How to Prove Similar Triangles

Step-by-Step Guide

1. 1

   Step 1: Define the angle-angle (AA) theorem.

   Two triangles can be proved similar by the angle-angle theorem which states: if two triangles have two congruent angles, then those triangles are similar.This theorem is also called the angle-angle-angle (AAA) theorem because if two angles of the triangle are congruent, the third angle must also be congruent.
   
   This is because the angles of a triangle must sum to 180°.
2. 2

   Step 2: Identify the measure of at least two angles in one of the triangles.

   Using a protractor, measure the degree of at least two angles on the first triangle.
   
   Label the angles on the triangle to keep track of them.
   
   Choose any two angles on the triangle to measure.
   
   Example:
   Triangle ABC has two angles that measure 30° and 70°. , Again, use a protractor to measure two of the angles on the second triangle.
   
   If both angles are identical on both triangles, then the triangles are similar to each other.
   
   Remember, if two angles of a triangle are equal, then all three are equal.
   
   Example:
   The second triangle, DEF, also has two angles that measure 30° and 70°. , Once you have identified the congruent angles, you can use this theorem to prove that the triangles are similar.
   
   State that the measures of the angles between the two triangles are identical and cite the angle-angle theorem as proof of their similarity.It is possible for a triangle with three identical angles to also be congruent, but they would also have to have identical side lengths.
   
   Example:
   Because both triangles have two identical angles, they are similar.
   
   Note:
   If the two triangles did not have identical angles, they would not be similar.
   
   For example:
   Triangle ABC has angles that measure 30° and 70° and triangle DEF has angles that measure 35° and 70°.
   
   Because 30° does not equal 35°, the triangles are not similar. , When a triangle has two sides that are in the same proportion to another triangle and their included angle is equal, these triangles are similar.Be careful not to confuse this theorem with the Side-Angle-Side theorem for congruence.
   
   For congruence, the two sides with their included angle must be identical; for similarity, the proportions of the sides must be same and the angle must be identical.For example:
   Triangle ABC and DEF are similar is angle A = angle D and AB/DE = AC/DF. , Using a ruler, measure two sides of triangle ABC and label them with that measure.
   
   Make sure triangle DEF is oriented in the same direction and measure the same two sides.
   
   Label these sides as well.Example:
   Measures of triangle ABC; side AB = 4 cm and side AC = 8 cm.
   
   Measures of triangle DEF; side DE = 2 cm and side DF = 4 cm. , Using a protractor, measure the included angle, or, the angle between the two sides that you already measured.
   
   For this theorem, the measure of the angle should be identical in both triangles.Example:
   Angle A in triangle ABC is 26°.
   
   Angle D in triangle DEF is also 26°. , To use the SAS theorem, the sides of the triangles must be proportional to each other.
   
   To calculate this, simply use the formula AB/DE = AC/DF.Example:
   AB/DE = AC/DF; 4/2 = 8/4; 2 =
   2.
   
   The proportions of the two triangles are equal. , Once you have determined that the proportions of two sides of a triangle and their included angle are equal, you can use the SAS theorem in your proof.
   
   Example:
   Because AB/DE = AAC/DF and angle A = angle D, triangle ABC is similar to triangle DEF.
   
   Note:
   If angle A did not equal angle D, the triangles would not be similar.
   
   Also, if the proportions were not equal, the triangles would not be similar. , Two triangles would be considered similar if the three sides of both triangles are of the same proportion.
   
   Sides measuring 2:4:6 and 4:8:12 would provide proof of similarity.Be careful not confuse this theorem with the Side-Side-Side theorem for congruence: when two triangles have three identical sides they are congruent.
   
   The theorem for similarity deals strictly with the proportions of the three sides.
   
   For example:
   In triangle ABC and DEF, the triangles are similar if AB/DE = AC/DF = BC/EF. , Using a ruler, measure all three sides of each triangle.
   
   Label each side to keep track of all the measurements.
   
   Be sure to use the same units for each measurement of the sides of the triangle.
   
   Example: triangle ABC has sides AB = 10 cm, BC = 15 cm, AC = 20 cm and triangle DEF has sides DE = 2 cm, EF = 3 cm, and DF = 4 cm. , For the SSS theorem to be applicable, the three sides of each triangle must be proportional to each other.
   
   Using the side measurements, calculate the proportions using the formula AB/DE = AC/DF = BC/EF.Example:
   AB/DE = AC/DF = BC/EF; 10/2 = 20/4 = 15/3; 5 = 5 =
   5. , If you have determined that the proportions of all three sides of the triangles are equal to each other, you can use the SSS theorem to prove that these triangles are similar.Example:
   Because AB/DE = AC/DF = BC/EF, triangle ABC and triangle DEF are similar.
   
   Note:
   If AB/DE ≠ AC/DF ≠ BC/EF then the triangles would not be similar. , A proof starts with a statement of given information which is known as the hypothesis statement.
   
   You will need to provide a list of relevant information as well as evidence to support each statement., You will need to make a chart, which generally has two columns.
   
   This first column will contain your statements, while the second will provide your evidence.Be sure that the final line in your statement column always matches the hypothesis statement.
   
   The middle rows will be where you show your work while you're solving the problem.
   
   All of the statements you provide, as well as your supporting evidence, should always refer back to the figures that are described by the hypothesis statement. , Use all of the details that are supplied by the hypothesis.
   
   Be sure to draw the figure big enough so that you can easily make out these details.
   
   Label all of the points that are described and be sure to include any information from the statement regarding parallel lines or congruent angles. , For any problem, you will be given some information about the measures of the angles and the sides of the two triangles you are trying to prove similar.
   
   The first step in identifying the correct theorem to use is writing down the information you already know.If no diagram is provided, draw the triangles and then label their angles and sides with the given information. , Once you have written down your given information and learned the three possible theorems that could apply, choose the one that matches the information given.
   
   It’s okay if multiple theorems apply, just choose one for your proof.
   
   If none of these theorems match the given information then the triangles are not similar. , Devise a strategy to solve the proof.
   
   There are three different postulates, or mathematical theories, which apply to similar triangles.
   
   Any one of these will provide sufficient evidence to prove that the triangles in question are similar.
   
   Gather your givens and relevant theorems and write the proof in a step-by-step fashion.
3. 3

   Step 3: Measure at least two of the angles on the second triangle.

4. 4

   Step 4: Use the angle-angle theorem for similarity.

5. 5

   Step 5: Define the Side-Angle-Side (SAS) Theorem for similarity.

6. 6

   Step 6: Measure the same two sides of each triangles.

7. 7

   Step 7: Identify the measure of the angle between those two sides.

8. 8

   Step 8: Calculate the proportion of the side lengths between the two triangles.

9. 9

   Step 9: Apply the Side-Angle-Side Theorem to prove similarity.

10. 10

    Step 10: Define the Side-Side-Side (SSS) Theorem for similarity.

11. 11

    Step 11: Measure the sides of each triangle.

12. 12

    Step 12: Calculate the proportions between the sides of each triangle.

13. 13

    Step 13: Apply the Side-Side-Side theorem to prove similarity.

14. 14

    Step 14: Study the format of a formal proof.

15. 15

    Step 15: Develop a hypothesis to solve the problem

16. 16

    Step 16: or complete the proof.

17. 17

    Step 17: Draw a diagram of the figures that are described in the hypothesis

18. 18

    Step 18: if an illustration has not already been provided.

19. 19

    Step 19: Write down the given information.

20. 20

    Step 20: Choose the theorem that fits the given information.

21. 21

    Step 21: Write the proof.

Detailed Guide

About the Author