Curriculum / Math / 10th Grade / Unit 2: Congruence in Two Dimensions / Lesson 17
Math
Unit 2
10th Grade
Lesson 17 of 18
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Prove that the opposite sides and opposite angles of a parallelogram are congruent.
The core standards covered in this lesson
G.CO.C.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.
The foundational standards covered in this lesson
8.G.A.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Below is parallelogram $${ABCD}$$ where $${\overleftrightarrow{BC} \parallel \overleftrightarrow{AD}}$$ and $${\overleftrightarrow{BA} \parallel \overleftrightarrow{CD}}$$.
What is the relationship between opposite angles of a parallelogram? Justify your reasoning.
In quadrilateral $${{DEFG}}$$ pictured below, $${\overline{DE} \cong \overline{FG}}$$ and $${\overline{DF} \cong \overline{EG}}$$.
From the given information, can we deduce that $${{DEFG}}$$ is a parallelogram? Justify your reasoning.
"Is this a parallelogram?", accessed on Aug. 10, 2017, 4:22 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
Find the values of $$x$$, $$y$$, and $$z$$ in the parallelogram shown below. State the reasons you used to find these angle measures.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Johnny declared that the following figure is a parallelogram because an opposite pair of sides is parallel, and an opposite pair of sides is congruent. Â
Describe why the following figure is not a parallelogram.
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
Lesson 16
Lesson 18
Topic A: Introduction to Polygons
Define polygon and identify properties of polygons.
G.CO.A.1 G.CO.C.11
Prove interior and exterior angle relationships in triangles.
G.CO.C.10
Describe and apply the sum of interior and exterior angles of polygons.
G.CO.C.11
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Topic B: Rigid Motion Congruence of Two-Dimensional Figures
Determine congruence of two dimensional figures by translation.
G.CO.A.2 G.CO.A.4 G.CO.A.5 G.CO.B.7
Reflect two dimensional figures on and off the coordinate plane.
G.CO.A.2 G.CO.A.3 G.CO.A.5 G.CO.B.7
Rotate two dimensional figures on and off the coordinate plane.
Describe a sequence of rigid motions that map a pre-image to an image (specifically triangles, rectangles, parallelograms, and regular polygons).
G.CO.A.5 G.CO.B.6
Describe single rigid motions, or sequences of rigid motions that have the same effect on a figure.
G.CO.A.3 G.CO.A.5 G.CO.B.6
Topic C: Triangle Congruence
Develop the Side Angle Side criteria for congruent triangles through rigid motions.
G.CO.B.7 G.CO.B.8
Prove angle relationships using the Side Angle Side criteria.
G.SRT.B.5
Prove and apply that the points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
G.CO.A.2 G.CO.C.10 G.CO.C.9
Develop Angle, Side, Angle (ASA) and Side, Side, Side (SSS) congruence criteria. Describe how the criteria develop from rigid motions.
Use triangle congruence criteria, rigid motions, and other properties of lines and angles to prove congruence between different triangles.
Prove triangles congruent using Angle, Angle, Side (AAS), and describe why AAA is not a congruency criteria.
G.CO.B.7 G.CO.B.8 G.CO.C.10
Develop the Hypotenuse- Leg (HL) criteria, and describe the features of a triangle that are necessary to use the HL criteria.
G.CO.B.7 G.CO.C.10
Use criteria for triangle congruence to prove relationships among angles and sides in geometric problems.
G.CO.C.9 G.SRT.B.5
Topic D: Parallelogram Properties from Triangle Congruence
Prove theorems about the diagonals of parallelograms.
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