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1 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. The student is expected to: G.2(B) derive and use Use the midpoint In the figure, quadrilateral MNPQ is a Congruent Include the derivation the distance, slope, and formula to verify parallelogram. Which of the following Distance formula of the midpoint 7.1, 7.2, 7.4, 8.3 midpoint formulas to geometric relationships. statements about the figures must be Lines formula. verify geometric Midpoint formula true? relationships, including Use the distance formula Parallel Include activities where congruence of segments and parallelism or perpendicularity of pairs of lines. Readiness Standard to verify geometric relationships, including congruence of segments. Use the slope formula to verify geometric relationships, including parallelism or perpendicularity of pairs of lines. Perpendicular Segments Slope formula student derive and then use the distance formula, slope formula, and midpoint formula. I. MNP PQM Misconceptions: The student may substitute the x-and y-values incorrectly when using the II. MR PR formulas. III. MQ MN The student may divide a value by 2 instead of taking the square root A. I only B. II only C. III only D. I and II only E. I, II, and III Correct answer: D Last Edit Page 1 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 vertical change.
2 G.(5) Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to: G.5(A) investigate Investigate patterns to The measures of three interior angles of Diagonal Patterns should include patterns to make make conjectures about a convex quadrilateral are 37, 110, Equilateral polygon numeric and geometric 7.1, 7.2, 7.4 conjectures about geometric relationships, and 78. Find the measure of the fourth Equiangular polygon properties. geometric relationships, including Regular polygon angle. including angles formed special Polygon Students should by parallel lines cut by a segments of Correct answer: 135 Convex investigate geometric transversal, criteria triangles Interior angles relationships. required for triangle interior angles Exterior angles congruence, special of polygons Quadrilateral Include some segments of triangles, exterior angles Segment bisector relationships with diagonals of of polygons algebraic expressions quadrilaterals, interior diagonals of representing properties. and exterior angles of quadrilaterals polygons, and special segments and angles of Use constructions to circles choosing from a investigate patterns and variety of tools. make conjectures. Readiness Standard 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. Last Edit Page 2
3 G.5(C) use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships. Use the constructions of congruent segments angles to make conjectures about geometric relationships. The diagram below shows the arcs and segments used to construct ABC, given line k. Angle bisector Bisect Congruent Congruent angles Congruent segments Constructions Perpendicular Perpendicular bisector As students construct figures, they should also describe what they see and explain why the construction works. Big Ideas Math Geometry 6.5, 7.3, 7.5, 8.2 Based on this construction, which statement is not true? A. AWC is complementary to CWB. B. CWB is a right triangle. C. ACB is isosceles. D. m CAB m CBA Correct answer : A Last Edit Page 3
4 G.5(D) verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems. Verify the Triangle Inequlatiy theorem using constructions. Apply the theorem to Decide whether it is possible to construct a triangle with the given side lengths Explain your reasoning 4ft, 9ft,10ft Answer: 4+9> 10 yes it can be constructed Constructions Side length Triangle Triangle Inequality Theorem Use angle legs to show triangle inequality properties Use a variety of tools such as compass, straight edge, patty paper, geometry software, technology or manipulatives to verify the Triangle Inequality Theorem. Big Ideas Math Geometry 6.5 Engaging p (43.pdf) Last Edit Page 4
5 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. 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, Verify theorems about the relationships in triangles, including of the midsegments. Apply these relationships to Two motorcycles start at the same point. One motorcycle travels 15 km due north and stops. The second motorcycle travels 32 km due west and stops. Which value is closest to the distance between the motorcycles when they stop? Right triangle Legs Hypotenuse Acute triangle Obtuse triangle Pythagorean triples Square root Pythagorean Theorem Remind students hypotenuse is always the C And can only have a hypotenuse if triangle is right Given a right triangle, 6.4, 6.6, 9.1 Engaging p.119 (47.pdf) midsegments, and F km use a b c. medians, and G km apply these H km relationships to solve J km Solve for side lengths. problems. Correct answer: H Given the side lengths Released EOC 2013 of a right triangle, Q#48 classify as acute, obtuse or right. Last Edit Page 5
6 G.6(E) prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals. Apply these relationships to Given LG WR, LG and WR bisect each other, prove WLRG is a rhombus. Sample answer: Since LG and WR are perpendicular, the four angles formed are right and also congruent. Because the segments bisect each other, WX XR LX XG and. By the S-A-S rule, the four right triangles must all be congruent. Then by CPCTC, WL LR RG GW is a rhombus.. So WLRG Diagonals Opposite angles Opposite sides Parallelogram Quadrilateral Rectangle Rhombus square Include standard geometric proofs: -two column -flow chart, -paragraph Instruction should include analysis of figures on a coordinate grid. 7.3, 7.4, 7.5 Last Edit Page 6
7 G.(7) Similarity, proof, and trigonometry. The student uses the process skills in applying similarity to G.7(A) apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles. Use properties of similar figures to find missing side lengths and angle measures. Isosceles trapezoid JKLM is shown below. Similar figures Corresponding parts Similarity transformation Proportion Scale factor dilation Verify triangles are similar by making sure sides are proportional and angles congruent If the dimensions of the trapezoid JKLM are multiplied by a scale factor of f to create trapezoid J K L M, which statement is true? Corresponding angles are always congruent Use example of 2 equilateral triangles since angles are always , 8.2 Engaging p.217 (85.pdf) F. Trapezoid J K L M contains two base angles measuring 30 each. G. The longer base of trapezoid J K L M is 56f units. H. The bases of trapezoid J K L M have lengths of 22 units and 39 units. J. Trapezoid J K L M contains two base angles measuring (120 f ) each. Correct answer: G Released EOC 2013 Q#2 Last Edit Page 7
8 G.7(B) apply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to Readiness Standard Verify that the triangles are simila by AA Find the length of missing sides. Explain why the triangles are similar and find the length of CE. Correct Answer: AA Similarity, CE = 13 Similar figures ratio Corresponding parts Similarity transformation Proportion Scale factor Dilation AA similarity Vertical angles Congruent Show examples where congruent angles might be from vertical angles, parallels cut by a transversal (prior knowledge) The sides in SSS and SAS must be proportional now (and not congruent like they were with proving triangles congruent) 8.2, 8.3 Engaging pp (83.pdf) Last Edit Page 8
9 G.(8) Similarity, proof, and trigonometry. 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. G.8(A) prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems. Prove theorems about similar triangles, including the Triangle Proportionality theorem. Apply these theorems to Cut out similar triangles and have students match up corresponding angles to show they are congruent 8.4 Determine if the given triangles are similar, if so, find x. Correct answer: Yes, x = 12 Correct answer: DF = 7.2 Similar figures ratio Corresponding parts Similarity transformation Proportion Scale factor Dilation Vertical angles Congruent Prove SAS similarity SSS similarity AA similarity Apply properties of similar triangles to Complete proofs of AA Similarity Postulate, SSS Similarity Theorem, SAS Similarity Theorem, and the Triangle Proportionality Theorem. Find missing sides and angles Last Edit Page 9
10 G.8(B) identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to Identify the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean to solve problems. Apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean to solve problems. 4 9 Find x, y, and z. Similar figures ratio Corresponding parts Similarity Proportion Geometric means Hypotenuse legs Use colored transparencies to demonstrate 3 similar triangles forms by altitude drawn to hypotenuse In a right triangle with an altitude drawn to the hypotenuse, Identify similar triangles and solve for missing lengths 9.3 Supporting STAAR Triangle Similarity (Geom11.pdf) Correct answer: x = 6, y = 2 13, z =3 13 Last Edit Page 10
11 G.9(B) apply the relationships in special right triangles and and the Pythagorean theorem, including Pythagorean triples, to Readiness Standard Apply the relationships in special right triangles to Apply the relationships in the Pythagorean Theorem, including Pythagorean triples, to Within a square section of land, a landscaper will build a path, as represented by the shaded section in the diagram below. Right triangle Hypotenuse Opposite Short leg Long leg Pythagorean theorem Pythagorean triples Isosceles triangle Having students memorize 3,4,5 and 5,12,13 triples will help Remind students that the properties of special right triangles can be derived from the Pythagorean theorem 9.1, 9.2 Engaging p.135 (53.pdf) Supporting STAAR Special Right Triangles (Geom4.pdf) Which measure is closest to the value of y? A. 5.5 ft B. 3.1 ft C. 4.3 ft D. 7.5 ft Correct answer: A Released EOC 2013 Q # 41 Misconceptions: The student may not correctly identify the short leg, long leg, and hypotenuse when solving problems involving a triangle. The student may multiply or divide by 2 instead of the 2 when solving problems involving a triangle. Last Edit Page 11
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