Encourage students to play with the compass and show them how to draw circles and arcs with it. This lesson introduces students to common pairs of angles, the linear angle postulate, and the vertical angle theorem.
Classifying Polygons
Reasoning and Proof
Inductive Reasoning
Conditional Statements
Deductive Reasoning
The best way for students to understand the Laws of Disjunction, Contrapositive, and Syllogism is to do a lot of practice. As in Examples 7 and 8, it may be helpful to have students put the statements into symbolic form.
Algebraic and Congruence Properties
This must be explicitly stated in the given or the angle marked. From the given statement, we know that 6 AGF∼=6 BGA, which can be marked on the drawing.
Proofs about Angle Pairs and Segments
Parallel and Perpendicular Lines
Lines and Angles
Explain that vertical angles and linear pairs use only two lines; however, these new angular relationships require three lines to be defined. You can explain that it exists (6 1 and 6 4 in the second picture), but it is not explicitly defined.
Properties of Parallel Lines
Proving Lines Parallel
Before you begin, demonstrate how to copy an angle (Investigation 2-2) and then allow students to work through the investigation.
Properties of Perpendicular Lines
The right angles are congruent/equal (all right angles are congruent or congruent linear pairs). Explain that they are writing a proof to someone who knows nothing about mathematics or the definitions of perpendicular lines or right angles.
Parallel and Perpendicular Lines in the Coordinate Plane
Apply it to real situations such as the slope of a mountain, or the part of a continent that drains into a particular ocean (Alaska's North Slope), the slope of a wheelchair ramp, etc. Of course, there are other ways to approach this. problems, but this method of clearing fractions will help students change slope-intercept form to standard form.
The Distance Formula
Triangles and Congruence
Triangle Sums
You can do the activity on the overhead and have students discover the sum of all the angles. If students forget the shortcut, they can still use the Triangle Transposition and the linear pair postulate.
Congruent Figures
This could be a discussion at the end of the lesson and lead into the next one. Although this is not a theorem, we will use it in proving that parts of triangles are congruent.
Triangle Congruence using SSS and SAS
If two triangles share a side or an angle, the Reflexive Property is the reason why this piece is congruent to itself. Students may feel like this is an unnecessary step, but just remind them that they need to straighten all three sets of congruent sides/angles to declare two triangles congruent.
Triangle Congruence using ASA, AAS, and HL
For SSS, there should be three steps, one for each set of congruent sides. For SAS, there should be two steps for the two sets of congruent edges and one step for the included angles.
Isosceles and Equilateral Triangles
Relationships with Triangles
Midsegments
Probe with students that if a line is parallel to a side of a triangle, that does not make it a midsegment. The parallel line must also join the midpoints, pass through the midpoints, or bisect the sides it passes through.
Perpendicular Bisectors and Angle Bisectors in Triangles
The intersection of the angle bisector would be equidistant from all the sides of the angles, which are also the sides of the triangle. The intersection of the angle bisector is always inside the triangle, so the circle will always be inside the triangles (inscribed).
Medians and Altitudes in Triangles
One way to help students remember which is to tell them that the point of intersection of the perpendicular bisectors can be outside the triangle, so the circle will go around the outside of the circle (circle). You can also use the figure above to help students determine properties of line segments.
Inequalities in Triangles
Extension: Indirect Proof
Polygons and Quadrilaterals
Angles in Polygons
Properties of Parallelograms
Proving Quadrilaterals are Parallelograms
As a way to introduce the proof of the Opposite Theorem Converse and the proof of Theorem 6-10 (Example 1), you can copy these theorems and then cut up the statements and reasons and put them in an envelope. Place the information, proof and image on one side of the envelope.
Rectangles, Rhombuses and Squares
So to prove that it is a rhombus, we can show that all sides are equal, or that the diagonals are perpendicular. It is easier to find the slopes of the diagonals than to perform the distance formula four times.
Trapezoids and Kites
Similarity
Ratios and Proportions
To show that these proportions are true, it may be helpful for students to see proof. Brainstorm other possible reconfigurations ofab=dc with students to see if there are any other possible "consequences".
Similar Polygons
Ask students to multiply their "true" proportions to check that they are valid.
Similarity by AA
Similarity by SSS and SAS
Proportionality Relationships
Using the letters from the theorem, the proportion would be BCAB=CDAD, which is a corollary of 7-1. Encourage students to use this layout if they have difficulty with the proportion given in the text.
Similarity Transformations
Extension: Self-Similarity
Right Triangle Trigonometry
The Pythagorean Theorem
Note that the distance formula in this section looks different than when it was introduced in Chapter 3.
Converse of the Pythagorean Theorem
Remember that the order doesn't matter as long as the corresponding xory value is first (x1 and y1 are first or x2 and y2 are first). If the integer is less than 8, then the triangle is obtuse, with 12 as the longest side.
Using Similar Right Triangles
Special Right Triangles
To find the hypotenuse, multiply the leg by √ 2. Students may need to simplify the square root. If they are given the hypotenuse, divide by 2, then they can multiply by √. 3 to get the longer leg.
Tangent, Sine and Cosine
To solve the 30-60-90 triangles, students must find the shortest leg, if not given. If they are given the longer leg, they will have to divide by √. 3 when appropriate) to get the shortest leg, then multiply it by 2.
Inverse Trigonometric Ratios
Circles
Parts of Circles & Tangent Lines
Therefore, students will need to recall all the information they learned in the previous chapter (special right triangles, the Pythagorean theorem and its converse, and the trig ratios). Example 6 is an important example because it uses the converse of the Pythagorean theorem to show that Bi is not a tangent point.
Properties of Arcs
The arc addition postulate should be familiar to students; it is very similar to the Segment Addition Postulate and the Angle Addition Postulate.
Properties of Chords
Inscribed Angles
For Statement 9-9, ask students what the measure of the intercepted arc is, then they should be able to determine the measure of the inscribed angle. You can choose whether to introduce this vocabulary or not to use what is in the text.
Angles of Chords, Secants, and Tangents
Walk around to answer questions and let students arrive at the Inscribed Angle Theorem on their own. Be sure to do lots of practice problems in class to ensure students are using the correct formula in the appropriate place.
Segments of Chords, Secants, and Tangents
Developing the correct formula will give students ownership of the material and help them retain the information. You can choose to put the formulas on the board, allow them to use annotations, or use nothing.
Extension: Writing and Graphing the Equations of Circles
Perimeter and Area
Triangles and Parallelograms
Here students will see that the area of a triangle is half the area of a parallelogram. This will show students that the triangles are congruent and that each is exactly half the parallelogram.
Trapezoids, Rhombi, and Kites
To show students the area of a parallelogram, cut out the figure (or draw a similar figure to cut out) of the parallelogram and then cut out the side and move it over so that the parallelogram becomes a rectangle. Explain to students that the line you cut is the height of the parallelogram, which is not a side of the parallelogram.
Areas of Similar Polygons
The diagonals bisect each other, so let's cut the diagonals in half and then connect the endpoints of the diagonals to form a rhombus.
Circumference and Arc Length
Students should make the connection between the fact that the arc length of the semicircle will be half the circumference. Using the same circle, see if the students can find the arc length of a 30◦ part of the circle 3π3 =π.
Area of Circles and Sectors
Surface Area and Volume
Exploring Solids
There can be multiple answers for a single solid, and depending on how the faces are arranged, the mesh may not work at all. The activity at http://illuminations.nctm.org/activitydetail.aspx?ID=84 gives students 20 possible grids of the cube and they have to find 11 that work.
Surface Area of Prisms and Cylinders
Surface Area of Pyramids and Cones
Students will always be given two of the three pieces of information required; h, r, orl (in the picture above). If the problem does not give the slant height, students will need to use the Pythagorean theorem to solve it.
Volume of Prisms and Cylinders
Regardless of order, the formula for the volume of a rectangular prism is length x width x height. The “height” in the formula is actually 2 meters, which is similar to the length of the base.
Volume of Pyramids and Cones
If you want, you can change the values given as slope height to the height of the pyramids. If you don't change the problems, students will have to find the height of the pyramids, using the slant height.
Surface Area and Volume of Spheres
This will be a difficult process because students must remember the properties of the center of gravity (the point where the vertical height hits the base). Therefore, the distance from the bottom of the inclined height to the center of gravity will be one third of the length of the entire height.
Extension: Exploring Similar Solids
Rigid Transformations
Exploring Symmetry
Translations
Students should know that these transformations never change the size or shape of the example and therefore will always create congruent images. For example, if the translation rule is (x,y)→(x+1,y−2) for a triangle, each vertex of the triangle is moved one unit to the right and two units down. a) If∆ ABC is the preimage, find the translation rule for image∆XY Z. b) If∆XY Z is the preimage, find the translation rule for image∆ABC. b) FromX toA the triangle is translated to the right side8 units and5 units higher.
Reflections
For a reflection on the x-axis (or horizontal line), the x-value will remain the same and the value will change. If a point is 5 units to the left of a reflection line, then the image is 5 units to the right of the reflection line.
Rotations
For a reflection on the they axis (or vertical line), the they value will remain the same and the x value will change. As we will see in the next chapter, a rotation of 180◦ is a composition of the double reflection about the axes.
Composition of Transformations
Also see if they can make a correlation between off-axis reflections and 180◦ rotation. From these coordinates we see that the double reflection over the xandias is equivalent to a rotation of 180◦.
Extension: Tessellating Polygons
Students can do the investigation with you, or you can just do the activity and the student writes down the necessary information. Now proceed to fill in the figures with either the original or the rotation.
C HAPTER
2 Basic Geometry TE - Common Errors
Chapter Outline
Basics of Geometry
Points, Lines and Planes
Segments and Distance
The last two expressions refer to segment length and/or distance from AtoBand can be used interchangeably. Adding the Segment - Students should be encouraged to always sketch the segment with separate end points and the point between them.
Angles and Measurement
Example 1 in the textbook illustrates the correct way to align a ruler with a section, and the diagram that follows this example shows how to read fractional inches and centimeters on a ruler. When counting these distances on the coordinate plane, students occasionally count the starting point as one and thus end up with a length that is one unit greater than the actual distance.
Midpoints and Bisectors
There is an easy way for students to check their work when measuring an angle with a protractor. It is imperative to practice these skills with students to avoid confusion later in the course.
Angle Pairs
Reasoning and Proof
If X is outside the outer circle, then it is outside the "is Saturday" circle. They think it's a waste of time because it's not part of the end result.
Parallel and Perpendicular Lines
Proofs of Converse Theorems - The textbook includes a proof of the Converse Theorem of the Alternate Interior Angles Theorem. Find the equation of the line that is perpendicular to the line that passes through the points (5, 7) and (12, 3) and passes through the second point.
Triangles and Congruence
For example, the congruence theorems tell the reader not only which triangles are congruent, but which parts of the triangle are congruent. Third Theorem by Proof - In the text an example is given to demonstrate the Third Theorem, this is inductive reasoning.
Triangle Congruence Using SSS and SAS
A deeper understanding of the theorem and different types of reasoning can be achieved by using deductive reasoning to write a proof. Using the distance formula to prove that triangles are congruent - It may be necessary to review the distance formula with students again and remind them that they cannot "distribute" the square root.
Triangle Congruence Using ASA, AAS and HL
Relationships with Triangles
The midline of the side of a triangle does not have to pass through a vertex. Construct the midline of the corner of the corner opposite the side with the midline.
Polygons and Quadrilaterals
Use the Venn diagram of the classification of squares to orient them in the chapter. The length of one base is twice the length of the other base, and the median is 9 cm.
Similarity
The original triangle and the smaller triangle created by the parallel segment are similar, as shown in the proof of the triangle proportionality theorem. The bottom half of the paper will have a paragraph explaining self-similarity in a fractal.
Right Triangle Trigonometry
Use similar triangles - Many students have trouble understanding that the sine, cosine, and tangent of a particular angle measure do not depend on the size of the right triangle used to take the ratio. How does the length of the hypotenuse compare to the length of the legs of a right triangle.
Circles
If the vertex is in the center of the circle, it is a central angle, and the measure of the arc and the angle are equal. The vertex of the angle is in the center of the circle, it is a central angle, and the arc and angle have the same measure.
Perimeter and Area
Identify the base and height of the parallelogram in terms of d1 end2, and then substitute these into the parallelogram area formula to derive the kite area formula. Also, the area of a rhombus can be found using the kite or parallelogram area formulas.
Area of Similar Polygons
This time, students must square the area ratio to find the ratio of the lengths of the sides. Let the students know if they should give one, the other or both forms of the answer.
Areas of Circles and Sectors
Surface Area and Volume
The area of the pyramid is just the lateral area because the base is not visible. The Pythagorean theorem (or recognition of a Pythagorean Triple) will be used to determine the height (4) of the cone.
Rigid Transformations
Have students practice this on the coordinate plane where distances between the vertices of the example and image can be easily calculated to verify the relationship. Students can also compare the rules for reflections with the rules for rotations to see that the combination of the two reflections results in the rule for the rotation.
3 Basic Geometry TE - Enrichment
Basics of Geometry
Points, Lines, and Planes
Reasoning and Proof
When all groups have finished, have them share their rules and see if the other groups can find any counterexamples. When you have finished, ask students if the meaning of the poem has changed and how conditional statements can affect different statements.
Parallel and Perpendicular Lines
Ask students how they can extend this investigation to the converse of the same side interior angle theorem. From here, use the slope of the rectangle to find the corresponding point ony=2x−1.
Triangles and Congruence
Make one side of the 50◦ angle 3 inches and the other side of the angle 2 inches long. The measure of a right angle can be determined if we know that both sides are perpendicular.
Triangle Congruence Using ASA, AAS, and HL
Relationships with Triangles
Extension example: If the midpoints of the sides of a triangle are A(1,5),B(4,−2), and C(−5,1), find the vertices of the triangle. Extension Example: Using the image above, find the equation of the median from A to the midpoint of BC.
