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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