9.3.3A Line & Angle Relationships
Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results.
For example: Prove that the perpendicular bisector of a line segment is the set of all points equidistant from the two endpoints, and use this fact to solve problems and justify other results.
Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results.
For example: Prove that two triangles formed by a pair of intersecting lines and a pair of parallel lines (an "X" trapped between two parallel lines) are similar.
Overview
Standard 9.3.3 Essential Understandings
The study of geometric figures might be considered one of the "main topics" in geometry, along with, among others, proportional thinking, reasoning and sense making. The focus of this standard is on geometric figures, with the others listed above also being involved to a great extent.
In this standard, students will work with a transversal that intersects two lines. They will study angles formed by the three lines, and determine whether or not the two intersected lines are parallel. They will see many types of triangles, including "special" triangles such as isosceles, equilateral, right, 30-60-90 and 45-45-90 triangles, focusing on the properties of each of these figures.
Students will study similarity and understand that congruence is a special case of similarity. They will work with scale factors in "real-life" applications of similar figures. They will look at quadrilaterals and put them into a hierarchy, with "special" quadrilaterals being "nested" within others. Also, in this standard, students will study circles and their properties, with special focus on their applications.
Benchmark Group A - Line and Angle Relationships
9.3.3
9.3.3.1 Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results.
9.3.3.2 Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Identify and differentiate between corresponding angles, alternate interior angles, same-side supplementary angles and, if necessary, alternate exterior angles, when two lines are cut by a transversal;
- Determine the relationship between the above angles when the two lines are parallel;
- Use the above information, and connections with algebra, to solve problems involving angle measures.
Work from previous grades that supports this new learning includes:
- Know that angles that form a linear pair are supplementary;
- Know that vertical angles are congruent;
- Know that parallel lines are coplanar lines that do not intersect;
- Know that perpendicular lines are lines that intersect to form right angles.
NCTM Standards
Geometry
- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:
- Analyze properties and determine attributes of two- and three-dimensional objects;
- Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
- Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;
- Use trigonometric relationships to determine lengths and angle measures.
- Use visualization, spatial reasoning, and geometric modeling to solve problems:
- Use geometric models to gain insights into, and answer questions in, other areas of mathematics;
- Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
Common Core State Standards (CCSM)
- HS.G-CO (Congruence) Prove geometric theorems.
- HS.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.
- HS.G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; 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.
- HS.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
- HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Define trigonometric ratios and solve problems involving right triangles.
- HS.G-SRT.7. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
- HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Apply trigonometry to general triangles.
- HS.G-SRT.10. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
- HS.G-C (Circles) Understand and apply theorems about circles.
- HS.G-C.1. Prove that all circles are similar.
- HS.G-C.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
- HS.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
- HS.F-TF (Trigonometric Functions) Extend the domain of trigonometric functions using the unit circle.
- HS.F-TF.3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6.
Misconceptions
- Students will need reminders and review of the different types of angles formed by two lines cut by a transversal.
- Students will sometimes mistake angles that are supplementary with angles that are congruent, when two parallel lines are cut by a transversal.
Vignette
In the Classroom
In this vignette, students investigate angles formed when two lines are cut by a transversal.
The teacher puts a diagram on the board.
Teacher: When two lines are cut by a transversal, the plane is split into three regions. The region between the two lines is the interior region, and the other two, outside the lines, are considered exterior regions. For example, in this case, lines t and e are cut by transversal m.
Or, as another example (again, lines t and e are cut by transversal m):
You'll notice that when these two lines, t and e, are cut by a transversal, m, the three lines form eight angles, as shown here:
Teacher: In this diagram, which angles are interior angles and which are exterior angles?
Student 1: Angles 3, 4, 5 and 6 are interior angles, and angles 1, 2, 7 and 8 are exterior angles.
Teacher: Correct. There are special names associated with these eight angles. Corresponding angles are two angles with different vertices on the same side of the transversal. One is an interior angle and the other is an exterior angle. Can you tell me a pair of corresponding angles?
Student 2: I think I can. Are angles 1 and 5 corresponding angles?
Teacher: Yes, they are. Are there any other pairs?
Student 2: Yes, angles 2 and 6, angles 3 and 7, angles 4 and 8 are all pairs of corresponding angles.
Teacher: Great. Alternate interior angles are two angles with different vertices on opposite sides of the transversal. They are both interior angles. Can you find a pair of alternate interior angles?
Student 3: I think angles 3 and 6 are alternate interior angles, and also angles 4 and 5.
Teacher: Correct. Same-side interior angles are two angles on the same side of the transversal, having different vertices. They are both interior angles. Can you find a pair of same-side interior angles?
Student 4: Yes, angles 3 and 5 are same-side interior angles, and angles 4 and 6 are same-side interior angles.
Teacher: You're right. Now, let's explore some properties of these angles. Here are two parallel lines cut by a transversal. Use patty paper to investigate possible conjectures you could make regarding the eight angles formed.
Some sheets will have diagrams that look like this:
While others have sheets with diagrams that look like this:
In this way, students can share and compare their conjectures with a colleague who has a different pair of parallel lines.
Student 5: You said we should use patty paper to investigate this. How would we use patty paper?
Teacher: I'll answer that with a question: What might be an important thing to investigate here?
Student 5: Segment lengths, maybe?
Teacher: Maybe, but since these are lines, and not just segments, could you measure their lengths?
Student 5: I guess not - they'd be infinitely long.
Teacher: Right. What else might be important in this diagram?
Student 5: Angle measures?
Teacher: That's a good start. I'll give you about five minutes or so to come up with a conjecture that might be true for your diagram.
The teacher walks around the room, observing and helping when appropriate. After a few minutes, the teacher calls the students back together.
Teacher: How did you use the patty paper to investigate the angles on these diagrams?
Student 6: I traced a couple of the angles from the diagram onto my patty paper, and moved the patty paper around to other angles on the diagram.
Teacher: What did you find?
Student 6: Well, is it OK if I number the angles like you had them numbered earlier?
Teacher: Sure.
Student 6: OK, so my diagram looks something like this:
And I found that angles 1 and 4 are congruent.
Student 7: But didn't we already know that? Those are vertical angles, and they're always congruent.
Student 6: Yes, I guess they are. I also found that angles 6 and 8 are supplementary.
Student 7: We also knew that. They form a linear pair, and they will always be supplementary.
Student 6: I also found that corresponding angles appear to be congruent.
Teacher: Great. Did anyone else find that?
Student 8: I did, and so did my partners.
Teacher: You are correct. Any time two parallel lines are cut by a transversal, the corresponding angles are congruent. Now, knowing that, you can justify other relationships. Tell me another.
Student 9: Well, if angle 6 and angle 8 are supplementary, and we just found out that angle 4 and angle 8 are congruent because they're corresponding angles, then angle 6 and angle 4 must be supplementary.
Student 10: Does that mean that any same-side interior angles will be supplementary?
Teacher: Not necessarily. What would be a restriction on that statement?
Student 10: The two lines would have to be parallel. Otherwise, that wouldn't be true.
Teacher: Right. What else do you think is true about this diagram?
Student 5: If we know that angle 6 and angle 4 are supplementary, and we know that angle 3 and angle 4 are supplementary, because they're a linear pair, then angle 3 and angle 6 must be congruent. Are all pairs of alternate interior angles congruent?
Teacher: What must be true in order for you to make that statement?
Student 5: Oh, yeah, the lines have to be parallel.
Teacher: Right. Have you found anything else?
Student 2: Is there any such thing as a pair of alternate exterior angles?
Teacher: Sure. What would those look like?
Student 2: Well, angle 1 and angle 8 would be a pair, and angle 2 and angle 7 would be another pair.
Teacher: Yes. Did you notice anything about them?
Student 2: Yes. If angle 3 and angle 6 are congruent, then angle 1 and angle 8 must be congruent, because they're supplements of congruent angles.
Student 3: I noticed something about these angles.
Teacher: What did you notice?
Student 3: It seems that when two parallel lines are cut by a transversal, if you pick any two of the eight angles formed, then those two angles would be either supplementary or congruent. There's no other relationship possible.
Teacher: It's great that you caught that. Way to go!
Teacher: Let's look at the diagram a bit differently now. If we know that lines t and e are parallel, and angles 1 and 2 are congruent, what does that mean? The diagram might look like this:
Student 5: It means that line t is perpendicular to line m.
Teacher: Right. How do you know that's true?
Student 5: We already know that angles 1 and 2 are supplementary, because they're a linear pair, so that their sum is 180. If they're also congruent, then each must have a measure of 90, so they're right angles. That means the lines are perpendicular.
Student 8: Then, wouldn't all eight angles be right angles?
Teacher: Yes.
Student 8: So, then line e must also be perpendicular to line m, right? So if a line is perpendicular to one of the parallel lines, then it must be perpendicular to the other one, as well.
Student 4: I don't think that's true.
Teacher: Why not?
Student 4: Well, look at the tiles on the floor, for example. There are parallel lines separating the tiles, right?
Teacher: It certainly looks that way. Let's say that's true, for the sake of this discussion.
Student 4: I can put a yardstick on one of these lines (call it line a), so that it's perpendicular to line a, but it wouldn't be perpendicular to any of the other lines that are parallel to line a. It would be skew to the lines on the floor that are parallel to line a.
Teacher: That's true.
Student 8: So, can I change what I said just a minute ago?
Teacher: Sure. How would you change it?
Student 8: I think that I should add "in a plane" to the start. So, in a plane, if a line is perpendicular to one of two parallel lines, then it must be perpendicular to the other one, as well.
Teacher: That sounds great.
Resources
Teacher Notes
- It will be helpful if you have access to some interactive geometry software, if only to show in "demo" mode.
- Some students might need a few reminders about which angles are supplementary and which are congruent, once they know the two parallel lines are cut by a transversal.
- Students may need support in further development of previously studied concepts and skills.
Dividing a Town into Pizza Delivery Regions
Students will construct perpendicular bisectors, find circumcenters, calculate area and use proportions to explore the following problem: You are the owner of five pizzerias in the town of Squaresville. To ensure minimal delivery times, you devise a system in which customers call a central phone number and get transferred to the pizzeria that is closest to them. How should you divide the town into five regions so that every house receives delivery from the closest pizzeria? Also, how many people should staff each location based on coverage area?
Angles formed by a transversal
Additional Instructional Resources
National Council of Teachers of Mathematics (NCTM). (2010). Focus in High School Mathematics: Reasoning and Sense Making in Geometry. Reston, VA: National Council of Teachers of Mathematics.
New Vocabulary
transversal: a line that intersects two or more other coplanar lines, in different points.
alternate interior angles: two interior angles on opposite sides of the transversal, having different vertices.
alternate exterior angles: two exterior angles on opposite sides of the transversal, having different vertices.
corresponding angles: two angles with different vertices, on the same side of the transversal. one is an interior angle, one is an exterior angle.
same-side interior angles (consecutive interior angles): two interior angles on the same side of the transversal.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- What other instructional strategies can be used to engage students' study of angles associated with parallel and perpendicular lines?
- How can manipulatives be used to help students visualize these abstract concepts?
- How can the instruction be scaffolded for students?
- What additional scaffolding is needed to provide ELL students?
- Do the tasks that have been designed connect to underlying concepts or focus on memorization?
- How can it be determined if students have reached this learning goal?
- How did the lesson be differentiated?
Assessment
Note to teachers: This assessment is written for students in a computer lab setting, but if there is none available, it can be done with worksheets, using patty paper and/or rulers with protractors, as well.
Instructions for students using the computer lab:
1. Put any triangle on your screen using the geometry software package available.
2. Construct the midpoints of each of the sides, and connect these midpoints to form a triangle inside your original triangle.
3. Move a vertex or a side of your original triangle and make some observations about what you see. Write these observations in the space below. Justify the conclusions you draw, either by measuring or by referring to a theorem you have learned, or both.
4. Draw a sketch of the diagram in the space below, and mark any parts that you find to be congruent.
5. How many trapezoids are in your diagram?
6. Write two or more things you notice about the trapezoids.
7. How many parallelograms are in your diagram?
8. Write two or more things you notice about the parallelograms.
9. Write at least one other observation about this diagram.
DOK Level: 4
Assessment hints/solutions:
Instructions for the computer lab.
1, 2. The figure drawn for instructions 1 and 2 might look like the one below, where triangle RUN is the original triangle, and triangle JOG is the midpoint triangle.)
3, 4. Students should note observations similar to the following:
- Segment JO is parallel to segment RN, after measuring angle R and angle UJO (corresponding angles). This will also be true for the other two sides of the midpoint triangle.
- Segment JO is half the length of segment RN. This will also be true for the other two sides of the midpoint triangle.
- The four small triangles are congruent. Students should justify that statement using SSS or SAS or ASA or AAS.
Their diagram might look something like this:
5. There are three trapezoids, RUOG, JUNG and RJON.
6. Students might note that the trapezoids are not congruent, nor will they have the same perimeter, but they do have the same area, as they are each made up of three of the four small triangles.
7. There are three parallelograms, RJOG, JUOG and NGJO.
8. Students might note that the parallelograms are not congruent, nor will they have the same perimeter, but they do have the same area, as they are each made up of two of the four small triangles.
9. Students might notice that three of the small triangles are translations of each other, or that the fourth small triangle (the midpoint triangle) is a 180-degree rotation of any of the others, about the midpoint of one of its sides. They might also notice that this diagram demonstrates that the sum of the measures of the angles of a triangle is 180. Notice that the three angles with a vertex at G form a straight angle, so that their sum is 180. These are a "one-tick angle," a "two-tick angle" and a "three-tick angle" (if a person uses tick marks to denote congruent angles). These three angles are also shown as the three angles of any of four small triangles, and also of the original triangle, thus showing that their sum is also 180.
Differentiation
- Before working on the Assessment, be sure to review terms like "trapezoid" and "parallelogram."
- Give frequent assessment checks throughout the process.
- Vocabulary will be an issue, especially in the assessment. "Trapezoid" and "parallelogram" will be terms that they might or might not remember.
- The instruction, "Write two or more things you notice about the parallelograms" (in the assessment) might have to be re-worded in order to be more specific.
- Tie this Framework's assessment activity (triangle constructions) in with similarity. The four small triangles are similar to the original triangle.
- This can be extended to show ratios relative to similar figures. The ratio of the sides of the small triangles to the largest triangle is 1:2, and the ratio of their areas is 1:4 (or, 12:22)
Parents/Admin
|
Students are: (descriptive list) |
Teachers are: (descriptive list) |
|
using patty paper or rulers/protractors to measure angles or using computer software to generate geometric representations. |
engaging students in exploring and problem solving with geometry either with computer generated representations or hand drawn models. |
|
working in groups to determine angle relationships. |
giving hints/suggestions when appropriate to scaffold student learning. |
|
identifying and differentiating between various angle relationships to determine the relationship between angles when the two lines are parallel. |
observing student work and asking probing questions to cause students to reflect on their mathematical learning. |
|
making connections with algebra, to solve problems involving angle measures.
|
|
Parent Resources
- Sophia geometry lessons
This social learning community website includes lessons on multiple topics. Most of these lessons were developed by teachers and reviewed.
- Khan Academy geometry lessons
This website offers a library of videos and practice exercises on multiple topics.
- Teacher Tube, YouTube
These websites include multiple uploaded lessons on most school topics.
- Mathematics textbooks
Many textbook publishers have websites with additional resources and tutorials. Check your child's textbook for a weblink.