ALEX Classroom Resource

  

Geometry Module 1, Topic E: Proving Properties of Geometric Figures

  Classroom Resource Information  

Title:

Geometry Module 1, Topic E: Proving Properties of Geometric Figures

URL:

https://www.engageny.org/resource/geometry-module-1-topic-e-overview

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 1, Topic E, students extend their work on rigid motions and proofs to establish properties of triangles and parallelograms. In Lesson 28, students apply their recent experience with triangle congruence to prove problems involving parallelograms. In Lessons 29 and 30, students examine special lines in triangles, namely, midsegments and medians. Students prove why a midsegment is parallel to and half the length of the side of the opposite triangle. In Lesson 30, students prove why the medians are concurrent.

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
10 ) Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180o, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point. [G-CO10]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.10- Given a measure of a leg or base angle of an isosceles triangle, identify the measure of the other leg or other base angle.


Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
11 ) Prove theorems about parallelograms. Theorems include opposite sides are congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each other; and conversely, rectangles are parallelograms with congruent diagonals. [G-CO11]


Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.11- Given the measure of one side or one angle of a parallelogram, identify the measure of the opposite side or opposite angle.


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
31. Justify whether conjectures are true or false in order to prove theorems and then apply those theorems in solving problems, communicating proofs in a variety of ways, including flow chart, two-column, and paragraph formats.

a. Investigate, prove, and apply theorems about lines and angles, including but not limited to: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; the points on the perpendicular bisector of a line segment are those equidistant from the segment's endpoints.

b. Investigate, prove, and apply theorems about triangles, including but not limited to: the sum of the measures of the interior angles of a triangle is 180?; the base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity.

c. Investigate, prove, and apply theorems about parallelograms and other quadrilaterals, including but not limited to both necessary and sufficient conditions for parallelograms and other quadrilaterals, as well as relationships among kinds of quadrilaterals.

Example: Prove that rectangles are parallelograms with congruent diagonals.

Unpacked Content
Evidence Of Student Attainment:
Students:
  • Make, explain, and justify (or refute) conjectures about geometric relationships with and without technology.
  • Explain the requirements of a mathematical proof.
    1. Present a complete mathematical proof of geometry theorems including the following: vertical angles are congruent. when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent. points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
      Critique proposed proofs made by others.
    2. Present a complete mathematical proof of geometry theorems about triangles, including the following: measures of interior angles of a triangle sum to 180o. base angles of isosceles triangles are congruent. the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. the medians of a triangle meet at a point.
      Critique proposed proofs made by others.
    3. Present a complete mathematical proof of geometry theorems about parallelograms, including the following: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
      Critique proposed proofs made by others.
Teacher Vocabulary:
  • Same side interior angle
  • Consecutive interior angle
  • Vertical angles
  • Linear pair
  • Adjacent angles
  • Complementary angles
  • Supplementary angles
  • Perpendicular bisector
  • Equidistant
  • Theorem Proof
  • Prove
  • Transversal
  • Alternate interior angles
  • Corresponding angles
  • Interior angles of a triangle
  • Isosceles triangles
  • Equilateral triangles
  • Base angles
  • Median
  • Exterior angles
  • Remote interior angles
  • Centroid
  • Parallelograms
  • Diagonals
  • Bisect
Knowledge:
Students know:
  • Requirements for a mathematical proof.
  • Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
  • Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
  • Generate a conjecture about geometric relationships that calls for proof.
Understanding:
Students understand that:
  • Proof is necessary to establish that a conjecture about a relationship in mathematics is always true, and may also provide insight into the mathematics being addressed.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.31.1: Define vertical angle, transversal, parallel lines, alternate interior angles, corresponding angles, perpendicular bisector, line segment, equidistant, endpoints, interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, median, intersection, opposite sides, opposite angles, diagonals, parallelogram, bisector, and converse.
GEO.31.2: Develop a process that demonstrates the logical order of properties to form a proof.
GEO.31.3: Arrange statements to form a logical order.
GEO.31.4: Identify measures of vertical angles, alternate interior angles, corresponding angles, measures of interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, and median.
GEO.31.5: Illustrate vertical angle, transversal, parallel lines, alternate interior angles, corresponding angles, perpendicular bisector, line segment, equidistant, endpoints, interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, median, intersection, opposite sides, opposite angles, diagonals, parallelograms, bisectors, and their properties.
GEO.31.6: Find the measure of the third interior angle of a triangle when given the measure of the other two interior angles.

Prior Knowledge Skills:
  • Define a right angle, Pythagorean Theorem, converse, and proof.
  • Define exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, corresponding angles, and transversal.
  • Identify attributes of triangles.
  • Identify exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, and corresponding angles.
  • Identify a transversal.
  • Apply properties to find missing angle measures.
  • Discover the Angle Sum Theorem (sum of the interior angles of a triangle equal 180 degrees).
  • Identify parallel lines.
  • Define supplementary angles, complementary angles, vertical angles, adjacent angles, parallel lines, perpendicular lines, and intersecting lines.
  • Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
  • Recognize and demonstrate that two right triangles make a rectangle.
  • Recognize polygons.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.


Tags: angles, lines, parallelograms, proof, prove, theorem, triangles
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Comments

There are three lessons in this topic.

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Author: Hannah Bradley