Tracy Carpenter

Angles of a Triangle

Section 4.2

 

Objectives

 

Alabama Course of Study

·        Students will be able to use relationships between pairs of angles:

o        Adjacent Angles

o        Vertical Angles

o        Complementary Angles – Theorem 4.4 – the acute angles of a right triangle are complementary

o        Supplementary Angles  (ACOS #14, page 89)

·        Students will be able to find the missing measures of triangles (ACOS #4, page 88 and #18, page 89 )

o        Triangle Sum Theorem – The sum of the measures of the interior angles of a triangle is 180 degrees.

·        Students will be able to find the measure of exterior angles of a triangle (ACOS #18, page 89)

o        Exterior Angle Theorem – the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote (nonadjacent) interior angles.

·        Students will be able to determine if triangles are congruent (ACOS # 19, page 89)

o        Third Angle Theorem – If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are also congruent.

 

Principles and Standards

·        Analyze properties and determine attributes of two dimensional objects.

·        Establish the validity of geometric conjectures by critiquing arguments made by others

 

Materials

·        Warm Up Transparencies

·        Example sheet

·        Geometers Sketchpad

·        T.O.W. Bundle (overhead/laptop)

·        Work sheets

·        Triangle cut outs

 

Prior Knowledge

·        Students must know the definitions of adjacent, congruent, complementary, and supplementary angles to use their relationships.

 

Rationale

One of the geometry standards by NCTM for 9-12 grades states “establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments make by others”.  This activity lets students work in groups, which allows them to listen to others’ solutions.  Students themselves must decide which is valid and which is not.  The NCTM also states that students should “understand and represent translations, reflections, rotations, and dilation of objects in the plane by using sketches, and use various representations to help understand the effects of simple transformations and their compositions”.  By using many examples students can see that the theorems hold for each example.  Also, the NCTM recommends using technology, and this activity is a great example of when technology should be used.  Geometer’s Sketchpad allows for many examples of rotations, reflections, and translations.  It allows for student involvement, where they can create their own triangle and analyze it themselves.

 

Procedures

 

Set Induction

1.  Warm Up

·        Students will complete 3 problems on the overhead from the AHSGE (Alabama High School Graduation Exam) Practice V-1

2.  We will begin by reviewing interior and exterior angles.

• I will begin by having students draw a triangle on their paper while I draw one on the board.  I will point to the inside and ask students what the angles are called that are where I am pointing. Interior Angles – the original three angles of the triangle.

• I will then have students extend one of the lines of their triangle.  I will point to the angle that it makes and ask them what is the name of the angle that is outside the triangle.  Exterior Angle- the angles that are adjacent to interior angles.

 

Lesson

1.  Students will get into groups of 4. 

• Each group will be handed a worksheet on triangles.  The worksheets all have the same questions, but different shapes of triangles.  (Group Worksheets)

• I will work through the first triangle on the board while the students follow along at their desks.  -  Remind students that an interior angle plus an adjacent angle are a linear pair, which means they equal 180 degrees. 

• Students will then work through the worksheet working with their group. 
• Each group will put their answers on the board for each question.  We will discuss the fact that all the answers for each group and each question are all the same.

2.  Each student will get a triangle cut out of paper. 

• I will ask students to tear all three corners of their triangle any size they want.

• We will then line up the angles.  I will ask students what they notice when the angles are put together.  They form a line which is 180 degrees, so the sum of all the angles will be 180 degrees.
• This leads to our first theorem

o        Triangle Sum Theorem – The sum measures of the interior angles of a triangle is 180 degrees.

3.  I will then open Geometer’s Sketchpad where there is a triangle already drawn.  Each angle measure and the angles sum will be present on the screen.

• I will let volunteers come up to the front of the class.
• Each volunteer will get to translate, rotate, or reflect, the triangle.  The students will be able to see as the volunteers move the triangle around that the angle sum stays 180 degrees no matter the shape.

4.   Pull up example triangle from Geometer’s Sketchpad.  Remind students that there are two triangles together to make on triangle.  Remind students that <BDC is 90°, and that it is a linear pair with <BDC. 

• So what would <BDC equal? 90°.
•  We also know that <BAD is congruent to <BCD so we can find what each angle actually equals.  2X = 60°, X = 30°.  Therefore <BAD = 60°. 
• So if we take triangle BAD and solve for <ABD we would get 60° + 90° + <ABD = 180°, so <ABD = 30°.  Then in triangle BCD we would get 60° + 90° + <CBD = 180°, so <CBD = 30°. 
• This leads to our next theorem  Third Angles Theorem – If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are also congruent.

5.  I will then get students to get their triangle back out.

• I will get students to put back one angle in the corner that it came from. 
• Students will then line up the other two missing angles.
• Students will be able to see that the other two torn angles make up the exterior angle.
• This leads to the next theorem.

• Exterior Angle Theorem – the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote (nonadjacent) interior angles.

6.   I will then ask for some more volunteers for Geometer’s Sketchpad.

• I will have more students rotate, reflect, and translate a triangle.
• I will have the measure of the exterior angle and compare it to the 2 nonadjacent interior angles.

7.  Students will then get out their worksheet and compare the measures of one of the exterior angles to the 2 nonadjacent interior angles.

8.  I will then ask students what they think if we compared the exterior angle to just one of the two nonadjacent interior angles.

• Well if it equals the two added together then it must be larger than just one.
• This leads to the last theorem
  • Exterior Angle Inequality – The measure of an exterior angle of a triangle is greater that the measure of either of the two remote (nonadjacent) interior angles.

Closure

1.  As a review of this lesson we will go over Example Sheet 4.2.

 

Assignment

1.  Have students complete example sheet, if they didn't finish in class.

 

Assessment

Informal during questioning, worksheet, and sketchpad.