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1.4: Congruent Triangles

Created by: CK-12

Triangle Sums

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to familiarize students with the polygonal sum theorem and its specific application, the triangle sum theorem. Students will incorporate algebra to find unknown polygons given an interior angle sum and find an interior angle sum given a specific polygon.

Physical Models! Using the triangle from the introduction, have students measure the interior angles of the triangle. Then, by extending segments \overline{AB}, \overline{BC}, \overline{AC}, students see three new exterior angles and should measure these too. Students should make the connection that the interior and exterior angles form a linear pair, and by the Linear Pair Theorem, are supplementary.

Extension! The Triangle Sum Theorem is a special case of the Polygonal Sum Theorem, in which the sum of interior angles of an n-gon is found by the following formula:

T = 180(n - 2), where n \ge 3

Ask students to brainstorm the reasoning behind n \ge 3. Students should remember that a polygon cannot be formed with less than three segments.

Physical Model! To demonstrate the explanation of the Triangle Sum Theorem found on page 209, students should draw a triangle and measure all three interior angles. Students can then rip or cut off any two angles and, like a puzzle, fit them with the third. The result is a straight line with a measurement of 180\;\mathrm{degrees.}

Technology Activity! Using a geometric software program, have students follow these steps:

  1. Place 3 noncollinear points on the plane, labeled A, B, C. Connect these three points to form \triangle ABC.
  2. Compute the measures of \angle A, \angle B, \angle C. How can we classify this triangle? Is it scalene, equilateral, or isosceles? Is it acute, obtuse, or right?
  3. Find the sum of all three angles. It should equal 180\;\mathrm{degrees.}
  4. Highlight points A, B (thus \overline{AB}), and point C. Using the appropriate menu, click on “construct a parallel line.” There should a line parallel to \overline{AB}.
  5. Locate points on the line parallel to \overline{AB}, calling them F and E.
  6. Measure \angle ABF and \angle CBE. Calculate the sum of these two angles and \angle A. The sum should equal 180\;\mathrm{degrees.}

Congruent Figures

Pacing: This lesson should take one class period

Goal: The goal of this lesson is to prepare students for the five triangle congruency theorems: Angle-side-angle, side-angle-side, side-side-side, angle-angle-side, and the special case of side-side-angle, the hypotenuse-leg theorem. This lesson provides a needed introduction by looking at congruent triangles in a non-formal manner.

Notation, Notation, Notation! Revisit congruence notation from earlier lessons: \cong Stress the importance of labeling each congruency statement such that the congruent vertices match. For example, ABCD \cong LMPQ shows \angle A \cong \angle L, \angle C \cong \angle P, and so forth.

Stress the tic mark notation in relation to the congruency statement. Simply because the letters used are in alphabetical order does not necessarily mean they will line up this way in a congruency statement. Students must follow the tic marks around the figure when writing congruency statements.

Look Out! Students begin to become confused with notation at this point. Be consistent with notation. Have groups of students create classroom posters regarding symbols.

Use the following mantra, “Distances are equal and side lengths are congruent.” While each lends to the other, students need to understand which value applies.

Ask students to determine if the below triangles are congruent and explain any reasoning. Use the following information: \overline{DE} \cong \overline{AB} and \overline{EF} \cong \overline{BC}. These are not congruent because the double tick marks do not match.

Proof Using SSS

Pacing: This lesson should take one class period

Goal: This lesson introduces students to the formal concept of triangle congruency. The easiest for students to visualize is the side-side-side (SSS) Congruence Postulate.

Differentiation! For students struggling with the distance formula, encourage them to create a right triangle using the \frac{\mathrm{rise}} {\mathrm{run}} of the line. Then students can use Pythagorean’s Theorem \mathrm{leg}^2+ \mathrm{leg}^2 = \mathrm{hypotenuse}^2 to find the length of the segment.

Arts and Crafts Time! Students can visualize the SSS Congruence Postulate in the following way. Using three 8.5” \times 11” sheets of paper, have students create three dowels by rolling tightly from corner to opposite corner. Cut the dowels to the following lengths: 4”, 6”, and 7.” Using tape, glue, or staples, the students should create a triangle and compare their figure with the figures of several classmates. Students should see that all triangles are congruent, helping to demonstrate the rationale behind the SSS Congruence Postulate.

Background Information! The SSS Congruence Postulate can be proved using the idea of congruence. In theory, as mentioned in the lesson, these two triangles represent a slide of 7\;\mathrm{units} right and 8\;\mathrm{units} down. A slide, or translation, is an isometry, preserving distance and angle measure. Thus, since the distances are equal, the lengths are congruent.

Proof Using ASA and AAS

Pacing: This lesson should take one class period

Goal: Students will learn how triangles can be determined congruent using Angle-Side-Angle and Angle-Angle-Side Theorems.

Look out! These two congruency postulates look identical to students. Use this method when explaining which to use. When moving left to right of a triangle, recite the information you have. If you have an angle-angle-side (or side-angle-angle), use the AAS Congruence Theorem. If you have an angle-side-angle, use the ASA Congruence Postulate. In-class activity: Have the students complete the following two activities:

Activity #1: Have students draw a 42\;\mathrm{degree} angle. Measure one ray 5\;\mathrm{cm}. Using the unmeasured ray, students will now draw a 70\;\mathrm{degree} angle. Connect both rays to form a triangle. Have students share their drawings. All drawings should be congruent, according to AAS.

Activity #2: Have students draw a 55\;\mathrm{degree} angle, measuring one ray 5\;\mathrm{cm.} Using this same ray, construct a second angle of measure of 85\;\mathrm{degrees.} Connect both rays to form a triangle. Have students share their drawings (all drawings should be congruent, according to ASA).

English Connection! There are three main types of proofs: two-column, flow charts, and paragraph form. Use all three methods when presenting this lesson (and subsequent lessons). Some students will be more comfortable with organizing information in a 2-column format and others will use a flow chart. Students with a strong language arts background will find that writing proofs using complete sentences in a paragraph will make the process of proving similar to drafting a persuasive essay.

Homework Check! Using the bonus question following #10 is a great review for students regarding the appropriate way to name angles. Be sure you review this question in class! 

Proofs Using SAS and HL

Pacing: This lesson should take one class period

Goal: This lesson is to complete the triangle congruency postulates and theorems. There is a brief description of the anatomy of a right triangle and Pythagorean’s Theorem. It also introduces the notion that AAA and SSA relationships will not produce congruent triangles.

Note Taking Time! Students need help organizing these congruency proofs. Use a table similar to the one shown below, projected on a digital imager or projector and take time to organize the material. Extend the organizer one more column and two more rows to accommodate the information gathered in the Using Congruent Triangles lesson.

Congruence Type Definition Diagram

Extension! Use the following diagram from example 1 to further assess your students’ understanding of Triangle Congruencies. Have students list the information needed for each of the four congruencies learned thus far: ASA, SSS, SAS, and AAS.

Triangle Anatomy! Understanding right triangle anatomy is crucial, especially once students move into trigonometry. Before discussing the Hypotenuse-leg Congruence Theorem, draw a blank right triangle. Have movable words of LEG, LEG, and HYPOTENUSE. Encourage students to correctly label the right triangle by moving the terms to the correct positions.

Pythagorean’s Theorem! This is one of the most useful theorems in mathematics; it is used for distance, finding missing side lengths in a right triangle, and is the basis of the Law of Cosines. Use this silly memory device. You will use Pythagorean’s Theorem enough times to cause Post-traumatic Stress Disorder (PTSD). Each time you say, “PTSD?” students should respond, “a^2 + b^2 = c^2!”

In-class Activity! Similar to the activities showing ASA and AAS congruencies, students will use the following two activities to show there are no such congruencies for AAA and SSA relationships.

Using Congruent Triangles

Pacing: This lesson should take one class period

Goal: The goal of this lesson to illustrate how congruent triangles can be used to determine congruent corresponding segments or vertices and find distance.

Extension! Using the graphic organizer from the SAS and HL lesson, include the information presented in the introduction.

Look out! This is approximately the lesson in which students will ask the question, “When will we ever need to know why triangles are congruent? How will this apply to my life?” Encourage students to research aviation, construction, manufacturing, and so forth to explore real world uses of triangles. Have students draft an essay with their findings and present it to the class. Here are some real life examples: Architecture such as bridge construction and roof rafters; proving properties of other figures, such as parallelograms, squares, rhombuses; determining congruent sails on sailboats; ensuring stairs are the same (the risers have congruent triangles cut from them).

Arts and Crafts Time! Students confuse “drawings” with “constructions.” Stress to students that a true construction can only be made with the following tools: compass and a straightedge. Constructions cannot be measured using degrees from a protractor or units from a ruler. Once these two tools come into play, the construction is now considered a drawing.

For a beginner warm up, encourage students to play with the compass. Many will make faces, animals, or flowers. Have students decorate their drawings and post them on a bulletin board.

Take time and go through the perpendicular bisector constructions as a class.

Extension! Have students continue to practice constructions by constructing an angle bisector using the following directions.

  1. Have students draw an angle of their choice, labeling it \angle {B}.
  2. Place the compass on B and draw an arc through both sides of the angle. Call the points of intersection A and C.
  3. Place the compass on point B and draw an arc across the interior of the angle.
  4. Without changing the radius of the compass, place the compass on point C and draw an arc across the interior of the angle.
  5. Label the intersection of the two arc as D. Draw \overrightarrow{BD}.
  6. \overrightarrow{BD} is the bisector of \angle ABC.

An example is show below.

Isosceles and Equilateral Triangles

Pacing: This lesson should take one class period

Goal: There is a natural progression from triangle congruencies to isosceles and equilateral triangles. This lesson illustrates the special properties that arise from these two types of polygons.

It is helpful for students to reproduce the isosceles triangle drawing. They will benefit from using this diagram when completing the exercises.

Another useful theorem of isosceles triangles has an especially long name. The Isosceles Triangle Coincidence Theorem states, “If a triangle is isosceles, then the bisector of the vertex angle, the perpendicular bisector to the base, and the median to the base are the same line.” Therefore, the perpendicular bisector to an isosceles triangle’s base is the same line generated by the angle bisector of the vertex.

Because of this theorem, step 5 of the proof in example 1 can be alternatively justified using the HL Congruence Theorem. \overline{AD} creates two right angles, \angle ADC and \angle ADB.

An equilateral triangle is a special type of isosceles triangle. Some definitions of isosceles state, “An isosceles triangle has at least two sides of equal length.” While this book does not use this exact definition, it is implied by using the Isosceles Triangle Base Angle Theorem with equilateral triangles.

Extension! Since the Isosceles Triangle Base Angle Theorem and its converse are true, have your students create a biconditional of this theorem. The base angles of a triangle are congruent if and only if the triangle is isosceles.

Extension! Students may fall into the trap of assuming figures must be both equiangular and equilateral. To show a counterexample to the belief, have students create equiangular polygons that are not equilateral. For example, students can draw a pentagon having five interior angle measurements of 108\;\mathrm{degrees} with varying side lengths.

Congruence Transformations

Pacing: This lesson should take one class period

Goal: This lesson introduces students to isometries, which can also be found in Chapter 12. The main focus of this lesson is on congruent triangle transformations.

Name That Transformation! Create slides or cards with images of transformations (preimage and image). Flash one at a time to the class. Have the students answer on a personal whiteboard or other monitoring system. Offer one point for each transformation the student correctly answers – offer double points if the student can explain why s/he chose that particular transformation. 

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