Lines, Angles, and Triangles: Lesson 6 - Congruent and Similar Triangles

Congruent and Similar Triangles is the sixth of eight self-paced lessons in the “Lines, Angles, and Triangles” section of KET’s GED^{®} Geometry Professional Development Online Course. This lesson covers the principles of congruence and similarity in triangles.

On the GED^{®} Mathematics Test, students may need to solve problems related to congruent triangles or similar triangles.

Students will need to recognize that triangles are congruent, even if they are rotated differently.

They will also need to recognize similar triangles and utilize proportions to find unknown sides.

This lesson covers the principles of congruence and similarity in triangles. Students often have difficulty applying these skills in complex drawings and realistic contexts.

Note: On the GED Mathematics Test, students will not be asked to produce answers that require recall of terminology.

Angles or side lengths are congruent when they have the same measures.

Congruent triangles have the same shape and size. Their corresponding sides and angles are equal. In diagrams, congruent parts are marked with the same symbols (I II III).

There are six corresponding parts in two congruent triangles. If you know that three of these parts are congruent, you can prove the triangles are congruent.

Side-Side-Side (SSS) Three corresponding sides are congruent.

Side-Angle-Side (SAS) Two sides and the angle between them are congruent.

Angle-Side-Angle (ASA) Two angles and the side between them are congruent.

Question Are these triangles congruent or not? Explain how you know.

Step 1 Look for pairs of congruent parts.

∠M = ∠R; side MO = side RO; ∠NOM = ∠POR

Step 2 Choose SSS, SAS, or ASA to prove the triangles are congruent.

The congruent sides are between the two pairs of congruent angles. This is angle-side-angle, or ASA, property.

Answer The triangles are congruent because of ASA.

3Similar Triangles

Watch the video for information about similar triangles.

Similar triangles have the same shape, but they are not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are in proportion to each other—they have the same ratio.

A proportion is an equation with two equal ratios. A proportion can be used to find a missing side length in a pair of similar triangles. Use the three given sides, and solve for the missing side length.

Question Henry wants to find the height of the tree in his front yard. With his son’s help, he measures his own shadow and the shadow of the tree. Henry is 6 feet tall.

What is the height of the tree?

Step 1 Organize the given facts in a table. Use h for the height of the tree. That is the number you need to find.

Step 2 Write a proportion based on the table.

Step 3 Find the cross products. Then divide by the remaining term to isolate h.

h× 10 = 6 × 60

h× 10 ÷ 10 = 360 ÷ 10

h = 36

Answer 36 feet

4Sample GED Questions

Directions: There are two questions on this page. Each will appear in the blue rectangle below. Click on Question 1 to see the first question, and then select your answer. Click on Question 2 to see the second question, and select your answer. As you solve these problems, also respond to the question below, which is related to building students’ skills.

How could you help students see that there are two similar triangles combined in the figure in Question 2?

Lines, Angles, and Triangles: Congruent and Similar Triangles Skill Review

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Printable Resource

In this lesson you have learned about congruent and similar triangles. This review consists of key terms and concepts with which you will need to be familiar. Click the view button on the left to access a review sheet.

Below you will also see a Classroom Connection with suggestions for linking this geometry content with your instruction.

CCSS.Math.Con.HSG-CO.B.7 ( High School - Geometry ): Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

CCSS.Math.Con.HSG-SRT.A.2 ( High School - Geometry ): Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

CCSS.Math.Con.HSG-SRT.A.3 ( High School - Geometry ): Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

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