the HL Triangle Congruence Theorem tell you about two
triangles?
How can you use triangle congruence to solve real-world problems?
Properties of Triangles can be used to solve real-world problems.
Module 8
Special segments in triangles can be used to solve real-world
problems.
Module 7
What can you say about the interior and exterior angles of a triangle
and other polygons?
What are the special relationships among angles and sides in
isosceles and equilateral triangles?
How can you use inequalities to describe the relationships among side
lengths and angle measures in a triangle?
How can you use the properties of triangles to solve real-world
problems?
Module 8
How can you use perpendicular bisectors to find the point that is
equidistant from all the vertices of a triangle?
How can you use angles bisectors to find the point that is equidistant
from all the sides of a triangle?
How can you find the balance point or center of gravity of a triangle?
How are the segments that join the midpoints of a triangle’s sides
related to the triangle’s sides?
How can you use special segments in triangles to solve real-world
problems?
Vertical Alignment Expectations
TEA Vertical Alignment Chart Grades 5-8, Geometry
Sample Assessment Question
Coming Soon....
Geometry Unit 2
Updated April 2015
The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper
depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the
suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the
district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material.
Ongoing TEKS
Math Processing Skills
G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding.
The student is expected to:
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(A) apply mathematics to problems arising in everyday life, society, and the
workplace;
(B) use a problem-solving model that incorporates analyzing given
information, formulating a plan or strategy, determining a solution, justifying
the solution, and evaluating the problem-solving process and the
reasonableness of the solution;
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(C) select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math,
estimation, and number sense as appropriate, to solve problems;
Focus is on application
Students should assess which tool to apply rather than trying only one or all
(D) communicate mathematical ideas, reasoning, and their implications using
multiple representations, including symbols, diagrams, graphs, and language
as appropriate;
(E) create and use representations to organize, record, and communicate
mathematical ideas;
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(F) analyze mathematical relationships to connect and communicate
mathematical ideas; and
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Students are expected to form conjectures based on patterns or sets of examples
and non-examples
(G) display, explain, and justify mathematical ideas and arguments using
precise mathematical language in written or oral communication
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Precise mathematical language is expected.
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Students should evaluate the effectiveness of representations to ensure they are
communicating mathematical ideas clearly
Students are expected to use appropriate mathematical vocabulary and phrasing
when communicating ideas
Knowledge and Skills with
Student Expectations
District Specificity/ Examples
G.2 Coordinate and
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Transformational Geometry.
The student uses the process
skills to understand the
connections between algebra
and geometry and uses the
one- and twodimensional
coordinate systems to verify
geometric conjectures.
(C) determine an equation of
a line parallel or
perpendicular to a given
line that passes through
a given point.
G.2C
Prior Knowledge: Students write an equation of a line that
goes through a given point that is parallel or perpendicular to
another line (Algebra 2E, 2F, 2G).
Teachers will still need to review the topic (point slope
formula, slope intercept formula) even though it was covered
in Algebra. Teachers will need to show how this will be
applied to figures on a coordinate grid.
Examples:
(1) A triangle with endpoints is given on a coordinate grid and
students need to find an equation of the altitude for one of
the sides.
(2) Find the equation of the tangent line to a circle given the
endpoint of the radius on the coordinate grid.
(3) Given Rectangle ABCD on a grid, find the equation for side
AB, given the slope of side CD is ½.
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Vocabulary
linear equations
parallel lines
perpendicular lines
point-slope form
slope
slope-intercept form
standard form
y-intercept
Suggested Resources
Resources listed and categorized to
indicate suggested uses. Any
additional resources must be aligned
with the TEKS.
HMH Geometry
Unit 2
Region XI: Livebinder
NCTM: Illuminations
G.5 Logical Argument and
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Constructions. The student
uses constructions to validate
conjectures about geometric
figures.
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G.5A
(A) The student is expected
to investigate patterns
to make conjectures
about geometric
relationships, including
angles formed by
parallel lines cut by a
transversal, criteria
required for triangle
congruence, special
segments of triangles,
diagonals of
quadrilaterals, interior
and exterior angles of
polygons, and special
segments and angles of
circles choosing from a
variety of tools.
Teachers should:
Show the relationships of parallel lines cut by a transversal &
patterns it creates
How patterns lead to theorems
Example: Pattern in interior angle sums
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Students should:
• Investigate patterns
• Discover theorems based on investigation type
activities
Misconceptions:
Angle relationships exist with any two lines at transversal.
Students must understand that lines cut by a transversal
Geometry Unit 2
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acute triangle
adjacent angles
alternate exterior
angles
alternate interior
angles
altitude
angle
angle bisector
bisect
central angle
chord
circle
congruent
congruent angles
congruent segments
constructions
corresponding
angles
diagonal
exterior angle
interior angle
linear pair
median
midsegment of a
triangle
obtuse triangle
parallel lines
perpendicular
perpendicular
bisectors
polygon
quadrilateral
right triangle
must be parallel of the angles relationships to exist.
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Examples:
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Angles 8 and 4 are Alternate exterior angles that are NOT
congruent.
VS.
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Angles 1 and 2 are Alternate exterior angles that ARE
congruent BECAUSE the two lines are parallel.
(B) The student is expected
to construct congruent
segments, congruent
angles, a segment
bisector, an angle
bisector, perpendicular
lines, the perpendicular
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G.5B
Teachers should:
Make formal geometric constructions with a variety of tools
and methods
Possible tools - compass, folding, patty paper, string,
reflective mirrors, geo software
same side interior
angles
secant
segment
side length
side-angle-side,
angle-side-angle,
hypotenuse-leg)
supplementary
tangent
transversal
triangle
triangle congruence
(side-by-side,
Triangle Inequality
Theorem
vertex
vertical angles
bisector of a line
segment, and a line
parallel to a given line
through a point not on a
line using a compass
and a straightedge.
Students should:
Copy a segment
Copy an angle
Bisect an angle
Construct perpendicular lines including perpendicular
bisectors
Construct parallel line
Misconceptions:
Assume one part is true (ex. 90 degrees)
Examples:
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(C) The student is expected
to use the constructions
of congruent segments,
congruent angles, angle
bisectors, and
perpendicular bisectors
to make conjectures
about geometric
relationships; and
(D) The student is expected
to verify the Triangle
Inequality theorem
using constructions and
apply the theorem to
solve problems.
Geometry Unit 2
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G.5C
Students:
• Prove theorems about lines & angles
• Vertical angles are congruent
• Alternate interior & corresponding are congruent
• Endpoints of a line segment that is cut by a
perpendicular bisector are equidistant from the point
of intersection
G.5D*
Prove:
The triangle inequality theorem states that any side of a
triangle is always shorter than the sum of the other two
sides.
Converse: A triangle cannot be constructed from three line
segments if any of them is longer than the sum of the other
two.
Examples:
1. Which of the following could represent the lengths of the
sides of a triangle? Choose one:
1, 2, 3
6, 8, 15
5, 7, 9
G.6 Proof and Congruence.
The student uses the process
skills with deductive
reasoning to prove and apply
theorems by utilizing a
variety of methods such as
coordinate, transformational,
axiomatic and formats such
as two-column, paragraph,
flow chart
G.6A
Confirm theorems about angles formed by
to verify theorems
about angles formed by Vertical lines, parallels cut by transversal
the intersection of lines Prove equidistance b/t endpoints of a segment & points
and line segments,
on its perpendicular bisector
including vertical angles, Apply these relationships to solve problems
angles formed by
Students should know how to:
parallel lines cut by a
• Prove vertical angles are congruent
transversal, and prove
• Prove when a transversal crosses parallel lines, alternate
(A) The student is expected
equidistance between
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Angle-Angle-Side
angles
Angle-Side-Angle
base angles
endpoints
equidistance
Hypotenuse-Leg
interior angles
intersection
isosceles triangle
line
line segment
median
mid-segment
parallel lines
perpendicular
bisector
Pythagorean
Theorem
Side-Angle-Side
the endpoints of a
segment and points on
its perpendicular
bisector, and apply
these relationships to
solve problems;
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interior angles are congruent and corresponding angles
are congruent.
Find midpoint and possibly use distance formula.
Understand labels & symbols in order to apply the
relationships.
Finding missing angles and measurements.
Identifying if lines are parallel
Every point on a point lying on the perpendicular bisector
of a segment is equidistant from the endpoints of the
segment.
Teachers should show:
• Vertical Angle Theorem
• Corresponding Angles
• Perpendicular Transversal
• Perpendicular Bisector Theorem
• Parallel Transversal (4) - Alt Int/ Ext, SS Int/Ext
• Examples on how to solve missing angles
Misconceptions
Vocabulary, proofs
Vertical angles are formed by 2 straight lines, not simply rays
that come together at the same point. For example:
Geometry Unit 2
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Side-Side-Side
Theorem
transversal
Triangle
vertical angles
Angle RPS is not a vertical angle to Angle QPU because Q and
S do not lie along the same line.
Example w/ proving angle measurements
Example proving theorems
Geometry Unit 2
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Example Equidistance with perpendicular bisector
(B) The student is expected
to prove two triangles
are congruent by
applying the Side-AngleSide, Angle-Side-Angle,
Side-Side-Side, AngleAngle-Side, and
Hypotenuse-Leg
congruence conditions;
G.6B
Big Idea - Use the definition of congruence to develop and
explain the triangle congruence criteria; ASA, SSS, and SAS.
Teachers should:
For Right Triangles: Explain postulates including Leg-Leg
(SAS),
• Hypotenuse-Angle (AAS)
**Reminder: “A’ is not the right angle, it must be one
of the acute angles
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Hypotenuse-Leg (SSS) - only true for right triangle bc since
it has a right angle you can use the Pythagorean theorem.
it only will give you one possible value for the 3rd side.
Creates SSS congruence
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Review special segments
Give examples proving congruence
Students should:
• Be able to prove two triangles congruent using
theorems.
Examples
Give any additional information that would be needed to
prove the triangles congruent by the method given.
Determine which method can be used to prove the triangles
congruent from the information given. For some pairs, it may
not be possible to prove the triangles congruent. For these,
explain what other information would be needed to prove
congruence.
Misconception: Part a - Students forget that a shared side b/t
two figures have the same length because it is the same side.
Students mistakenly use Angle-side-side as a congruence
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theorem.
HL - is not a version of SSA because you cannot use the 90
degree angle as your ‘A’
(C) The students is expected
to apply the definition
of congruence, in terms
of rigid transformations,
to identify congruent
figures and their
corresponding sides and
angles.
G.6C
Big Idea - identify congruent figures and identify their
corresponding parts
Rigid Transformation - when the size of the figure does NOT
change (translation, rotation, reflection, NOT dilation)
Teachers should show:
• Explain what a rigid transformation is and how the size of
the figure will not change
• Explain definition of congruence & corresponding parts
• Relate terminology of rigid transformations (reflection,
translation, rotation) to congruent transformations CPCTC
**Review how to find the length of a segment in the
coordinate plane, slope of parallel/perpendicular lines
Students should do:
• Look at two figures & identify corresponding parts
• Prove congruence based on congruent corresponding
parts
Misconceptions:
Students should understand that figures can be congruent
Geometry Unit 2
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even when they are rotated or flipped. Congruence is true
even if figures have different orientation.
Examples:
Geometry Unit 2
Updated April 2015
(D) The student is expected
to verify theorems
about the relationships
in triangles, including
proof of the
Pythagorean Theorem,
the sum of interior
angles, base angles of
isosceles triangles, midsegments, and medians
and apply these
relationships to solve
problems
G.6D
Big Idea - verify theorems about relationships in TRIANGLES
and apply to solve problems
Teaches need to show:
• Proof Pythagorean theorem - need to show an
Algebraic proof and concrete proof
• Sum of interior angles - (180)
• Base angles of isosceles - base angles are congruent
Midsegment of a triangle is a segment connecting the
midpoints of two sides of a triangle. This segment has two
special properties. It is always parallel to the third side, and
the length of the midsegment is half the length of the third
side.
Median of a triangle is a line segment that extends from one
vertex of a triangle to the midpoint of the opposite side. All 3
will intersect to form centroid. The centroid splits the
medium in a two to one ratio.
Students need to:
• Understand relationships in theorems in order to
apply and solve problems
• Find midpoint of a side, midsegment, median
• Find length midsegment given parallel side
• Apply Pythagorean formula
• Find missing angles knowing angle sum is 180 or find
missing angle given a single base angle
Geometry Unit 2
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Example - Pythagorean proof
Pythagorean Theorem Application
Geometry Unit 2
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Median Application
⅔(EM)=ET
Midsegment Examples
Geometry Unit 2
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Geometry Unit 2
Updated April 2015