Mathematics Course: Geometry Unit 5: Congruent Triangles (cont) Unit 6: Relationships with Triangles (begin) TEKS Guiding Questions/ Specificity

Designated Grading Period: 3rd Days to teach: 12 Assessment

Vocabulary

Instructional Strategies

Resources/ Weblinks

G.(2) Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the oneand two-dimensional coordinate systems to verify geometric conjectures. G.2(B) derive and Use the distance formula to Graph the quadrilateral with the given Distance -Use Pythagorean Big Ideas Geometry use the distance, slope, verify geometric vertices in a coordinate plane. Then Slope Theorem (previous 5.8, 6.1 and midpoint formulas relationships, including show that the quadrilateral is a Midpoint knowledge) to derive to verify geometric congruence of segments. parallelogram. Length Distance Formula relationships, Parallel -Relate concept of including congruence Use the midpoint formula A(0,0), B(1,4), C(6,6), D(5,2) Perpendicular midpoint to the of segments and to verify geometric Quadrilateral concept of average parallelism or relationships. Parallelogram perpendicularity of Rectangle pairs of lines. Rhombus Square Readiness Standard Misconception:  The student may substitute the x- and y-values incorrectly when using the formulas. (ie: substitute the y value for x)  The student may divide a value by “2” instead of takin gthe square root when using the distance formula.  The student may add the x-value to the y-value, instead of computing the sum of the x-values and computing the sum of the y-values before dividing by 2 in the midpoint formula.  The student may incorrectly write the ratio of the slope of a line as the ratio of horizontal change divided by the vertical change (ie. x2  x1 ). y2  y1

2016-2017

Page 1

Mathematics Course: Geometry Unit 5: Congruent Triangles (cont) Unit 6: Relationships with Triangles (begin) TEKS Guiding Questions/ Specificity

Designated Grading Period: 3rd Days to teach: 12 Assessment

Vocabulary

Instructional Strategies

G.(5) Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. G.5(A) investigate Investigate patterns to Jordan provided the information to Diagonal Connect formula for patterns to make make conjectures about Equilateral polygon sum of the interior prove that LMN  NQP using conjectures about geometric relationships, Equiangular polygon angles of a triangle to the SAS Congruence Theorem. Is his geometric including: Regular polygon the formula for sum of information correct? relationships, Exterior angle the interior angles of a  criteria for triangle including angles Interior angle polygon congruence formed by parallel Exterior angle  special segments of lines cut by a Convex triangles. transversal, criteria Parallelogram required for triangle Quadrilateral congruence, special Diagonal segments of triangles, Segment bisector diagonals of Perpendicular quadrilaterals, interior Correct answer: and exterior angles of Yes; Two pairs of sides and the polygons, and special included angles are congruent. segments and angles of circles choosing from a variety of tools. Readiness Standard

Resources/ Weblinks Big Ideas Geometry 5.3, 5.5, 6.3 www.khanacademy.org Engaging Mathematics p. 149 (58.pdf) Engaging Mathematics p. 183 (71.pdf)

Misconceptions:  The student may make a conjecture based on limited investigation of patterns.  The student may randomly state a conjecture without investigating and recognizing patterns.  The student may not know how to use a construction to make a conjecture.  The student may not be able to perform constructions correctly.  The student may not state a conjecture using precise geometric vocabulary.

2016-2017

Page 2

Mathematics Course: Geometry Unit 5: Congruent Triangles (cont) Unit 6: Relationships with Triangles (begin) TEKS Guiding Questions/ Specificity G.5(C) use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships. Supporting Standard

Use the constructions of angle bisectors to make conjectures about geometric relationships.

Designated Grading Period: 3rd Days to teach: 12 Assessment

In the figure shown,

QP  2 x  9 and Angle bisector

QM  5x  3 . Find QN.

Use the constructions of perpendicular bisectors to make conjectures about geometric relationships.

Vocabulary

Bisect Congruent

Instructional Strategies Tracing paper is helpful when explorations are done by paper folding.

Resources/ Weblinks Big Ideas Geometry 6.1, 6.2

Consider using a local map that your students would be familiar with. Take a screen shot and import it into your dynamic geometry software. Correct Answer: QN= 17

2016-2017

Page 3

Mathematics Course: Geometry Unit 5: Congruent Triangles (cont) Unit 6: Relationships with Triangles (begin) TEKS Guiding Questions/ Specificity

Designated Grading Period: 3rd Days to teach: 12 Assessment

Vocabulary

Instructional Strategies

Resources/ Weblinks

G.(6) Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to: G.6(A) verify theorems Prove equidistance Find the measure of GH. Angles Use coordinate Big Ideas Geometry about angles formed by between the endpoints of Endpoints geometry to 6.1 the intersection of lines a segment and points on Equidistance demonstrate and line segments, its perpendicular bisector. Intersection relationships to prove including vertical Line that points on the angles, and angles Apply these relationships Line segment perpendicular bisector formed by parallel lines to solve problems. Parallel lines of a segment are cut by a transversal and Perpendicular bisector equidistant from the prove equidistance “Prove” by using formal Theorem endpoints of the between the endpoints proof to be shown in Transversal segment. of a segment and points Vertical angles  Paragraph on its perpendicular  Flow chart bisector and  Two-column Correct Answer: apply these formats. relationships to solve 4.6, because GK  KJ and problems. HK  GJ , point H is on the Readiness Standard Misconceptions: perpendicular bisector of GJ . So, by  The student may not use logical reasoning correctly to work through the Perpendicular Bisector Theorem, proofs. GH  HJ  4.6 .  The student may not apply justification to support statements in a twocolumn proof.

2016-2017

Page 4

Mathematics Course: Geometry Unit 5: Congruent Triangles (cont) Unit 6: Relationships with Triangles (begin) TEKS Guiding Questions/ Specificity G.6(B) prove two triangles are congruent by applying the SideAngle-Side, AngleSide-Angle, Side-SideSide, Angle-AngleSide, and HypotenuseLeg congruence conditions. Readiness Standard

Prove two triangles are congruent by applying the  Side-Angle-Side  Angle-Side-Angle  Side-Side-Side  Angle-Angle-Side  Hypotenuse-Leg congruence condition.

Designated Grading Period: 3rd Days to teach: 12 Assessment

Given AC  EC , BC  DC Prove

ABC  EDC

Correct answer:

Given: Can you use ASA to prove RST  TUR? If yes, what theorem(s), postulate(s), or property(ies) did you use besides ASA? If no, what other information would make it possible to prove the triangles congruent by ASA?

Vocabulary

Instructional Strategies

Resources/ Weblinks

Corollary Have students keep all Big Ideas Geometry Corresponding Angles the related definitions, 5.3, 5.5, 5.6, 5.7 Corresponding postulates and Polygons theorems together. www.khanacademy.org Corresponding Sides Included Angle Use facts definitions Engaging Mathematics Included Side postulates theorems Geometry Interior and properties to prove p.89-90 (35.pdf) Triangle Rigidity statements true or SAS false. SSS ASA Analyze and produce AAS proofs to solve AL problems Misconceptions:  The student may think Side-Side-Angle may be a congruence condition that proves two triangles are congruent.

Correct answer: a) yes b)if parallel lines then alternate interior angles are congruent (∠URT ≅ ∠STR, ∠SRT ≅ ∠UTR) Reflexive property of congruence ̅̅̅̅ RT ≅ ̅̅̅̅ RT

2016-2017

Page 5

Mathematics Course: Geometry Unit 5: Congruent Triangles (cont) Unit 6: Relationships with Triangles (begin) TEKS Guiding Questions/ Specificity G.6(D) 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. Supporting Standard

2016-2017

Verify theorems about the relationships in triangles, including medians.

Designated Grading Period: 3rd Days to teach: 12 Assessment

Vocabulary

Point P is the centroid of LMN . The measure of QN = 30. Find PN and QP.

Base Angle Interior Angles Isosceles Triangle Median Mid-Segment Pythagorean Theorem Theorem Triangle Centroid Orthocenter

Apply these relationships to solve problems. Methods for proving may include coordinate, transformations, axiomatic, and formats such as two-column, paragraph, or flow chart.

Instructional Strategies Use rules for congruent or similar triangles to prove relationships.

Resources/ Weblinks Big Ideas Geometry 6.2, 6.3

Include activities to apply these relationships to solve problems.

Correct Answer: 20, 10

Page 6