Lesson 4.2: Angle Relationships in Triangles
Page 223 in text
Learning Objectives:
The learners will be able to measures of interior and exterior angles of triangles
The learners will be able to apply theorems about the interior and exterior angles of triangles.
Common Core Standards: Prove geometric theorems
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.
Continuity:
Previous Lessons This Lesson Next Lesson Yesterday we learned how to classify triangles according to side length and angle measure.
Today, we will prove that the angle measures of a triangle add up to 180 degrees, and we will prove that the measure of an exterior angle of a triangle is equal to the sum of the two nonadjacent angles of the triangle
Next, we’ll add to our knowledge by exploring congruence in the context of triangles.
Lesson Overview
Opening: Hand back tests from Chapter 3
Launch: Warm Up
Review Angle Measures
Review Triangle Classifications
Review Interior and Exterior Angles
Explore:
Triangle Sum Activity
o Tearing triangles angles to show they add up to 180o
o Folding triangle into a rectangle to show angle sum is 180o
Triangle Sum Proof
o Corollaries
What is a corollary anyways?
Exterior Angle Proof
o Why does this make sense?
o Reflect back to Triangle Sum Activity
Third Angles Theorem
Reflect: Exit Slip
How are the three main theorems we learned today connected?
What did you learn?
What are you still confused about?
Homework:
2-5, 6-14E, 24, 28,
Detailed Outline
Warm UP Objective Activity Teacher Timing Recall Angle Measures to connect what learners know about angles to what they will learn about classifying triangles by angle measure.
Acute Right Obtuse Straight mA = 90 mA = 180
Use Effective Questioning from PBS sheet.
2 min
Classify Triangles by Angle Measure and Side Length by identifying properties and attributes of the figures.
(By Angles): Acute Right Obtuse Equiangular 3 acute ’s 1 right 1 obtuse 3 congruent ’s (2 acute ’s) (2 acute ’s) (all acute ’s) (By Sides): Scalene Isosceles Equilateral No sides 2 sides 3 sides
Draw visual on board that looks like picture for learners who need organization and visual.
2 min
Recall Parallel Postulate
How many lines can be drawn through N parallel to N M P Answer: Exactly 1 by the Parallel Postulate.
Introduce auxiliary line. If learners cannot recall have them flip back in their textbook.
3 min
Review Exterior and Interior Angles
Have students label all angle relationships.
3 min
Explore Part 1: Triangle Sum Theorem
Dialogue for Intro.
Objective of Activity
Outline of Activity Teacher Timing Assessment
Proving
Triangles Sum
is 180 degrees.
This is clearly
something
we’ve known
and accepted
for awhile but
now we are
going to prove
it!
Students will “discover” or convince themselves that the Triangle Sum Theorem should in fact be true. Students will help with the proof of the Triangle Sum Theorem (sketch of the proof), thus working on their proof skills.
Have students (with partners) rip
corners of each triangle (each
type of triangle by angle) and line
them up on a straight line on
their paper (straight angle) –
have them make conjectures [the
interior angles in a triangle add
up to 180 degrees].
Sketch the proof for The Triangle
Sum Theorem – refer them to
Page 223 in their book to see the
full proof. Mentions use of
Auxiliary line.
***(SEE SKETCH BELOW)
Mentions corollary to the
Triangle Sum Theorem [in a right
triangle, the acute angles are
complementary] CHECK 2 in text
Sketches proof out loud with
students.
Gives students a non-precise justification of the Triangle Sum Theorem. Allows them to learn in a concrete, hands-on way. Allows students to see the idea of the proof, but puts responsibility on them to discover it. Once they understand the purpose, they can delve into the formal proof and really make sense of the idea being discussed.
15 minutes
Students will work with partners to make conjectures. They will make the correct conjecture, or if they make an incorrect conjecture, this allows me to target misconceptions.
Sketch of Proof (with guided questing from teacher)
Triangle Sum Proof: With what we
know about parallel lines and
alternate interior angles, it's pretty
straight forward:
Construct Auxiliary Line: a line that is added to a figure to aid in a proof.
How can we justify the auxiliary lines existence? That is, why are we allowed to
construct this line?
Through any two points there is exactly one line.
But how can we make our auxiliary parallel to ?
By the parallel postulate!!
What kind of Angles did we create? Label Interior and Exterior
What would the transversal be in each case?
Does this make sense in terms of the activity we did when we tore angles from the
triangles?
CHECK 2 in text COROLLARIES for triangle sum theorem: What is a corollary anyways? A theorem
whose proof follows directly from another theorem. So basically, we get these for free, well
almost free.
Part 2: Exterior Angles Theorem
Dialogue for Intro.
Objective of Activity Outline of Activity Teacher Timing Assessment
Part 2:
Exterior
Angles. In the
first activity we
were able to create
and examine the
features of a
triangle. In doing
this, we learned
how to use
deductive
reasoning to
formally prove the
sum of the
measures of the
interior angles is
equal to the
measure of a
straight angle
(180o) of a line
drawn through one
of the vertices. We
will now use that
information to
examine the
exterior angles of a
triangle.
Students will use what they know to help them sketch the Exterior Angle theorem. Students will see why this theorem seems logical and then they will formulate a proof.
Recall what we know about
triangles
Define terms: remote interior
angles, exterior angles, and
interior angles.***See Definition
Whiteboard Activity BELOW
Read the Exterior Angles
Theorem from book on page 225.
Do CHECK 3 in book with
partners.
Teacher uses GEOGEBRA to
“convince them” in a non-precise
sense that the theorem is true.
Then sketch the proof for The
Exterior Angle Theorem on the
board– refer them to Page 225 in
their book to examples of the
Theorem. ***See Geogebra
sketch BELOW
Allows students to see visual representation of the material and build off prior knowledge by connecting familiar concepts with the new ideas. Includes technology in the classroom for the purpose of making a conjecture.
20 min Students will be enthusiastic, or at least engaged in watching Geogebra. They will have ideas of how to sketch the proof of this theorem and will believe it to be true. If the students aren’t asking the right kinds of questions or are applying the wrong vocabulary, I can stop the demonstration and review the main ideas we are using to construct the proof.
Whiteboard Organization Interior and Exterior Angles in
Triangle
CHECK 3
An interior angle is formed by two sides of a triangle.( inside the figure)
In figure:
An exterior angle is formed by one side of the triangle and the extension of the
adjacent side. (outside the figure)
IN figure:
Each exterior angle has two remote interior angles.
In figure:
A remote interior angle is an interior angle that is not
adjacent to the exterior angle. (Interior and away from exterior)
Exterior Angle: A better visual of why the proof make sense
Geometers Sketchpad Proof (tying everything back together)
Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Using Geogebra, try 2 examples to see if the theorem is true.
In the figures above the exterior angle ∠ABP is equal to the sum of the remote interior angles ∠BAC and ∠ACB.
Now that the conjecture is believed to be true, work through a formal proof with students.
PART 3: Third Angles Theorem Dialogue for
Intro. Objective of
Activity Outline of Activity Teacher Timing Assessment
Next we are going to look at the Third Angles Theorem. We will discuss why it makes sense and how we can use it.
The students will understand how to compare angles amongst two triangles and will understand how to find the third angle of a triangle by applying the previous theorems.
Read the Third Angles Theorem
Aloud to the class.
Let them work on CHECK 4 in the
text in partners.
As a class discuss our findings
and connections.
Gives the students freedom to explore the third theorem, which is fairly straightforward, with one another. Asks questions that force the learners to use the exterior angles theorem and triangle sum theorem.
10 min Students will be given the freedom to explore examples that force them to apply all of the theorems they learned to find the third angle. The teacher can walk around and ask questions to ensure learners understand the material.
Reflection Objective Activity Teacher Timing Summarize lesson. Complete Exit Slip Apply understanding to homework.
Today we learned how to classify triangles, we learned that the angle measures of a triangle add up to 180 degrees, and we learned that the measure of an exterior angle of a triangle is equal to the sum of the two nonadjacent angles of the triangle. Now, it’s time to try some problems that apply what you know from today’s lesson and from your previous experience with angles, side lengths, and triangles, to some homework problems. Tomorrow we’ll add to our knowledge of triangles by exploring congruence in the context of triangles.
Wraps up what was covered. Assesses student learning.
10 min
………
.
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