High School Geometry

Proving Congruence Using Theorems

Alana Linde

Standard:

High School Geometry: Similarity, Right Triangles, and Trigonometry

CCSS.MATH.CONTENT.HSG.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove

relationships in geometric figures.

Objectives:

Students will be able to

Solve a task incorporating strategies previously learned (definitions, congruence theorems, etc.)

Articulate and justify their reasoning

Collaborate with peers to work towards a goal

Math Task:

Given triangle BAC is congruent to triangle ABD, prove that triangle ABG is isosceles.

Prior Knowledge:

In order for students to solve this task, they will need exposure to and knowledge of

what comprises an isosceles triangle

the relationship between the parts of congruent triangles

triangle congruence theorems (SSS, SAS, ASA, AAS, HL)

angle addition postulate (and/or)

concept of dropping a perpendicular

Vertical bisect

Vertical angle theorem

http://www.corestandards.org/Math/Content/HSG/SRT/B/5/

Plan for Implementation:

1. Introduce students to the task a. Tell students they will be incorporating what they have been learning to solve this

task.

b. Provide each student a copy of the task c. Have colored pencils and graph paper available

2. Independent practice a. Allow students independent time to engage with the task b. Monitor throughout the room to ensure students are on task.

3. Group work a. Place students in groups to discuss their thought processes. b. This will encourage students to articulate their thoughts and be able to provide

justification to ensure they understand what they are doing.

c. Students that are struggling will be given some ideas to steer them toward solving. d. Continue to monitor students progress in their groups and ask questions from the

following list based on what the students are doing or conversing about.

4. Whole class discussion a. Sequence groups to present in a logical way that enables connections between

proofs to be made, demonstrating that there are several ways to prove the same

task.

b. Provide the figure to be projected so that the groups can mark up the figure and write out their proof.

c. Have the students take notes and ask questions of their peers while they are presenting.

Questions to ask students while monitoring:

What do you know about congruent triangles?

What components are you seeking to prove?

Could you think of this figure in a different way?

What do you know about overlapping triangles?

What do you know about the transversal line theorem?

How could you break down this image into smaller parts?

What do you know about bisecting angles and line?

How else could you prove this?

Key Reminders for Students:

Because this is a task to enhance their critical thinking and proof writing skills,

1. Write out each step proving the process. 2. Use specific explanations words (vertical angles, congruent sides, etc.) and statements

(used ASA, etc.).

Provisions for Diversity:

All Students

1. What components are you seeking to prove? 2. What do you know about congruent triangles?

Students Who Are Struggling

1. How could you break down this image into smaller parts? 2. Is that way you could cut out the pieces to help you understand congruent triangles? 3. What do you know about angles in a triangle and the sides of differs types of

triangles?

4. What do you know from these two triangles being congruent? (mark up their paper with congruent marks)

Students Who Are Ready for More

1. How many congruent triangles are in the image? 2. Is there another way that you can prove this?

Extension of the problem:

Given line DC is parallel to line AC, prove triangle CDG is similar to triangle ABG.

Alana Linde Given triangle BAC is congruent to triangle ABD, prove that triangle ABG is isosceles.

To prove that triangle ABG is isosceles, you have to show that

Given triangle BAC is congruent to triangle ABD, prove that triangle ABG is isosceles.

Drop a perpendicular from point G to side AB. Construct point H at the intersection of the line perpendicular to side AB.

Consider triangle AHG and triangle BHG