TEACHER GUIDE TO CLARIFICATION
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.5 Use informal arguments to establish facts about the angle sum and exterior angles of triangles, about the angles created when parallel lines are cut by a transversal and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and gives an argument in terms of transversals why this is so.
Using Informal Arguments to Establish Facts about Angles in Triangles
Students use exploration and deductive reasoning to determine relationships that exist between a) angle sums and exterior angle sums of triangles, b) angles created when parallel lines are cut by a transversal, and c) the angle- angle criterion for similarity of triangle.
Stop! Students need to be constructing lines, angles and triangles.
Students construct various triangles and find the measures of the interior and exterior angles. Students make conjectures about the relationship between the measure of an exterior angle and the other two angles of a triangle. (The measure of an exterior angle of a triangle is equal to the sum of the measures of the other two interior angles) and the sum of the exterior angles (360°). Using these relationships, students use deductive reasoning to find the measure of missing angles.
Students construct parallel lines and a transversal to examine the relationships between the created angles. Students recognize vertical angles, adjacent angles and supplementary angles from 7th grade and build on these relationships to identify other pairs of congruent angles. Using these relationships, students use deductive reasoning to find the measure of missing angles.
Students construct various triangles having line segments of different lengths but with two corresponding congruent angles. Comparing ratios of sides will produce a constant scale factor, meaning the triangles are similar.
Students are constructing angles !
Students can informally prove relationships with transversals.
Show that if l and m are parallel lines and are transversals.
Angle 1 and Angle 5 are congruent because they are corresponding angles.
because alternate interior angles are congruent.
Students can informally conclude that the sum of a triangle is 180º (the angle-sum theorem) by applying their understanding of lines and alternate interior angles. In the figure below, line x is parallel to line yz:
Angle a is 35º because it alternates with the angle inside the triangle that measures 35º. Angle c is 80º because it alternates with the angle inside the triangle that measures 80º. Because lines have a measure of 180º, and angles a + b + c form a straight line, then angle b must be 65 º (180 – 35 + 80 = 65). Therefore, the sum of the angles of the triangle are
35º + 65 º + 80 º.
Kansas Association of Teachers of Mathematics (KATM) Flipbooks. Questions or to send feedback: email@example.com.
Retrieved from Math Flipbooks.
In this standard, teachers should be incorporating Math Practice 5, use appropriate tools strategically. Students should have rulers, protractors and tracing paper available to help in their learning of transversals and the angle-angle criterion. Remember students must CHOOSE their appropriate tool. If a teacher tells or suggest a tool then MP.5 is not addressed.
Math Practice 5: Use Appropriate Tools Strategically
Smarter Balanced http://www.smarterbalanced.org/practice-test/
Coherence and Connections: Need to Know
In seventh grade, students learned about adjacent, vertical, congruent, complementary and supplementary angles. They also learned about similar triangles. Students use this knowledge to support their informal arguments about triangles in 8th grade.
Examples of Major Within-Grade Dependencies
An important development takes place in grade 8 when students make connections between proportional relationships, lines, and linear equations (8.EE, second cluster). Making these connections depends on prior grades’ work, including 7.RP.2 and 6.EE.9. There is also a major dependency within grade 8 itself: The angle-angle criterion for triangle similarity underlies the fact that a non-vertical line in the coordinate plane has equation y = mx + b. Therefore, students must do work with congruence and similarity (8.G.1-5) before they are able to justify the connections among proportional relationships, lines, and linear equations. Hence the indicated geometry work should likely begin at or near the very start of the year.
PARCC Model Content Frameworks: Mathematics Grades 3-11 (version 5). (2017, November).
Retrieved from https://files.eric.ed.gov/fulltext/ED582070.pdf
Illinois Assessment of Readiness Mathematics Evidence Tables.
Evidence Statement Text
Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures.
Content Scope: Knowledge and skills articulated in 8.G.5
Also check out Student Achievement Partners Coherence Map.
Mathematics Assessment Project: A formative assessment lesson on Identifying Similar Triangles
NCTM Illuminations: Interactive activity finding sums of triangles (and other polygons)
A video from TeacherTube that shows how a teacher introduces and explores standard 8.G.5. Note: she is using direct instruction. We encourage you to consider alternate ways to introduce and teach 8.G.5 http://www.teachertube.com/video/8g5-angle-relationships-288717
1. In the figure below the measure of1 is 125°. Name the angles congruent to angle 1.
Angles 4, 5, 8
2. How many angles are in the figure below are congruent? Can you classify them?
Vertical angles: 1 and 4, 2 and 3, 5 and 8, 6 and 6
Alternate interior: 3 and 6, 4 and 5
Alternate exterior: 1 and 8, 2 and 7
Corresponding: 1 and 5, 2 and 6, 3 and 7, 4 and 8
3. What is the measure of the exterior angle? Justify your answer.
2a = 75 + (a – 5) since 2a is supplementary to the non-labeled angle. And since 3 angles in a triangle equal 180°,
then 2a is equal to 75 + (a – 5)°. The exterior angle is 160°.
4. Determine whether the statement below is always, sometimes or never true.
a. Given 2 angle measures, you can construct a triangle. True
b. The acute angles of a right triangle are complementary. True
c. If you know the triangle has an exterior angle of 80°, you can figure out all three interior angles. False, the other 2
angles have a sum of 100°, but it could be any combination of 100°.
d. The three angles in a triangle are supplementary. True
e. The sum of the exterior angles in a triangle is 180°. False, it is 360°.
f. When a transversal intersections a set of parallel lines, 8 congruent angles are formed. False. 2 sets of 4 congruent
angles are formed. Not all 8 angles are congruent.
5. Are the two triangles similar?