Curriculum Geometry B/ Algebra II
UNIT 2: SIMILARITY, CONGRUENCE, AND PROOFS
UNDERSTAND SIMILARITY IN TERMS OF SIMILARITY TRANSFORMATIONS |
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MGSE9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor.
MGSE9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
- The dilation of a line not passing through the center of the dilation results in a parallel line and leaves a line passing through the center unchanged.
- The dilation of a line segment is longer or shorter according to the ratio given by the scale factor.
MGSE9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
KEY IDEAS
- A dilation is a transformation that changes the size of a figure, but not the shape, based on a ratio given by a scale factor with respect to a fixed point called the center.
When the scale factor is greater than 1, the figure is made larger.
When the scale factor is between 0 and 1, the figure is made smaller.
When the scale factor is 1, the figure does not change.
When the center of dilation is the origin, you can multiply each coordinate of the original figure, or pre-image, by the scale factor to find the coordinates of the dilated figure, or image. - When the center of dilation is not the origin, you can use a rule that is derived from shifting the center of dilation, multiplying the shifted coordinates by the scale factor, and then shifting the center of dilation back to its original location. For a point (x, y) and a center of dilation (xc , yc ), the rule for finding the coordinates of the dilated point with a scale factor of k is (xc + k(x – xc ), k(y – yc ) + yc).
When a figure is transformed under a dilation, the corresponding angles of the pre-image and the image have equal measures.
When a figure is transformed under a dilation, the corresponding sides of the pre-image and the image are proportional.
So, when a figure is under a dilation transformation, the pre-image and the image are similar. - When a figure is dilated, a segment of the pre-image that does not pass through the center of dilation is parallel to its image.
When the segment of a figure does pass through the center of dilation, the segment of the pre-image and image are on the same line.
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PROVE THEOREMS INVOLVING SIMILARITY |
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MGSE9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, (and its converse); the Pythagorean Theorem using triangle similarity.
MGSE9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
MGSE9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
KEY IDEAS
- When proving that two triangles are similar, it is sufficient to show that two pairs of corresponding angles of the triangles are congruent. This is called Angle-Angle (AA) Similarity.
- When a triangle is dilated, the pre-image and the image are similar triangles. There are three cases of triangles being dilated:
• The image is congruent to the pre-image (scale factor of 1).
• The image is smaller than the pre-image (scale factor between 0 and 1).
• The image is larger than the pre-image (scale factor greater than 1). - When two triangles are similar, all corresponding pairs of angles are congruent.
- When two triangles are similar, all corresponding pairs of sides are proportional.
- When two triangles are congruent, the triangles are also similar.
- A two-column proof is a series of statements and reasons often displayed in a chart that works from given information to the statement that needs to be proven. Reasons can be given information, can be based on definitions, or can be based on postulates or theorems.
- A paragraph proof also uses a series of statements and reasons that work from given information to the statement that needs to be proven, but the information is presented as running text in paragraph form.
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UNDERSTAND CONGRUENCE IN TERMS OF RIGID MOTIONS |
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MGSE9-12.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
MGSE9-12.G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
MGSE9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (Extend to include HL and AAS.)
MGSE9-12.G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
MGSE9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (Extend to include HL and AAS.)
KEY IDEAS
- A rigid motion is a transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations (in any order). This transformation leaves the size and shape of the original figure unchanged.
- Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, reflections, and translations). Congruent figures have the same corresponding side lengths and the same corresponding angle measures as each other.
- Two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent. This is sometimes referred to as CPCTC, which means Corresponding Parts of Congruent Triangles are Congruent.
- When given two congruent triangles, you can use a series of translations, reflections, and rotations to show the triangles are congruent.
- You can use ASA (Angle-Side-Angle) to show two triangles are congruent. If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
- You can use SSS (Side-Side-Side) to show two triangles are congruent. If three sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent.
- You can use SAS (Side-Angle-Side) to show two triangles are congruent. If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
- You can use AAS (Angle-Angle-Side) to show two triangles are congruent. If two angles and a non-included side of a triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
- If two sides and a non-included angle of one triangle are congruent to two sides and a non-included angle of a second triangle, the triangles are not necessarily congruent. Therefore, there is no way to show triangle congruency by Side-Side-Angle (SSA).
- If two triangles have all three angles congruent to each other, the triangles are similar, but not necessarily congruent. Thus, you can show similarity by Angle-AngleAngle (AAA), but you cannot show congruence by AAA.
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PROVE GEOMETRIC THEOREMS |
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MGSE9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: 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.
MGSE9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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.
MGSE9-12.G.CO.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.
MGSE9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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.
MGSE9-12.G.CO.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.
KEY IDEAS
- A two-column proof is a series of statements and reasons often displayed in a chart that works from given information to the statement that needs to be proven. Reasons can be given information, can be based on definitions, or can be based on postulates or theorems.
- A paragraph proof also uses a series of statements and reasons that work from given information to the statement that needs to be proven, but the information is presented as running text in paragraph form.
- It is important to plan a geometric proof logically. Think through what needs to be proven and decide how to get to that statement from the given information. Often a diagram or a flow chart will help to organize your thoughts.
- An auxiliary line is a line drawn in a diagram that makes other figures, such as congruent triangles or angles formed by a transversal. Many times, an auxiliary line is needed to help complete a proof.
- Once a theorem in geometry has been proven, that theorem can be used as a reason in future proofs.
- Some important key ideas about lines and angles include the following:
• Vertical Angle Theorem: Vertical angles are congruent.
• Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles formed by the transversal are congruent.
• Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then corresponding angles formed by the transversal are congruent.
• Points on a perpendicular bisector of a line segment are equidistant from both of the segment’s endpoints. - Some important key ideas about triangles include the following:
• Triangle Angle-Sum Theorem: The sum of the measures of the angles of a triangle is 180°.
• Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are also congruent.
• Triangle Midsegment Theorem: If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and half its length.
• Points of Concurrency: incenter, centroid, orthocenter, and circumcenter - Some important key ideas about parallelograms include the following:
• Opposite sides are congruent and opposite angles are congruent.
• The diagonals of a parallelogram bisect each other.
• If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
• A rectangle is a parallelogram with congruent diagonals.
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MAKE GEOMETRIC CONSTRUCTIONS |
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MGSE9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
MGSE9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle.
MGSE9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle.
KEY IDEAS
- Copy a segment
- Copy an angle
- Bisect an angle
- Construct a perpendicular bisector of a line segment
- Construct a line perpendicular to a given line through a point not on the line
- Construct a line parallel to a given line through a point not on the line
- Construct an equilateral triangle inscribed in a circle
- Construct a square inscribed in a circle
- Construct a regular hexagon inscribed in a circle
Review_Examples_5.pdf | |
File Size: | 78 kb |
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USE COORDINATES TO PROVE SIMPLE GEOMETRIC THEOREMS ALGEBRAICALLY |
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MGSE9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, sqrt{3} ) lies on the circle centered at the origin and containing the point (0, 2).
KEY IDEAS
- To prove properties about special parallelograms on a coordinate plane, you can use the midpoint, distance, and slope formulas.
- You can use properties of quadrilaterals to help prove theorems
• To prove a quadrilateral is a parallelogram, show that the opposite sides are parallel using slope. • To prove a quadrilateral is a rectangle, show that the opposite sides are parallel and the consecutive sides are perpendicular using slope.
• To prove a quadrilateral is a rhombus, show that all four sides are congruent using the distance formula.
• To prove a quadrilateral is a square, show that all four sides are congruent and consecutive sides are perpendicular using slope and the distance formula. - You can also use diagonals of a quadrilateral to help prove theorems.
• To prove a quadrilateral is a parallelogram, show that its diagonals bisect each other using the midpoint formula.
• To prove a quadrilateral is a rectangle, show that its diagonals bisect each other and are congruent using the midpoint and distance formulas.
• To prove a quadrilateral is a rhombus, show that its diagonals bisect each other and are perpendicular using the midpoint and slope formulas.
• To prove a quadrilateral is a square, show that its diagonals bisect each other, are congruent, and are perpendicular using the midpoint, distance, and slope formulas.
- When using the formulas for midpoint, distance, and slope, the order of the points does not matter. You can use either point to be (x1 , y1 ) and (x2 , y2 ), but be careful to always subtract in the same order.
- Parallel lines have the same slope. Perpendicular lines have slopes that are the negative reciprocal of each other.
Sample_Problems_6.pdf | |
File Size: | 40 kb |
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RESOURCES |
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